ESSAYS ON APPLIED ECONOMICS

By

Mingli Zheng

A thesis submitted in conformity with the requirements

For the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

University of Toronto

©Copyright of Mingli Zheng (2002)

ABSTRACT

ESSAYS ON APPLIED ECONOMICS

By

Mingli Zheng

DOCTOR OF PHILOSOPHY

Department of Economics

University of Toronto

This dissertation consists of three essays on applied economics.

In the first essay, I provide an empirical assessment of competing auction theory.

Specifically, it uses an extensive new data set containing detailed information about

bids placed on eBay computer CPU auctions to explore bidding strategies in the

presence of competing auctions. The evidence indicates that a significant proportion

of bidders bid across several competing auctions at the same time and that bidders

tend to submit bids on auctions with the lowest standing bid. We also find that

winning bidders who bid across competing auctions pay lower prices than winning

bidders who do not cross-bid. These findings jointly amount to the first evidence

lending empirical support to competing auction theory.

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In the second essay, I consider the convergence properties of behavior under

a comparative negligence rule (CN) and under a rule of negligence with contributory

negligence (NCN), assuming bilateral care with three care levels. Using an

evolutionary model, we show that CN reduces the proportion of the population

using low care more rapidly than does NCN. However NCN increases the

proportion of the population using high (efficient) care more rapidly than does CN.

As a result, the mean care level increases more rapidly and the mean social cost falls

more rapidly under CN than under NCN.

In the third essay, I provide a model for rational legal decision-making by

considering the consistency of social decision-making on resource allocations. In the

model, legal decision-making is not based on individual utilities. The purpose of law

is described by a subjective social welfare function, with a social value judgment

and an attitude towards distributional inequality as its parameters. Thus a legal

decision-making always involves a social value judgment. Social efficiency can be

different from the maximization of social wealth. In tort law, the determination of

the efficient precaution level and the award of compensation depend on the social

value judgment, and the compensation scheme implicitly transfers wealth from the

less socially valued party to the more socially valued party. The model also

provides another explanation for the award of punitive damage.

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ACKNOLEDGEMENTS

I am deeply indebted to my supervisor Michael Peters, committee members Don

Dewees and Robert McMillan. Their generous help, stimulating suggestions and

constant encouragements helped me in all the time of research for and writing of this

thesis. My gratitude to them cannot be expressed by words.

I could not have survived the PHD study without the understanding and the support

by my wife Fang Xiao. I would like to give her special thanks.

I also thank Mohr Siebeck for authorizing me to include the paper published in JITE

in my dissertation.

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Table of Contents

Abstract……………………………………………………………………………….iiList of Tables……………………………………………………….………………..viList of Figures……………………………………...………………………………..vii

Chapters

1. Bidding Behavior with Competing Auctions: Evidence from ebayAbstract………………………………………………………………………11. Introduction………………………………………………………………..32. Theory of Competing Auctions………………………………………..…..83. Mechanism in eBay and Summary of Data………………………………124. Bidding Behavior with Competing Auctions in ebay…………………….205. Does Cross Bidders Pay Lower Price?…………………………………...286. Conclusion………………………………………………………………..30References…………………………………………………………………..31

2. Liability Rules and Evolutionary DynamicsAbstract……………………………………………………………………..451. Introduction………………………………………………………………462. Liability Rules and Nash Equilibrium……………………………………493.Evolutionary Dynamics…………………………………………………...524. Conclusion………………………………………………………………..70References…………………………………………………………………...72

3. Rational Legal Decision-Making, Value Judgment and the Efficient Precaution in Tort Law

Abstract…………………………………………………………………….741. Introduction…………………………………………………………… 752. Rationality, Subjective Social Welfare Function and Value Judgment

……………………………………………………………………...813. The Determination of Efficient Precaution Level and the Damage

Compensation in Tort Law……………………………………...……..944. Summary……………………………………………………..…….…..105References………………………………………………………..……….106

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List of Tables

Table Page

1.1 Sample statistics for CPU auctions ………………………………………34

1.2 Sample statistics for samples of competing auctions……………………..35

1.3 Statistics of Cross Bidding in the Whole Process…………………………38

1.3B Percentage of Cross Bidders in All bidders………………………………..39

1.4 Statistics of Cross Bidders for the Last Day……………………………….40

1.5 Result on Bidding on the Auction with the Lowest Standing Bid…………41

1.6 Result on Bidding on the Auction with the Lowest Standing Bid

(For groups with cross bidders only)………………………………42

1.7 Result on Bidding on Group with Zero Bid Auctions……………………...43

1.8 Average Number of Bids Submitted in a Group by a Bidder……………....43

1.9 Price Paid by Cross Bidders and Non-Cross Bidders………………………44

2.1 Payoff Under CN………………………………………………………...…51

2.2 Simulation Results………………………………………………………….65

3.1 Sensitivity of the Award of the Punitive Damage to the Estimation

Bias of …………………………………………………………104

vi

List of Figures

Figure Page

1.1 A Bidding History Page From ebay………………………………….33

1.2 Histogram of Bidders’ Feedback…………………………………….36

1.3 Histogram of Bids Submission Time for Daily Sample……………..37

2.1 Proportion of Individuals Taking High care…………………………66

2.2 Proportion of Individuals Taking Medium Care……………………..66

2.3 The Composition of the Population in the Simulation……………….67

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Chapter 1

Bidding Behavior with Competing Auctions:

Evidence from ebay

Abstract

The existing auctions literature treats on-line auctions as running independently of one

another with each bidder choosing to participate in only one auction. This characterization is

less than perfect: in on-line auctions, many substitutable goods are auctioned concurrently,

and bidders can bid in several auctions at the same time. Some recent theoretical research by

Peters and Severinov (2001) is more relevant to the study of bidding behavior in on-line

auctions, showing that bidders can gain from the existence of competing auctions.

Specifically, a strategy in which bidders bid on the auction with the lowest standing (or

prevailing) bid is a Bayesian Nash equilibrium. In the light of this work, the current paper

provides the first empirical assessment of competing auction theory. Specifically, it uses an

extensive new data set containing detailed information about bids placed on eBay computer

CPU auctions to explore bidding strategies in the presence of competing auctions. The

evidence indicates that a significant proportion of bidders bid across several competing

auctions at the same time and that bidders tend to submit bids on auctions with the lowest

standing bid. We find that for homogeneous items, as the difference in ending time across

competing auctions becomes smaller, so more bidders bid across competing auctions and bid

on the auction with the lowest standing bid. We also find that winning bidders who bid

across competing auctions pay lower prices than winning bidders who do not cross-bid.

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These findings jointly amount to the first evidence lending empirical support to competing

auction theory.

1

1. Introduction

There is a tremendous amount of interest in auctions as a means of selling items,

both for vendors and theorists. The appeal of auction is understandable: as shown in

the mechanism design literature (e.g., Myerson (1981)), when a seller wants to sell

an object to several buyers, an auction is the best way to do it.

In standard auction theory, the typical assumption is that there is a single seller and

several bidders. The seller acts as a monopoly and gets all the information rent from

bidders. In practice, sellers often do not have monopoly power, but rather have to

compete against other sellers. For example, in on-line auctions, many sellers sell

their goods at the same time and some of the items are almost indistinguishable.

Thus buyers can choose among many auctions and decide whether to buy from one

among many sellers.

A few papers in the literature consider the case in which sellers compete

against each other (see for instance McAfee (1993), and Peters and Severinov

(1997)). But these papers assume that bidders can only choose to buy from one

seller, and the only equilibrium involves buyers randomizing over available sellers.

In this case, it is natural that some auctions have many bidders while other auctions

have few or no bidders, and consequently that some profitable trades may not be

realized.

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For on-line auctions, such as those on eBay, bidders are not constrained to

participate only one auction. eBay has evolved to act as a clearinghouse for a large

number of homogeneous goods. At any time, there are many similar items on sale,

the bidding cost for on-line auctions is very low, and bidders can easily monitor

several auctions at the same time. Thus it is possible for bidders to bid across several

competing auctions simultaneously.

The central question addressed in this study is: Are bidders responsive to the

existence of competing auctions? If they can choose among competing auctions,

how do bidders bid? Suppose that bidders bid across several competing auctions at

the same time, even if they only need one item, as they search for the best deal.

Doing so exposes them to the risk that they may win more than one item. But this

consequence can be avoided if bidders use a specific strategy: always bid on an

auction with the lowest ‘standing’ (or prevailing) bid, and bid with the minimum

increment; if the bidder becomes the highest bidder in one auction, pause bidding

until other bidders outbid him/her.

This strategy ensures that a bidder never wins more than one auction.

Another advantage of this strategy is that bidders are never trapped in very

competitive auctions. For example, suppose there are two competing auctions and

four bidders, sellers’ valuations are all 0, and bidders’ valuations are 10,10, 7, and 6,

respectively. If all bidders choose one auction and bid their true valuation, those two

bidders with valuation 10 may end up bidding on the same auction; whoever wins

the auction has to pay 10. If bidders bid across competing auctions and bid with the

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minimum increment, these two high valuation bidders will always end up winning

two different auctions and paying a much lower price.

Most existing work on on-line auctions treats them as many independently

running auctions and allows bidders to bid on only one auction. The emphasis in

prior work has typically been on studying the strategic behavior of market players.

For instance, Roth and Ockenfels (2000) explain the phenomenon of late bidding in

eBay by the existence of a fixed auction ending time. Bidders bid late because very

late bids have a positive probability of not being successfully submitted, and this

provides a way for bidders to implicitly collude and avoid bidding wars. Bajari and

Hortacsu (2000) study costly entry for bidders and the choice of reserve price on the

part of sellers. Both models assume that bidders only bid on one auction.

A recent paper by Peters and Severinov (2001) studies the market equilibrium

with competing auctions similar to those in eBay. If there is no bidding cost and no

fixed ending time for auctions, the paper proves that the strategy in which bidders

always submit a bid on an auction with the lowest standing bid and bid with the

minimum increment is actually a (weak) perfect Bayesian equilibrium. Contrary to

standard second-price auctions, in this environment bidding once and bidding one’s

true valuation is no longer equilibrium. Intuitively, if bidders bid their true valuation

and bid only once, they may be trapped in very competitive auctions and not have

opportunity to switch to other less competitive auctions. Consequently, the final

price of one auction is affected by the existence of other auctions. Further, prices

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tend to be uniform for competing auctions, and in addition, the price is the same as

the price under a double auction.

The strategy needs two assumptions: that there is no bidding cost and no

fixed ending time for auctions. In eBay, these assumptions are not perfectly met.

There is a bidding cost, though it is very low; and all auctions have a fixed ending

time. For identical or very similar items, auctions with almost the same ending time

compete against each other, while auctions with different ending times do not

perfectly compete with each other. Obviously, those auctions ending early cannot

compete with those ending later once they are finished, especially when many bids

are clustered at the very late period, as documented by Roth and Ockenfels (2000)

and Bajari and Hortacsu (2000).

Despite this discrepancy, eBay provides a valuable opportunity to see how far

actual bidding behavior in the presence of competing auctions corresponds to the

strategy prescribed in the theory. We have assembled data on competing auctions

for CPU’s taking place in one month, the period of September 20 to October 19.

Each group of competing auctions consists of auctions with the same description, the

same starting price and delivery method, and with a similar ending time. We classify

auctions into three categories: auctions ending in the same day, ending within the

same hour, and auctions ending within the same minute. Doing so allows us to

identify the effect of increasing the degree of substitutability on the behavior of

auction participants.

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Our results provide convincing evidence that bidders bid across competing

auctions (or “cross-bid”), and they tend to bid on the auction with the lowest

standing bid, as the theory would predict. Further, this tendency becomes stronger as

auctions become closer substitutes. Thus we find that for competing auctions that

end within the same minute, bidders are more likely to cross-bid and bid on auctions

with the lowest standing bid than is the case for auctions that end further apart. We

also find that, on average, bidders revise their bids more often when the auctions

they bid on end closer together; bidders also revise their bids more often when they

cross-bid. To assess the potential gains from following the strategy outlined in the

theory, we compare the winning price for winning bidders who bid across competing

auctions and for winning bidders who do not, finding that bidders who bid across

competing auctions pay lower prices on average than those bidders who do not. In

total, these results provide the first compelling evidence in support of competing

auction theory.

The paper is organized as follows: We first briefly summarize a theory with

competing auctions. Then in section 3, we describe the data. Section 4 reports

results of bidding behavior in eBay and in section 5, we compare the winning price

for bidders who bid across competing auctions with those for bidders who do not.

Section 6 concludes.

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2. Theory of competing auctions

There are few papers on auctions with many sellers and many bidders. When many

bidders with independent valuations simultaneously choose among many sellers, the

only equilibrium (as noted above) has buyers randomizing over available sellers (see

McAfee (1993) and Peters and Severinov (1997)). When there are many competing

auctioneers, if bidders cannot bid across auctions, independently run auctions lead to

inefficient trades, in the sense that the sum of all agents’ welfare is not maximized:

some very low valuation sellers do not successfully sell their goods while some high

valuation buyers cannot buy a good, because of the mismatch between buyers and

sellers.

In a double auction, the outcome is much more efficient. There, potential

buyers and sellers of a single good move simultaneously, with buyers submitting

bids and sellers submitting asking prices. An auctioneer then chooses a price that

clears the market: all sellers who ask less than sell, all buyers who bid more than

buy, and the total number of units supplied at price equals the number demanded.

In Wilson’s (1985) research, buyers and sellers’ valuations are drawn independently.

When the number of buyers and the number of sellers are large enough, a double

auction yields an efficient allocation; the sum of all agents’ welfare is maximized,

and all sellers with low valuations and all buyers with high valuations successfully

make trades.

The assumption that bidders have to choose one and only one auction

simultaneously is critical for the efficiency outcome in independent auctions with

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many buyers and many sellers. For on-line auctions, this assumption is unlikely to

hold. Peters and Severinov (2001) ask the question whether independently organized

auctions in a centralized exchange such as eBay can overcome the inefficiency of

random matching. In their model, there are many sellers and many bidders. Each

seller has a single good for sale and all goods for sale are identical. Each bidder only

needs one good, so that winning more than one auction is undesirable – the

additional good provides no additional utility. Auctions follow a similar pattern to

eBay: the standing bid is the second highest bid and the highest bid is never revealed.

However, bidders are not required to confine their attention to only one auction.

Under the assumption that bidding is costless and there is no fixed ending time for

auctions, the paper shows that competing auctions can overcome the inefficiency of

random matching. The paper gives a symmetric strategy for bidders and proves that

this strategy is a perfect Bayesian equilibrium. In equilibrium, all trades occur at the

same price and the price is the same as that in a double auction.

The main results of Peters and Severinov (2001) are repeated here:

Lemma: The symmetric equilibrium is defined as follows:

(a) if the buyer is the current high bidder at any auction, or if the buyer’s valuation is

less than or equal to the lowest standing bid, the buyer should pass;

(b) otherwise, if there is a unique lowest standing bid, the buyer should submit a bid

with the seller offering the lowest standing bid. The bid should be equal to the

smallest valuation that exceeds this lowest standing bid;

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(c) otherwise, let be the set of sellers who have the lowest standing bid. Let

be the subset of sellers in whose current high bidder submitted his bid

while the standing bid was strictly below its current level; let be the set of

sellers in who have not received a bid. If is not empty, the buyer should

bid with equal probability with every sellers in . If is empty but is

non-empty, the buyer should bid with equal probability with every seller in .

Otherwise, if both subsets are empty, the buyer should bid with equal probability

with every seller in .

Suppose there are sellers and bidders. Let be the lowest

valuation of all traders. Their theorem states:

Theorem: The outcome in which all buyers use the strategy is a (weak) perfect

Bayesian equilibrium. Buyers trade if and only if their valuation is above .

Sellers whose reserve prices are lower than trade for sure. All trades occur at the

price .

Bidders’ strategies can be summarized as follows: if a bidder is currently the

highest bidder in any auction, he pauses until he is outbid. Otherwise, he always bids

on an auction with the lowest standing bid and bids with the minimum increment.

The essential idea is that with cross-bidding, bidders bid up prices with each seller as

slowly as possible. They try to pick sellers where they can become the highest

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bidders by bidding the minimum increment. In this way, high valuation buyers are

never trapped into bidding a high price by having another high valuation buyer

accidentally bid against them. Cross bidding ensures that any mismatch between

buyers and sellers can be solved by giving bidders the opportunity to bid on other

auctions with low standing bids. As a consequence, all trade occurs at the same

price. Most importantly, the outcome of trading is the same as in a double auction.

We are interested to know whether bidders behave as the theory predicts.

Economists usually seek clean theoretical results, which lie in the territory of a

complete market and rational individuals. In case there is incomplete information,

players are required to make probabilistic calculations and forecast future outcomes.

Each player is assumed to be fully aware of the strategic value of his own private

information and to know the structure of other players’ strategies. Such analysis

seeks to characterize the Bayesian Nash equilibrium of the incomplete information

game defined by the trading institution and the trading environment. Yet research in

psychology and behavioral science is replete with examples in which individuals do

not, or perhaps even cannot, perceive circumstance objectively. This raises serious

questions about whether markets can achieve efficiency in practice.

There are more and more experiments in economics. Most results seem to

support that the efficient outcome of the market can be achieved. One import result

gaining support is the Hayek Hypothesis: markets economize on information in the

sense that strict privacy together with the public information (about prices) in the

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market are sufficient to produce efficient competitive equilibrium outcomes.1 How

and why this is true, there is no general answer. The information flow and the

contracting rule in the market may be more important than some traditional structural

characteristics in determining the market outcome. However, whether players use

strategies in Bayesian Nash equilibrium is dubious. In research by Forsythe et al.

(1992), a market worked extremely well, yet traders in the market exhibited

substantial amount of judgment bias.

3. Mechanism in eBay and Summary of Data

eBay provides a rich resource for the empirical study of auctions, and since there

exists a large quantity of similar auctions at any given time, eBay also provides an

excellent resource for the study of competing auctions.

eBay is a list e-commerce site. It provides a central market for buyers and

sellers to meet each other by way of auctions. The income eBay earns comes from a

fee charged to sellers, which varies from a fixed fee per listing, or a small proportion

of the final sale price. Buyers on eBay do not pay anything to participate.

Sellers choose an auction type in which they sell their good.2 They set a

starting bid, a bid increment, and the duration of the auction. Sellers also provide a

detailed description of the item, which usually also includes the method of delivery

1 This may refer to an efficient market allocation and a competitive price, or simply refer to an efficient market allocation with price different to the competitive price.2 There are 6 other types of auctions in eBay: reserve price auctions, Buy It Now, NEW eBay Store Buy It Now listings, Private Auctions, Dutch Auctions, Restricted-Access Auctions, see http://pages.ebay.com/help/sellerguide/selling-type.html.

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and method of payment. Sellers have the option to set a secret reserve price. If they

do so, during the process of the auction, eBay will indicate whether the reserve is

met or not. At the end of the auction, if the reserve is not met, sellers have the right

not to sell the item with the final price.

eBay use the mechanism of a second price auction. At any time, eBay shows

the current standing bid of the auction, which is the current second highest bid (if

there is no bid or there is only one bid, the current bid is the starting bid). When a

bidder submits a bid, he has access to the level of present standing bid, the identity of

the seller (with the seller’s “feedback,” discussed below), the starting and ending

time and the description of the item. He also has access to information as to how

many bids have already been submitted, the identity of the bidders, and the time of

the bids. However, the exact amount of each bid is not revealed until the end of the

auction. The final price is the second highest bid plus the bid increment.

At the end of an auction, eBay does not intervene in the actual transaction

between the seller and the winner of the auction; the seller and the winner contact

each other themselves to complete the transaction. Since bidders cannot inspect the

good directly, sellers have incentives to provide false information; winners may

regret the high price they have to pay and may not contact the seller to finish the

purchase. To promote the faithful implementation of the transaction, eBay uses a

feedback system. After each transaction (whether successful or not), the seller and

the bidder can send feedback about the other party to eBay, marked as positive,

neutral and negative, with values of +1, 0, -1 respectively, plus brief comments. A

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trader is given a feedback number, which is the sum of value of each feedback. The

feedback information is public, and always associated with the trader, though eBay

cannot prevent the traders from changing identity. Indeed, traders with negative

feedbacks have a strong incentive to change their identities in subsequent trades,

while traders with high feedbacks and good comments own an asset of good

reputation. There is some evidence that the reputation associated with the feedbacks

of sellers have an effect on the final price of auctions (see Houser and Wooders

(2001)).

When an auction has ended, eBay provides detailed information about the bid

history.3 Figure 1 is an example of an auction history extracted from eBay’s website.

The first half page shows the basic information about the auction. It is a three-day

auction, started at Oct-31-01 22:51:27 PST and ended at Nov-03-01 22:51:27 PST.

The seller has feedback with value 11. The starting bid set by the seller is $10 and

the increment is set to $0.50. The auction received 10 bids from 4 different bidders.

There was a shipping cost of $5 and optional shipping insurance $5.

The second half page shows the detailed bidding history. The bid history is

sorted by the amount of each bid. The auction received its first bid of $17.50 by

planetorb around 23 hours after the start of the auction. Then 4 hours before the end

of the auction, bidder raheem112 started to bid. He could observe that there was

already a bidder, but would not know the exact amount of bid. Since there was only

one bidder at that time, the standing bid was still $10. Bidder raheem112 first bid

3 In 2000, eBay provided only the highest bid of a bidder and his last bidding time.

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$11, and found that the standing bid increased to $11 and he was not the highest

bidder. Then he increased his bid subsequently in 2 minutes to $13, $14, $15, until at

last he became the highest bidder with bid $20. The standing bid became $17.50. In

the last hour of the auction, the bidder iteachcomputers submitted a bid $20,

increasing the standing bid to $20. Bidder raheem increased his bid to $21, followed

by another bidder who bid $23.99. Bidder planetorb finally won the auction with an

unknown bid. Since the minimum increment is $0.50, the final price was $24.49,

which is the second highest bid $23.99 plus the bid increment $0.50. There was no

bid retraction or cancellation for this auction. eBay keeps the information of

completed auctions public for one month.

Competing auctions in our sample should satisfy the following conditions: 1.

they should be reasonably homogeneous in quality (including warranty) and have

similar delivery method and shipping cost, 2. they should end at approximately the

same time. As documented by Roth and Ockenfels (2000) and Bajari and Hortacsu

(2000), bids on eBay tend to be clustered towards the end of the auction.

We use eBay CPU auctions data from a one-month period - September 20 to

October 19. These are drawn for the category of “Computer, Component, CPUs” in

eBay, which includes subcategories “AMD”, “Cyrix”, “Intel” and “Others”, and each

subcategory includes further subcategories. At the time of writing in November

2001, there are 800 to 900 new CPU auctions every day. We choose those auctions

with only one CPU for sale, and only those auctions with the method of standard

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auction.4 Most of the CPUs are second hand. Table 1 reports the basic statistics of the

sample.

In the sample, there are 7910 auctions involving a single CPU for sale. Not

all items were sold. Among all auctions, 1452 of them did not receive any bids. 899

of them, which is more than 10% of the auctions, have a secret reserve price. In all

899 auctions with secret reserve price, 515 auctions had the reserve price met.

The CPUs in the sample are very different. The mean final price is $60.60,

with standard deviation of $82.34. The number of bids received is 6.66 per auction,

with standard deviation of 6.91. The maximum number of bids in the sample is 49.

Also the starting bids of these auctions are very different, with a mean of $29.25 and

standard deviation of $61.60.

Though the conditions of properly working CPUs in the same specific

category (such as Pentium III 800 retail box) are largely the same, they may be very

different in many respects: some are new and never opened, others are used for

several months or years; some are still under warranty, others aren’t; some are with

both box and complete manual, others without. And the method of the delivery, the

shipping cost and the method of payment can differ.

To overcome this complication in obtaining the competing auctions, we use a

sample consisting of groups of auctions with the same product description, the same

4 eBay has introduced the ‘Buy it Now’ features for auction. Sellers can set a fixed Buy It Now price so that if any buyer wants of buy at that price, he can buy it immediately without go through the auction price. Bidders can bid on auction with Buy It Now feature before the item is sold. Since the Buy It Now method and the standard auction are very different, we exclude all CPUs sold by Buy It Now from the sample. (We will study the buy it now feature in a separate paper).

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delivery method and the same shipping cost. This is done by choosing the auctions

sold by the same sellers.5 Some big second CPU sellers sell many items with the

same description, the same delivery method and the same shipping cost. Without

considering the different ending time, these auctions are completely

indistinguishable to buyers.

Such auctions with identical items started and ended at different time.

Auctions with almost the same ending time compete directly against each other, and

auctions with large differences in ending time compete less directly. We get three

different samples for competing auctions. Each observation in the samples is a group

of auctions, which consists of 2 or more competing auctions. In the first sample (the

daily sample), each group of competing auctions consists of homogenous auctions

ending on the same day. In the second sample (the hourly sample), each observation

is a group of competing auctions consisting of homogenous auctions with the

difference of ending time being less than 1 hour. In the third sample (the minute

sample), each observation is a group of competing auctions consisting of

homogenous auctions with the difference of ending time being less than 1 minute.

Auctions appearing in the minute sample must appear in the hourly sample

and auctions appear in the hourly sample must appear in the daily sample. The

minute sample consists of groups of auctions that compete directly against each

5 This method of selection may not get all possible competing auctions. To get exhausted set of competing auctions, we have to include all similar items, including those sold by different sellers. However, such selection does not affect the validity of our result.

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other, and the hourly and daily data consists of groups of auctions that compete

against each other less directly. Table 2 is a sample description of the three samples.

In the daily sample, there are 550 groups of competing auctions, consisting of

1247 different auctions. Among all auctions, 305 (24%) of them do not receive any

bids and 106 auctions have secret reserve price (40 of them with the reserve price not

met). The hourly sample has a relatively smaller size, with 321 groups of competing

auctions, consisting of 748 different auctions. Among these auctions, 196 (26%) of

them do not receive any bids and 66 of them have secret reserve price (30 of them

with reserve price not met). The minute sample is less than half of the size of the

hourly data, with 139 groups of competing auctions, consisting of 346 different

auctions. Among them 115 auctions (33% of the sample) do not receive any bids and

24 have secret reserve price (19 of them with reserve price not met).

We find that in each sample, there are bidders who are winners for more than

one auction in a group of competing auctions. In the daily sample, there are 24

groups of competing auctions with bidders winning more than one auction,

representing 4% of the total groups. In the hourly sample and minute sample, the

number of the groups with a bidder winning more than one auctions is 18 and 10,

representing 6% and 7% of the total groups.

These bidders may happen to need more than one item for themselves, or

they can be professional dealers. Buyers with multiple demands can bid across more

than one auction at the same time. If we consider all bidders with single unit demand,

the existence of buyers with multiple demands will exaggerate the result that bidders

17

bid across auctions. However, the proportion of such bidders is low, and the number

of group in which such bidders win more than one auctions is low. In the following,

we will address this problem in more detail and distinguish true cross bidders from

the multiple demand bidders. Our results are not affected significantly by the

existence of those multiple demand buyers.

One may be concerned that bidders, especially the novice, may not fully

understand the mechanism in eBay auction. They might not use optimized strategies.

We use the feedback as an indicator of traders’ experience in eBay6. eBay also uses

heavily the number of feedback in daily trade. For example, to use the Buy It Now

feature, sellers must have a feedback greater than 10 or be ID verified7. Figure 2 is

the distribution of bidders’ number of feedbacks. In the daily sample, there are 2286

bidders and 21 bidders with negative feedback (1 bidder with feedback –4, 2 bidders

with feedback –3, 8 bidders with feedback –2, the rest 10 with feedback –1). 60% of

the bidders have feedback greater than 8. Most of the bidders have history of

transactions in eBay. We can be confident that the observed behavior is not very

different from the optimal one for most individuals.

4. Bidding behavior with competing auctions in eBay

Since we have very detailed bidding history for each auction, we can use it to

study how bidders bid in face of competing auctions.

6 The feedback number in eBay is the sum of positive and negative feedbacks.7 See http://pages.ebay.com/services/buyandsell/buyitnow-seller.html.

18

The most interesting result we observe is that bidders, even with single unit

demand, bid across competing auctions.

In table 3 we report the statistics of the bidders and cross bidders in each

sample. For each group of competing auctions, we use a Java program to get all

different bidders and all bidders that bid on more than one auction.

We only consider the group of competing auctions with positive bid for at

least one auction. In the daily sample, there are 458 such groups of competing

auctions. On average, there are 2.28 auctions in each group and the maximum

number of auctions in a group is 9. On average, there are 6.56 bidders for each

group and 1.53 bidders bid across auctions in the group. On average, 23% of them

bid across competing auctions in the group they bid.

In the hourly sample, there are 258 groups of competing auctions with

positive bids for at least one auction. The number of different bidders per group is

almost the same as that in the daily sample, and slightly few numbers of bidders who

bid across auctions in a given group. 21% of the bidders bid across auctions in the

group they bid.

In the minute sample, there are only 101 groups of competing auctions with

positive bids for at least one auction. On average, there are fewer bidders in each

group, with an average of 4.868. And there are 1.48 bidders who bid across auctions

in a given group. The proportion of bidders who cross bid is higher than the other

8 It may be a little puzzle to see that there are fewer bidders on average for the minute sample. This is because that in the minute sample, a higher proportion of auctions have high starting bid and receive only one or zero bids.

19

two samples, but it is not statistically significant. (The t-statistics for hourly sample

and minute sample is 1.33 and the t-statistics for daily sample and minute sample is

1.14).

The data shows that a significant proportion of bidders bid across auctions in

a competing group. For auctions with all most the same ending time, significantly

higher proportion of bidders bid across competing auctions. The proportion of

bidders who bid across competing auctions is not very different for daily sample and

hourly sample (with a t-statistics of 0.54).

For competing auctions ending at all most the same time, the strategy for

bidders to bid across auction is more effective in avoiding in trapping in highly

competitive auctions. However, if there is a significant time difference in ending

time, bidders cannot effectively coordinate bidding across auctions.

As documented at Roth and Ockenfels (2000) and Bajari and Hortacsu

(2000), bids are clustered at the ending period of the auction. This is also true for our

competing auction data. We have information about all the bids and the time the bids

are submitted. Figure 3 shows the bid submission time for all auctions in the daily

sample.

Bids are clustered at the ending period. Almost all auctions received their last

bid in the last several hours. In addition, 28% of the samples receive their last bid

within the last 60 seconds.

In Roth and Ockenfels (2000) the reason for delaying bids is that eBay

auction has a fixed ending time. If the bids are submitted at the last minute, there is

20

some probability that bids may not be submitted successfully. This can explain the

very last minute bidding, but cannot explain late bidding. Roth and Ockenfels (2000)

also find that even in auctions like Amazon auctions that have no fixed ending time,

there are significant late biddings. Simply bidding late does not have effect that bids

might not be submitted successfully. Bajari and Hortacsu (2000) argue that in a

common value environment, bidders refrain from bidding earlier to avoid revealing

their private information.9

Therefore, it is sometimes argued that early biddings are not serious, and only

late biddings are serious. We look at the data of the last day of each auction, with all

different bidders in the last day and the different bidders who bid across competing

auctions. We use the sample consisting of the groups with at least one auction

receiving positive bids in the last day. We find that for all bidders bidding in the last

day of the auctions, significantly more bidders bid across competing auctions when

the difference in ending time is less than 1 minute. (See Table 4.) For the minute

sample, on average, 27% of the bidders in the last day bid across competing auctions,

compared to the percentage of 0.20 in the hourly sample and 0.17 in the sample. For

the hypothesis of having the same percentage of bidders who bid across competing

auctions, the t-test for the minute data and the hourly data is 1.65 and the t-test for

9 The existence of competing auction can provide an explanation for late bidding. With the existence of competing auctions, bidders bid across competing auctions to avoid being trapped in an auction if other high valuation bidder happens to bid against them. If there is no bidding cost, bidders should bid the minimum increment each time and bid again and again. With bidding cost, it is too costly to bid very often. Bidders can bid with a significant jump and bid less frequently. However, this increases the risk of being trapped in a very competitive auction. Another alternative for bidders is to bid only at the late period, and bid with small increment. In this way, they do not need to bid too frequently, at the same time they can effectively cross bid and avoid being trapped in too competitive auctions.

21

the minute data and daily data is 2.52, which is significant at the 0.01 level. And

though there is slightly higher percentage of bidders who bid across competing

auctions for hourly data than for daily data, the result is not statistically significant,

with t-test 1.23.

We may wonder if the cross bidding observed here is only because of the

existence of bidders who want more than one item at the same time. There is not a

simple way to check directly if a bidder wants more than one item. We use the

following methodology: bidders who need more than one item is very likely to be the

highest bidders of more than one competing auction. In the first case, we define true

cross bidders as all those bidders who bid across competing auctions and who are

never the highest bidder of more than one auction at any time (type I). In the second

case, we consider true cross bidders as all those bidders who bid across competing

auctions and who are not the highest bidders of more than one competing auction in

the last day (type II).

The first criterion for true cross bidder is stricter than the second criterion.

The second criterion has already excluded all bidders who win more than one

competing auctions. Some bidders excluded by the first criterion may still be true

cross bidders, since at the early stage of the auctions, bidders may feel safe to be

high bidders of more than auction even if they only need one item.

Table 3B reports the result of such considerations. When we exclude those

possible bidders with more than one demand, we still observe high percentage of

bidders who bid across competing auctions. For the minute data, 28 bidders are high

22

bidders for more than one competing auction in some point of time of auction

process and 21 are high bidder for more than one competing auction in some pint of

the last day of the auctions. The bidders who truly bid across auctions are 25% in the

first criterion and 26% in the second criterion. Similarly, for the hourly data, we

observe that among all bidders, 21% of them bid across competing auctions. Apart

from those who need more than one item, the percentage of true cross bidder is 18%

(first criterion) and 19% (second criterion). For the daily data, we observe 23% of

bidders who bid across competing auctions. The percentage of true cross bidders is

21% (under the first criterion) and 22% (under the second criterion).

Another feature of bidding with competing auctions is that bidders tend to bid

on the auction with the lowest standing bid. This strategy not only guarantee that a

bidder never wins two auctions, it also let bidders to avoid being trapped in very

competitive auction. (This also makes the price of the auction become uniform.) We

report the result on whether bidders bid on the auction with the lowest standing bid

in table 6. 10

When all auctions in a competing auction group are ended except the last

one, bids submitted on this last auction thereafter are considered as bidding on

auction with the lowest standing bid. This way of calculation tend to increase the

number of bids submitted on the auction with the lowest standing bid for group of

auctions with big difference in ending time, such as in the daily and hourly sample.

10 It is not trivial to see if a bid is submitted on the auction with the lowest standing bid. We use some data structure such as TreeSet in Java. Interested reader may have a look at the appendix of Zheng (2001).

23

In the daily sample, the average number of bids a group of competing

auctions received is 13.29 and the maximum number of bids a group received is 79.

The average number of bids submitted on auctions with the lowest standing bid is

8.95 and the maximum number of such bids is 37. For the hourly sample, the average

number of bid in a competing group and the average number of bids on auction with

the lowest standing bid is roughly the same. The proportion of the bids submitted on

the auction with the lowest standing bid is also roughly the same, representing 79%

for daily sample and 77% for hourly sample (the t-test for the proportion for daily

and hourly sample is 1.10).

For the minute sample, the average number of bids a group receives is

relatively less, with a value of 10.89, and the maximum number of bids that a group

receives is 7.86. The proportion of the bids submitted on the auction of the lowest

standing bid is significant higher, with an average of 87%. The t-test for the minute

data and the hourly data is 4.49 and the t-test for the minute data and the daily data is

4.33.

Bidder who bid across competing auctions are more likely to bid on the one

with the lowest standing bid. In table 6, we report the result about the proportion of

bids submitted by bidders who bid across competing auctions.

In all three samples, there are groups in which some auctions do not receive

any bids while others receive positive number of bids. It happens usually that if there

is any auction does not receive bids, then other competing auctions either receive no

bids or receive only one bid. For same items, there is no reason that bidders bid on

24

auctions with positive bids and not bid on auctions with zero bids. It is interesting to

see how often it happens that in a group of competing auctions, some auctions

receive no bids while other auctions receive more than one bid. Table 7 reports the

result for three samples.

In the daily sample, there are 74 groups of competing auctions with positive

bids and with zero bid auctions. 30% of them with auctions receiving more than 1

bid. In the minute sample, there are 20 groups of competing auctions with positive

bids and with zero bid auctions. However, only 15% of them have auctions receiving

more than 1 bid.

It is obvious from the above analysis that bidders are not following the

strategy as in independent second price auction: bid once and bid the true valuation.

Bidders are not following the advice form eBay to submit the true valuation and let

the proxy to bid for them. The proxy bid mechanism in eBay is believed to save

bidders from revising their bid. eBay’s help page about proxy bid explains the proxy

bid as the following:

A proxy bid and a maximum bid are the same thing. To place a

proxy bid, just enter the maximum amount you are willing to pay. eBay

will then automatically bid up to your maximum amount for you.

(http://pages.ebay.com/help/basics/e_item1.html)

25

In table 8, we report the result of the average number of bids a bidder

submitted in each group. We only consider the sample consisting of groups with

bidders who bid across competing auctions. With competing auctions, bidders do not

bid on one auction, but rather on a group of competing auctions. We calculate the

bids each bidder submitted on each competing group. There is a tendency that

bidders bid more frequently with competing auctions with almost the same time. In

the minute sample, on average, each bidder submit 2.24 bids on a group of

competing auctions (On average, there are 2.6 auctions in each group. Each bidder

bid less than 1 bid on each auction of a group). However, for bidders bidding cross

auctions, the average number of bids they submitted on a group is significantly

higher. For the minute sample, each bidder who bid across auctions submits 3.97

bids on a given group of competing auctions.

Bid revising can be a consequence of the existence of competing auctions. If

there is no bidding cost, bidders should bid with the minimum increment and bid

many times. If bidders bid true valuation and bid only once, they may be trapped in

very competitive auctions and do not have opportunity to switch to other less

competitive auctions. Even if with bidding cost, bidders still revise their bids very

often if the cost is not too high compared to the risk of bidding with another high

bidding bidder. (They may not always bid the minimum increment, which is too

costly).

26

5. Does cross bidders pay lower price?

Some winners of auctions bid across competing auctions, while others do not.

Bidding across competing auctions make bidders have more choice in bidding and in

trying to find auction with low price. The theory predicts that bidders who bid across

competing auctions can win auction with lower price.

To test this result, we divide each group of competing auctions into two

subgroup: a group consisting of auctions in which winners bid across competing

auctions and with single unit demand, another group consisting of the rest of

auctions. Here we define a bidder who only needs one item as in the sense that he

was never the highest bidder for more than one auction in any time of the auction

process (criterion 1). Then we compare the average final price in these two sub-

groups by calculating the ratio of price paid by cross bidder and the price paid by

non-cross bidder. In some group all winners bid across competing auctions while in

other groups neither of the winner bid across competing auctions. These groups are

excluded from the analysis. We keep the groups in which there are winners who bid

across competing auctions and winners who do not bid across auctions. Table 9

reports the result.

In the minute data, there are only 17 groups of competing auctions with both

winners who bid across competing auctions and those winners who do not. The ratio

of the price paid by cross bidder and by non-cross bidder is 0.89, with a standard

deviation of 0.19. We consider the test ratio<1. For minute data: we have

27

. At level 0.01, the winners who bid across competing auctions pay a

price statistically lower than those winners who do not. On average, the winners who

bid across competing auctions pay a price that is only 90% of the price paid by those

winners who do not.

This is also true for the daily data. There are 103 groups of competing

auctions with winners who bid across competing auctions and winners who do not.

For the test ratio<1, we have , and at level 0.01, the winners

who bid across competing auctions pay a price statistically lower than the winners

who do not.

However, for the hourly sample, the price paid by the cross bidding winner

and non-cross bidding winner is not significantly different. For the test ratio<1,

we have .

Therefore, it is strictly benefit for bidders to bid across competing auctions

even when some other bidders do not.

6. Conclusion

This paper has provided the first related pieces of evidence in the literature that

together lend strong support to competing auctions theory. First, in the presence of

competing auctions, bidders use a very different strategy than they would with

independent auctions. Bidders tend to bid across competing auctions and bid on the

auction with the lowest standing bid. We further demonstrate that when homogenous

28

auctions end at almost the same time, more bidders are likely to bid across

competing auctions than when auctions end at different times. We also find that

winners who bid across competing auctions pay a lower price than those winners

who do not bid across competing auctions. Therefore, since there are many

homogenous auctions competing against each other in on-line auctions, it is

inappropriate to consider on-line auctions as independent, ignoring the existence of

other competing auctions.

In this paper, we do not consider whether the auction prices are uniform with

the existence of competing auctions. According to the theory, competing auctions

tend to make prices more uniform. We will study such price effect of the existence of

competing auction on eBay in a separate paper. In the current paper, we also take the

behavior of sellers as exogenous. It will be interesting to see what the equilibrium

behavior of sellers is given the existence of competing auctions. This will be

especially useful for those sellers who want to sell many identical items one by one

in a short time.

29

Reference

Bajari, Patrick and Ali Hortacsu. (2000) “Winner’s curse, Reserve Prices and

Endogenous Entry: Empirical Insights from eBay auctions”, Working paper,

Stanford University.

Hooser, Daniel and Daniel Wooders. (2001) Reputation in Auctions: Theory and

Evidence from eBay, working paper, University of Arizona.

Lucking-Reiley, David. (1999) “Auctions on the Internet: What’s being auctioned,

and How?” Working paper, Vanderbilt University.

Myerson, R. B. (1981) Optimal Auction Design. Mathematical of Operations

Research, 6, 58-73.

McAfee, P. (1993) “Mechanism Design by Competing Sellers,” Econometrica 61

(1993) no.6, 1281-1312.

Peters, Michael and Sergei Severinov. (1997) “Competition among sellers who offer

auctions instead of prices,” Journal of Economic Theory, 75, 141-197.

30

Peters, Michael and Sergei Severinov. (2001) “Internet Auctions with Many

Traders,” Working paper, University of Toronto.

Forsythe, Robert et al. (1992) “Anatomy of an Experimental Political Stock Market,”

American Economic Review, vol. 5, 1142-1161.

Roth, Alvin and Axel Ockenfels. (2000) “Last Minute Bidding and the Rules for

Ending Second Price Auctions: Theory and Evidence from a Natural Experiment in

eBay,” Tech. Report, Harvard University

Wilson, R. (1985) “Incentive efficiency of double auctions,” Econometrica 53, 1101-

1117.

Zheng, Mingli (2001) Real time simulations of eBay auctions with competing

auctions, working paper, University of Toronto.

31

Figure 1.1

A bidding History Page from ebay

EBay Bid History forIntel Pentium II Xeon 450MHz 512k - Used (Item # 1292121814)

Currently $24.49 First bid $10.00

Quantity 1 # of bids 10

Time left Auction has ended. Started Oct-31-01 22:51:27 PSTEnds Nov-03-01 22:51:27 PSTSeller (Rating) nhahmad (11)

View page with email addresses (Accessible by Seller only)   Learn more.

Bidding History (Highest bids first) User ID Bid Amount Date of Bid planetorb (9) $24.49 Nov-03-01 22:30:37 PSTibgeek (4) $23.99 Nov-03-01 22:18:01 PSTraheem112 (0) $21.00 Nov-03-01 22:03:35 PSTiteachcomputers (1) $20.00 Nov-03-01 22:06:49 PSTraheem112 (0) $20.00 Nov-03-01 18:33:27 PSTplanetorb (9) $17.50 Nov-01-01 21:11:05 PSTraheem112 (0) $15.00 Nov-03-01 18:33:16 PSTraheem112 (0) $14.00 Nov-03-01 18:32:50 PSTraheem112 (0) $13.00 Nov-03-01 18:32:39 PSTraheem112 (0) $11.00 Nov-03-01 18:31:52 PSTRemember that earlier bids of the same amount take precedence. Bid Retraction and Cancellation History There are no bid retractions or cancellations.

Table 1.1

33

Sample Statistics For CPU Auctions (Number of auction=7910)

Variable Mean Std Dev Minimum Maximum

Number of bids 6.66 6.91 0 49

Last bid (unit: $) 60.60 82.34 0.01 981

Starting Bid (unit: $) 29.25 61.60 0.01 899

1452 auction receive no bids899 auctions have secret reserve price515 auctions have met reserve prices

34

Table 1.2

Sample Statistics for Samples of Competing Auctions

Daily sample Hourly sample Minute sample

Number of auctions 1247 748 346

Number of groups 550 321 139

Number of auction with reserve met 66 36 5

Number of auctions with reserve not met 40 30 19

Number of auctions not receiving bids 305 196 115

Number of bidders winning more than one item in a

group 24 18 10

35

Figure 1.2

Histogram of Bidders’ Feedback

36

Figure 1.3

Histogram of Bids Submission Time for Daily Sample

0

200

400

600

800

1000

1200

1400

1600

1800

Fraction of auction duration

Freq

uenc

y

37

Table 1.3

Statistics of Cross Bidding in the Whole Process

Groups Auctions per group Cross bidder per group Bidder per group ProportionMean Std Min Max Mean Std Min Max Mean Std Max Min Mean

Daily 458 2.28 0.87 2 9 1.53 1.78 0 10 6.56 5.14 32 1 0.23Hourly 258 2.36 0.98 2 8 1.41 1.90 0 10 6.59 5.32 32 1 0.22Minute 101 2.60 1.29 2 8 1.48 1.96 0 10 4.86 4.34 20 1 0.27

(For groups with positive bids only.)Proportion: proportion of cross bidders.

T-statistics for the proportion of hourly sample and minute sample is:

T-statistics for the proportion of dailyly sample and minute sample is:

T-statistics for the proportion of hourly sample and daily sample is:

38

Table 1.3 B

Percentage of Cross Bidders in All Bidders

Percentage of cross bidder Percentage of true cross bidder I Percentage of true cross bidder II

Daily 0.23(700/3007) 0.21(627/3007) 0.22(648/3007)Hourly 0.21(363/1700) 0.18(312/1700) 0.19(324/1700)Minute 0.30(149/491) 0.25(121/491) 0.26(128/491)

True cross bidder I: bidders who bid across competing auctions and who are never the highest bidder of more than one auctions at any

moment.

True cross bidder II: bidders who bid across competing auctions and who are never the highest bidder of more than one auctions at any

moment in the last day.

39

Table 1.4

Statistics of Cross Bidders for the Last Day

Groups Auctions per group Last day cross bidder per group Last day bidder per group Proportion

Mean Std Min Max Mean Std Min Max Mean Std Max Min Mean StdDaily 419 2.28 0.87 2 9 0.63 1.02 0 6 3.61 2.44 12 1 0.17 0.28

Hourly 235 2.37 0.97 2 8 0.71 1.11 0 6 3.62 2.55 12 1 0.20 0.31Minute 88 2.60 1.26 2 8 0.93 1.23 0 6 3.06 2.19 10 1 0.27 0.35

(For groups with positive bids in the last day only.)

Proportion: proportion of cross bidders.

T-statistics for the proportion of hourly sample and minute sample is:

T-statistics for the proportion of daily sample and minute sample is:

T-statistics for the proportion of hourly sample and daily sample is:

40

Table 1.5

Result on Bidding on the Auction with the Lowest Standing Bid(for all bidders)

GroupsBids on the auction with lowest

standing bid Total bids Proportion

Mean Std Max Min Mean Std Max Min Mean StdDaily 458 8.95 7.68 37 0 13.29 12.51 79 1 0.79 0.20

Hourly 258 8.45 7.85 37 0 13.26 12.97 79 1 0.77 0.25Minute 101 7.86 8.32 36 1 10.89 13.17 63 1 0.87 0.16

(Groups with positive bids only.)Proportion: proportion of bids submitted on auction with the lowest standing bid.

T-test for proportion of minute and hourly sample:

T-test for proportion of minute and hourly sample:

T-test for proportion of hourly and daily sample:

41

Table 6

Result on Bidding on the Auction with the Lowest Standing Bid(For groups with cross bidders only)

GroupsBids on the lowest

(cross bidder)Total bids

(cross bidder)Proportion

(Cross BidderBids on lowest

(all bidders)Bids (all Bidders)

Proportion

(all bids)Mean Std Max Min Mean Std Max Min

Daily 286 5.97 4.94 31 0 8.78 7.76 47 2 0.76 8.78 18.60 0.74Hourly 138 6.54 5.50 27 0 9.56 8.51 41 2 0.77 12.42 18.36 0.76Minute 55 7.69 6.54 31 1 10.75 10.19 47 2 0.81 12.44 17.82 0.78

Proportion: proportion of the bids on lowest standing bid by cross bidders among all bids by cross bidders

42

Table 1.7

Result on Bidding on Group with Zero Bid Auctions

Groups with positive bids and with zero bid auctions

Groups with auctions receiving more than 1 bids

Proportion

Daily 74 20 30%Hourly 27 6 22%Minute 20 3 15%

Table 1.8

Average Number of Bids Submitted in a Group by a Bidder

For all bidders For cross bidders only1 day 2.02 3.581 hour 2.01 3.63

1 minute 2.24 3.97

43

Table 1.9

Price Paid by Cross Bidders and Non-Cross Bidders

Price paid by cross bidder/price paid by non cross bidderMean Standard deviation

Daily (obs =103) 0.91 0.17Hourly (obs =41) 1.08 0.95Minute (obs =17) 0.89 0.19

Cross bidders: defined as those cross bidders who are never the highest bidder of more than one auctions (type I)

ratio<1.

For minute data:

For hourly data:

For daily data: At 0.01 level, for minute data and daily data, winners who bid across competing auctions pay lower price than other bidders. Those winners who cross bid pay a price that is 10% lower. For hourly data, the price paid by winners who cross bid and who do not cross bid are not significantly different.

44

45

Chapter 2

Liability Rules and Evolutionary Dynamics11

(Published inJournal of Institutional and Theoretical Economics, 157 (4) Dec 2001.)

Abstract

We consider the convergence properties of behavior under a comparative negligence

rule (CN) and under a rule of negligence with contributory negligence (NCN), assuming

bilateral care with three care levels. Using an evolutionary model, we show that CN reduces

the proportion of the population using low care more rapidly than does NCN. However NCN

increases the proportion of the population using high (efficient) care more rapidly than does

CN. As a result, the mean care level increases more rapidly and the mean social cost falls

more rapidly under CN than under NCN. (JEL: K 13, C 79)

11 I am very grateful to Donald N. Dewees for extensive help and support. Without his help the paper could never be the present form. Myrna Wooders introduced me to the evolutionary game theory and provided many support. Joanna Robert and Michael Peters gave me valuable comments and encouragements. I am also very grateful to two anonymous referees for their very detailed comments that greatly improve the paper.

46

1 Introduction

An economic analysis of law considers tort liability as a tool that can induce injurers

to internalize the costs they impose on others. An efficient liability rule should

provide incentives for a causative contributor to an accident to minimize the sum of

accident and avoidance costs by taking cost-justified precautions. Many authors have

discussed the equilibrium and efficiency of different liability rules. (BROWN

[1973], LANDES AND POSNER [1987], SHAVELL [1987], ARLEN [1992]).

An important issue is how to choose the best liability rule. In recent years the

comparative negligence rule (CN hereafter) has spread widely, replacing the rule of

negligence with contributory negligence (NCN hereafter). Eight US states had

adopted comparative negligence by 1971, but an additional 34 adopted it between

1971 and 1985. CN was argued to be inferior to NCN because the court must decide

on the degree of the negligence by both parties (WHITE [1988]). POSNER [1992]

stated that “the modern movement to substitute comparative for contributory

negligence” is one of the three “most important counter examples to the efficiency

theory of law”. WHITE [1989] tested empirically whether the incentive to take care

to avoid accidents is stronger under NCN than under CN.

The above literature only considered whether the rules provide an efficient incentive

to take care. WITTMAN, FRIEDMAN, CREVIER AND BRASKIN [1997]

47

addressed another consideration in choosing among liability rules: the speed of

convergence to equilibrium levels of care. When behavior is not at equilibrium, Nash

equilibrium is seldom achieved instantaneously. WITTMAN, FRIEDMAN,

CREVIER AND BRASKIN [1997] undertook an experimental test of convergence

to equilibrium under different liability rules. In their laboratory experiment,

convergence to equilibrium (measured by mean care level) is much more rapid under

comparative negligence than under contributory negligence. They gave no

theoretical explanation for their result.

From time to time, society is away from the equilibrium level of care for different

reasons. Individuals may not be fully rational. In auto accidents, there may be new

drivers who do not correctly perceive the risks and costs, such as when a

demographic bulge hits driving age, or a wave of immigration produces a stock of

new drivers. At any time some individuals may experiment with new strategies.

When a court or legislature changes the rules, it will take time before drivers adapt

themselves to that change. Changing technology can change the risk and cost of

driving significantly12. Any of these factors can require drivers to adjust their

behavior to a new optimum, incurring high social costs if the adjustment is slow.

The main contribution of this paper is that we use an evolutionary approach to

analyze the speed of convergence under different liability rules, using a simple

setting of bilateral care with three care levels. Homogenous drivers decide whether to

12 For example, anti-lock brakes and air bags may reduce the cost of aggressive driving. The increase in the proportion of large sport-utility vehicles and pickup trucks may reduce the benefits of careful driving of their owners and increase the benefits of careful driving for owners of small cars.

48

take a high, medium or low levels of care. The high level of care is the social

optimum. In this setting of three care levels, the Nash equilibrium under both CN

and NCN is the social optimum.

The evolutionary approach assumes that a strategy that does well is imitated, while a

strategy that does badly is rejected. We assume that in every period an individual

reflects on the payoff from his strategy and shares strategy and payoff information

with others. In every period a fraction of those individuals with a lower payoff

change their current strategies to more profitable strategies. This eventually leads to

the convergence to the equilibrium strategy. Because of inertia and uncertainty the

fraction of individuals that change their strategy in any period is relatively small.

The greater is the payoff difference, the greater is the incentive to change the

strategy.

We show that CN reduces the proportion of the population using low care more

rapidly than does NCN. However NCN increases the proportion of the population

using high (efficient) care more rapidly than does CN. As a result, the mean care

level increases more rapidly and the mean social cost falls more rapidly under CN

than under NCN during the early periods. This is consistent with the result in

WITTMAN, FRIEDMAN, CREVIER AND BRASKIN [1997].

49

The paper is organized as follows: section 2 states the basic assumptions and the

Nash equilibrium under different liability rules, section 3 considers evolutionary

dynamics and section 4 concludes.

2 Liability Rules and Nash Equilibrium

We consider a framework of bilateral care with three care levels. Suppose the

population has infinitely many drivers. There are only three levels of care for each

driver, high ( ), medium ( ) and low ( ), with the costs of taking care as , , and

respectively. For simplicity, we also use to denote the amount of care under

three care levels, . Once an accident occurs, a total damage is

incurred.

A driver does not know what kind of other drivers he will encounter during the day.

We model these encounters as random matches. The probability of an accident in a

match between two drivers is: and , where is the probability

of an accident when a driver taking care meets another driver taking care

. We assume that:

Assumption 1. Social cost is minimized if all drivers take high care. Social cost is

maximized if all drivers take low care. Thus:

50

. (2.1)

From this assumption, we have: , and

other similar inequalities.

We further assume that:

Assumption 2. An increase in the care level reduces more effectively the probability

of an accident when the care level of the other driver is relatively low.

This implies that and other similar inequalities.

Because taking high care minimizes social costs, any driver in an accident who does

not take high care is negligent (or contributorily negligent).

Under CN, both drivers share the losses according to their relative negligence, or

care shortfall. For example, if an accident happens between a driver taking medium

care and a driver taking low care, then the driver taking medium care will bear

portion of the total loss13, while the other party will bear the rest of the

loss. Since , we have . The following is the payoff matrix of the game.

Table 2.1

13 This is a typical assumption in comparative negligence, see, e.g., COOTER and ULEN (1997). It may take a more general form as a function of the care levels and (or) the cost of care.

51

Payoff Under CN

H M L

H

M

L

Under NCN, the parties share the liability in the same way as under CN except for

the cases when both parties are negligent. In those cases, they both incur half of the

total damage. The payoff of the game is the same form as that for CN, with .

Under NCN drivers taking low care and medium care are considered equally

negligent.

Proposition 2.1. If taking high care is socially efficient, then under both CN and

NCN, taking high care is a Nash equilibrium.

Proof: Given the assumption about the parameter values, it is easy to check from the

payoff matrix that taking high care is a Nash equilibrium.

As we can see from the payoff matrix, the only difference between CN and NCN is

the value of . When there is an accident between a party taking low care and a

party taking medium care, the party who takes low care has to incur a larger fraction

52

of the cost of the accident under CN than under NCN. This provides a greater

incentive to abandon low care under CN, as we will discuss in the next section.

3 Evolutionary Dynamics

In reality it is rare that Nash equilibrium is achieved instantaneously. Nash

equilibrium requires that players are rational and know the payoff functions of all

players, that they know their opponents are rational and know the payoff functions,

that they know their opponents know, etc. In actual life, these requirements may not

be met14.

This poses a problem: will the Nash equilibrium always be closely approximated at

least in the long run? If this is the case, does the outcome converge to the Nash

equilibrium rapidly and what is the path of the convergence?

To answer this question, we use an evolutionary approach (MAYNARD SMITH

[1982]). The idea of evolutionary games began with the idea that animals are

genetically programmed to play different pure strategies, and that the genes whose

strategies are more successful will have higher reproductive fitness. The population

fractions of strategies whose payoff against the current distribution of opponents'

play is relative high will tend to grow at a faster rate, and any steady state must be

14 REA [1987] points out that individual may be judgment proof or they could misperceive both the risks and costs. They may not choose the Nash equilibrium care level (they are unresponsive). The author suggests that, in comparison to the negligence rule with contributory negligence, the negligence rule with comparative negligence is more robust to the presence of these unresponsive individuals.

53

Nash equilibrium. There is no need for the strong requirement of rationality and

common knowledge among players.

Evolution can be taken as a metaphor for learning in economics. Individuals respond

to different payoffs by modifying their strategies. If we assume inertia in human

behavior and costs associated with switching strategies, then the proportion of the

population choosing each strategy changes smoothly. In the following, the

proportion of drivers taking care is subject to evolutionary pressure over time. The

fraction of the population using better performing strategies will increase relative to

those using lower payoff strategies. Our main focus will not be on the steady state of

evolution, but on the relative speed and the path of convergence to the steady state

under different liability rules.

We denote and the proportion of drivers taking high, medium and

low level of care at time , . Given the population composition ( ,

), a driver will meet drivers taking high care with probability and will meet drivers

taking medium and low care with probability . Under CN (and also NCN,

which corresponds to ), at any time , the expected payoff for a driver who

takes high care is:

. (3.1)

54

The expected payoff for a driver who takes medium care is:

. (3.2)

The expected payoff for a driver who takes low care is:

. (3.3)

The mean payoff of the population is:

. (3.4)

The value of is the social cost.

When , some drivers will find it profitable to switch from the strategy of taking

low or medium care to the strategy of taking high care, and will increase. The

payoff differential exerts evolutionary pressure on the population composition. The

standard model of the movement of the composition of the population in

evolutionary game theory is that of the replicator dynamics15, (TAYLOR AND

JONKER [1978]), defined as:

, (3.5)

where is the time derivative of , and is a constant. The rate of the

growth (decline) of the proportion of the population using a strategy is proportional

15 This is only for simplicity of the analysis. As we will see later, most of the analysis is still true if we use more general evolutionary dynamics such as a growth monotone dynamics.

55

to the amount by which that strategy's payoff exceeds (falls below of) the average

payoff of the whole population. The standard replicator dynamics can be derived

from different models of individual learning behavior (for example, NACHBAR

[1990])16. The NACHBAR [1990] model can be reasonably used in the driving

environment.

Evolutionary game theory is widely used in economics, for example in choosing the

most likely equilibrium from all possible Nash equilibriums. Replicator dynamics

allows us to compare the convergence properties under different liability rules in a

given social environment and with a same learning pattern.

Simple calculations using expressions (3.1)-(3.4) show that the dynamics of a

population taking high care is exactly the same17 under both CN and NCN, and can

be written as:

,

with

16 In this model, individuals meet randomly somebody else to exchange information about each other's strategy and payoff. The individual with a lower payoff switches his strategy if the switching cost is less than the payoff difference. Assuming the switching cost is independently determined across individuals and is uniformly distributed on [0,M], we get the exact form of replicator dynamics as in (3.5).17 Notice that at any given population composition is always the same under CN and NCN.

56

(3.6)

Under Assumption 1 and Assumption 2, we can check that each term in the above

equation is always positive, i.e., at any population composition with . The

proportion of drivers taking efficient cares always strictly increases.

For the dynamics of a population taking medium care, under CN (also under NCN,

which corresponds to ),

(3.7)

Since , , (by Assumption

2), if , we have:

(3.8)

As increases to a certain extent, becomes small and we have , and

will begin to decrease monotonically.

57

Similarly, will decrease monotonically as increases to a certain extent.

From the above analysis, we have:

Corollary 3.1. Under evolutionary dynamics, both CN and NCN lead to

convergence to the social optimum in the long run.

Proof: As in the above analysis, always strictly increases, and will decrease

as increases to a certain extent. Therefore, the population composition (

must converge to (1,0,0), which is the social optimum.

Given any population composition, the difference between the payoff of individuals

taking low care and the average payoff of the population is bigger under CN than

under NCN. Therefore under CN, the proportion of individuals taking low care

decreases faster under CN at any given population composition (the proportion of

individuals taking medium care decreases more slowly). We have the following

lemma:

Lemma 3.2. At any population composition ( ), the instantaneous growth

rate of the proportion of individuals taking high care is the same under CN and NCN.

The instantaneous rate of decrease of the proportion of individuals taking low care is

greater under CN than under NCN by . The instantaneous rate of

58

decrease of the proportion of individuals taking medium care is greater under NCN

than under CN by .

Proof: It is easy to check by using expressions (3.1)-(3.4), and by the definition of

replicator dynamics (3.5).

Unfortunately, the above lemma is only a local property and it assumes that we are at

the same population composition under CN and NCN. Once the system begins to

evolve, the evolution of the system will follow different paths under the two liability

rules and the local comparison becomes meaningless. Now we begin to discuss the

global convergence property and the convergence path under the two liability rules.

We can look at the path of the dynamics of population composition by looking at the

( ) plane (since ). From the same starting point at time , since the

increase rate of is the same under both liability rules and the decrease rate of is

slower under a comparative negligence rule, we have:

Lemma 3.3. Starting from the same point, the path (when is small) under CN is

above the path under NCN.

We may wonder if this is always true, or the two paths may cross at a later point.

Lemma 3.4. After starting from the same point, the path under CN will always be

above the path under NCN.

59

Proof: Suppose that at a later point, the two paths reach a same composition .

Starting from this point, the path under CN will again be above the path under NCN.

So the two paths can never cross.

Therefore the path of convergence under CN in the plane is always above the

one under NCN. See an example in figure 3 of the simulation.

Lemma 3.5. For two population compositions and

, if and , then the increase rate of at point

under CN is less than the increase rate of at point under NCN.

Proof: The rate of change of has exact the same expression under both CN and

NCN at any given population composition. From the form of the dynamics and the

fact that the path under CN is always above the path under NCN in the plane,

it is sufficient to prove that for a fixed , (which is now a function of only

because of the constraint ) is a decreasing function of .

60

Since , , and

(by Assumption 1 and Assumption 2), we have:

Intuitively, when increases (so decreases), the mean payoff increases.

Therefore, there is less evolutionary pressure for individuals to take a high level of

care. According to the lemma, at any level of , the increase rate of is larger

under NCN than under CN. We have:

Proposition 3.6. Globally, the proportion of population taking the efficient level of

care increases faster under NCN than under CN.

Locally, under CN there is a stronger evolutionary pressure on the individuals who

take low care; while under NCN there is a stronger evolutionary pressure on the

individuals who take medium care. CN is more effective in reducing the number of

individuals taking low care while NCN is more effective in reducing the number of

individuals taking medium care. Such a difference is significant when the proportion

of individuals taking medium or low care is high. When society is close to the social

optimum, this difference almost disappears. Globally, since at any given level of

there are more individuals taking low care and the mean payoff of the population is

lower under NCN, increases at a faster rate with NCN.

61

Which rule is better? It seems that a model of replicator dynamics suggests that NCN

is better, since under NCN more individuals take the efficient care at any time. But

that is not the whole story. Society cares about minimizing the total present value of

social cost. For our case, the social cost at a given time can be measured by

and the present value of social cost is:

(3.9)

which depends not only on how many individuals take efficient care, but also on the

care taken by other individuals. Under NCN more individuals take efficient care, but

at any level of more individuals take low care, which might be more costly to the

society.

Another measure used in WITTMAN, FRIEDMAN, CREVIER AND BRASKIN

[1997] is the mean care level of the population. In our case, the mean care level is:

. Though this measure ignores the social cost associated with each

population composition, it is better than looking at only one component of the

population composition.

We can compare the speed of the change of social cost and of the change of

the mean care level of the population under CN and NCN.

62

Proposition 3.7. At a given population composition, the social cost falls more

rapidly and the mean care level of the population increases more rapidly under CN

than under NCN.

Proof: Under both CN and NCN, we have:

(3.10)

Therefore,

(3.11)

At a given population composition , is the same under both CN and

NCN. By equations (3.2) and (3.3), the difference in payoff between taking medium

care under CN and under NCN is: , and the difference in payoff

between taking low care under CN and under NCN is: .

Using the replicator dynamics, we have that the difference of the time derivative

of the mean social cost between under CN and under NCN is:

(3.12)

63

By Assumption 1 and Assumption 2, , and:

.

Therefore, increases at a greater rate under NC than under NCN, i.e., the social

cost decreases at a greater rate under NC than under NCN.

Similarly, for the mean care level , we have:

. (3.13)

The difference of the time derivative of the mean care level between under CN

and under NCN is:

.

Therefore, the mean care level increases at a greater rate under CN than under

NCN.

Since CN is more effective in reducing the number of individuals taking low care

and NCN is more effective in reducing the number of individuals taking medium

care, and taking low care is more costly for the society, at a given population

composition, CN minimizes the social cost more effectively (at least for a short

period of time). Also for a period of time the mean care level of the society increases

more quickly under CN.

We cannot generalize the result in proposition (3.7) (which is a local property) to a

global property. After a very long period of time, the result might not be true any

more. However in the real world, the inefficiency is caused by periodic shocks, so

64

we may not observe an undisturbed convergence for very long period. The result in

proposition 3.7 can be applied in most cases.

We use simulations to further illustrate the difference between CN and NCN. The

parameter values are chosen to make it socially optimal for all individuals to take

high care under both CN and NCN. The values of the parameters are:

and . We take 9 and 1. We use the discrete counterpart of the

replicator dynamics, that is:

(3.14)

Table 2 displays the time path of the composition of population, the social cost and

the mean care level under CN and NCN with an assumed initial condition

0.7, . In this dynamics, both liability rules lead to the

social optimum. The convergence paths display the features as we discussed above.

The proportion taking high care increases faster under NCN than under CN (figure

1). The proportion taking low care decreases faster under CN. The proportion taking

medium care decreases more slowly under CN (figure 2). See figure 3 for the path of

( .

65

Table 2.2

Simulation Results

Comparative Negligence Negligence with Contributory Negligenceh m l hsocial cost mean care h m l social cost mean care

0 0.700 0.150 0.150-3.7000 -4.4403 8.500 0.700 0.150 0.150 -4.4403 8.5001 0.752 0.146 0.103-3.7518 -4.3108 8.932 0.752 0.138 0.110 -4.3290 8.8702 0.794 0.138 0.068-3.7938 -4.2159 9.248 0.795 0.126 0.079 -4.2411 9.1633 0.827 0.128 0.045-3.8273 -4.1492 9.471 0.831 0.114 0.055 -4.1743 9.3874 0.854 0.117 0.029-3.8540 -4.1036 9.624 0.859 0.103 0.038 -4.1250 9.5535 0.875 0.106 0.018-3.8753 -4.0729 9.728 0.882 0.092 0.026 -4.0896 9.6746 0.893 0.096 0.012-3.8926 -4.0524 9.799 0.900 0.082 0.018 -4.0644 9.7607 0.907 0.086 0.007-3.9069 -4.0385 9.847 0.915 0.073 0.012 -4.0468 9.8228 0.919 0.076 0.005-3.9188 -4.0291 9.881 0.927 0.065 0.008 -4.0344 9.8659 0.929 0.068 0.003-3.9289 -4.0225 9.905 0.937 0.058 0.005 -4.0257 9.896

10 0.938 0.060 0.002-3.9376 -4.0179 9.922 0.945 0.051 0.003 -4.0196 9.91911 0.945 0.054 0.001-3.9451 -4.0145 9.935 0.952 0.045 0.002 -4.0153 9.93512 0.952 0.048 0.001-3.9517 -4.0120 9.946 0.958 0.040 0.001 -4.0121 9.94713 0.957 0.042 0.000-3.9574 -4.0101 9.954 0.964 0.036 0.001 -4.0098 9.95614 0.962 0.037 0.000-3.9625 -4.0085 9.960 0.968 0.031 0.001 -4.0081 9.96315 0.967 0.033 0.000-3.9669 -4.0073 9.965 0.972 0.028 0.000 -4.0067 9.969

Present value of social cost: -33.7903 Present Value of social cost -33.8837

h, m, l: proportion of population taking high (10), medium (9), low (1) level of careDiscount factor =0.9

Figure 2.1

Proportion of Individuals Taking High Care

66

Figure 2.2

Proportion of Individuals Taking Medium Care

Figure 2.3

The Composition of the Population in the Simulation

67

We also give the value of mean social cost at each period. Until period 12 the social

cost is smaller under CN than under NCN for each period. After that the mean social

cost becomes greater under CN for each period. The mean care level is also greater

under CN until period 11. After that, the trend also reversed. In the simulation, as

time goes on, the proportion of individuals taking low care is very small, so NCN

outperforms CN (even though not significantly) because it reduce the proportion

taking medium care more effectively.

We calculate the present value of the expected social cost with the discount factor

:

(3.15)

In 16 periods, the present value of the expected total social cost is -33.7903 under

CN and –33.8837 under NCN. The social cost is about 3% lower under CN. With the

parameters we have randomly chosen, the cost saving is not very significant.

In reality, we may not see such monotonic convergence. From time to time there will

be random shocks to the composition of the population, either because of new

entrants or some other factors that drive society away from the equilibrium. The

shocks may be very small or relatively large. The evolutionary pressure through

imitation and learning leads society to the Nash equilibrium. The constant

68

appearance of shocks makes the rate of convergence and the path of convergence

very important, as it leads to different social cost.

Our results are consistent with the experimental results in WITTMAN, FRIEDMAN,

CREVIER AND BRASKIN [1997]. The experiments in that paper showed the

convergence of the mean care level to the Nash equilibrium under both CN and

NCN, and they showed that CN promotes a faster convergence to the Nash

equilibrium than NCN. In our analysis, starting from a same population composition,

the mean care level increases more rapidly under CN than under NCN for a period of

time. The result may not be maintained in the long run without further shocks, but it

is usually true in reality when inefficiency is often caused by periodical shocks.

In WITTMAN, FRIEDMAN, CREVIER AND BRASKIN [1997], individuals are

assumed to choose their best response, given the behavior of other individuals in the

population. They considered an adjustment dynamics using model

, where is the state at time , is the best response function,

and is a forecast of state at time using information at time ( is a proxy

for all other influences that vary slowly). They consider several possible models by

choosing different and . One of their findings is that when estimating the

population care level ( as the population mean care level), the model in which

players give their best response to the mean care level of the last period ( )

provides a good fit of the data (with ). The model in which players give

their best response to the distribution of previous care levels also provides a good fit

69

for the data. Our evolution model considers the distribution of care levels at time as

a function of last period's distribution of care levels, which is more reasonable for a

large population in a social environment.

Instead of using the simple replicator dynamics, we can use a more general

evolutionary dynamics model such as a growth monotonic system (see VEGA-

REDONDO [1996]):

, (3.16)

with the condition: if , then .

Intuitively, in a growth monotonic system, strategies that “do better” grow faster.

Under a growth monotonic system, most of our results persist. After starting from the

same point, the path in the ( plane under CN is always above the path under

NCN, using the same reasoning as in Lemma 3.4. (Therefore, at any given there

will be fewer individuals taking low level of care under CN than under NCN). If we

further assume that 18 at any given , we can get the

result that grows faster under NCN as in proposition 3.6. The results about the

comparison of mean social cost and mean care level require more restriction on the

form of the evolutionary system.

18 The condition is quite natural: at a given , if increases a little bit (so decreases a little bit), the average payoff of the population will increase. The benefit of switching to taking high level of care will decrease and the increasing rate of will decrease.

70

The method we used can also be applied to more than three care levels, and it is still

true that CN is more effective in reducing the proportion of individuals taking the

most inefficient care levels and NCN is more effective in reducing proportion of

individuals taking the medium care levels. This is beyond the scope of this paper.

4 Conclusion

In this paper we use a simple replicator dynamics model to study the evolutionary

dynamics of comparative negligence and negligence with contributory negligence.

We compare analytically the convergence properties under these two rules, finding

that the proportion of the population taking high care increases more slowly under

CN than under NCN. However CN reduces the proportion of the population using

low care more rapidly than does NCN. At a given population composition, the mean

social cost falls more rapidly and the mean care level increase more rapidly under

CN than under NCN. This may explain the modern movements to substitute

comparative for contributory negligence.

Intuitively, the advantage of CN is that this rule is more effective in inducing the

very careless individuals to abandon their current strategy and to take a more

efficient strategy. Very careless individuals impose more cost on society than those

slightly careless individuals do. Under comparative negligence the cost is shared

according to relative negligence. Very careless individuals always have to incur

more cost of an accident under comparative negligence rule than under negligence

71

with contributory negligence rule. This is why comparative negligence reduces social

cost more effectively and is better than negligence under contributory negligence.

It is in society's interest that liability rules lead to optimal care and that the optimum

be approached rapidly when out of equilibrium. Therefore, when comparing the

effects of different liability rules, in addition to considering the efficiency of

equilibrium, it is also necessary to consider how rapidly the society converges to that

equilibrium.

References

ARLEN, J. [1992], “Liability for Physical Injury When Injurers As Well As Victims

Suffer Losses,” Journal of Law, Economics and Organization, 8, 411-426.

BROWN, J. P. [1973], “Toward an Economic Theory of Liability,” Journal of Legal

Studies, 2.2, 323-350.

COOTER, R., AND T. ULEN [1997], Law and Economics, Reading, Mass.:

Addison-Wesley.

LANDES, W. M., AND R. A. POSNER [1987], The Economic Structure of Tort

Law, Harvard University Press: Cambridge, MA.

72

NACHBAR, J. H. [1990], “Evolutionary Selection in Dynamic Games,”

International Journal of Game Theory, 19, 59-90.

MAYNARD SMITH, J. [1982], Evolution and the Theory of Games, Cambridge

University Press: Cambridge.

POSNER, R. A. [1992], Economic Analysis of Law, 4th ed., Little, Brown.

REA, S. [1987], “The Economics of Comparative Negligence,” International Review

of Law and Economics, 7, 149-162.

SHAVELL, S. [1987], Economic Analysis of Accident Law, Harvard University

Press: Cambridge, MA.

TAYLOR, P., AND L. JONKER [1978], “Evolutionarily Stable Strategies and

Game Dynamics,” Mathematical Biosciences, 40(2), 145-56.

VEGA-REDONDO F. [1996], Evolution, Games, and Economic Behavior, Oxford

University Press.

WHITE, M. J. [1988], “The Economics of Accident Law,” Michigan Law Review,

86, 1217-1231.

- - [1989], “An Empirical Test of the Comparative and Contributory Negligence

Rules in Accident Law,” Rand Journal of Economics, 20, 308-330.

73

WITTMAN, D. AND D. FRIEDMAN AND S. CREVIER AND A.BRASKIN

[1997], “Learning Liability Rules,” Journal of Legal Studies, 26, 145-164.

74

Chapter 3

Rational Legal Decision-Making, Value Judgment and the Efficient

Precaution in Tort Law

Abstract

In this paper, we provide a model for rational legal decision-making by considering the

consistency of social decision-making on resource allocations. We emphasize that legal

decision-making is not based on individual utilities. The purpose of law is described by a

subjective social welfare function, with a social value judgment and an attitude towards

distributional inequality as parameters of the social preference. Thus a legal decision-making

always involves a social value judgment and an attitude toward distributional inequality.

Social efficiency can be different from the maximization of social wealth. In tort law, the

determination of the efficient precaution level and the award of compensation depend on the

social value judgment, and the compensation scheme implicitly transfers wealth from the

less socially valued party to the more socially valued party. We provide another explanation

for the award of punitive damage and for the arbitrariness of the dollar value of the punitive

damage award.

75

1. Introduction

In this paper, we provide a model for rational legal decision-making. In the

model, the efficiency is described by a subjective social welfare function, with a

social value judgment and a social attitude toward distributional inequality as

parameters of the social preference. As an application of the model, we analyze the

determination of the due precaution levels and of the damage compensations in tort

law, and provide an explanation for the award of punitive damage and for the

arbitrariness in the dollar value of punitive damage award.

The purpose of law and the role of moral value in legal decision-making are

always in hot debate. Most of the debates are based on philosophical foundations19.

Many legal commentators and morality theorists emphasize the role of value

judgment20. Moral theorists suppose that there is universal justice and fairness that is

based on exogenously preset social norms21. For example, utilitarianism is based on a

social welfare function, which is the (weighted) average of individual utilities. As

pointed out by Posner (1997), this approach has its defects: the weights in the social

welfare function are arbitrary, and the aggregation of individual utility will be

19 See a debate in Harvard Law Review (May 1998) by Posner et al and the work by Posner (1979) and Deworkin (1980).20 An interesting example is Calabresi's analysis of tort law. In Calabresi (1972), it is argued that liability should be placed in such a way to minimize accident avoidance costs and thus maximize wealth. However, in Calabresi (1985), he argues: “who is the cheapest avoider of a cost, depends on the valuations put on acts, activities, and beliefs by the whole of our law and not on some objective or scientific notion”. “What is efficient, or passes a cost-benefit test, is not a ‘scientific’ notion separated from beliefs and attitudes, and always must respond to the question of who we wish to make richer or poorer”.21 In Rawls' justice theory, a society cares about the welfare of the worst individual.

76

vulnerable to the abuse in the measurement and aggregation of these utilities. The

concepts of justice, fairness and the social value judgment have not yet played any

role in the standard law and economic analysis. The economic analysis of law ruled

out the role of value judgment by assuming that the purpose of law is to maximize

total social wealth (or to minimize total social cost). According to Posner (1984),

wealth maximization is the only social value that courts can do much to promote.

Courts should take the distribution of wealth for granted, as a responsibility that the

political system has allocated elsewhere, and should seek to maximize wealth in their

judgments. Dworkin (1980) criticizes that wealth is neither a value nor an instrument

of social value22.

In actual life, legal decision-making is not based on individual utilities. In

eminent domain cases, which is defined as ‘the power to take private property for

public use by a state, municipality, or private person or corporation authorized to

exercise functions of public character, following the payment of just compensation to

the owner of that property’, the compensation is not determined by the individual

utilities. The measure of damages is the fair market value of the property harmed or

taken for public use. The market value is commonly defined as the price that could

22 Mercuro and Samuels (1986) also challenged Posner on the empirical assertion: “The issue remains that of whether courts affect the distribution of rights and of wealth. We affirm that in a slow gradual, incremental manner, the courts, through the precedential and appellate system are making law and that in making law they are determining who will have what rights and that this constitutes determining the distribution of wealth in society. Surely Posner does not think that with respect to rights litigants are indifferent as to what courts decide. The only question then is whether court decisions affect prices through affecting the distributional basis of prices, namely rights. We affirm that they do.”

77

have reasonably resulted from negotiations between an owner who was willing to

sell and a purchaser who desired to buy23.

In this paper, we do not base the social decision-making on individual utility.

Individual decision-making on consumption and risk taking is based on the

individual utility. On the contrary, social decision, as in many cases of legal decision,

deals with activities involving trading and exchange. Social preferences cannot be

measured by individual preferences. Collectively determined exchange values of

commodities (usually their market values) are imposed in legal decision. Marx

recognized quite early that social preferences couldn’t be measured in terms of

individual preferences. One of the foundations of Marxist analysis is the separation

of use-value and exchange-value. Commodities can fulfill human needs. The

concrete and qualitative character of need in relation to each commodity is reflected

in its use-value. But once in the exchange, commodities lose their qualitative

differences and become quantitatively equivalent to other commodities, thus

masking their use-value. In social decision-making, the exchange-value is involved,

as in the eminent domain case. Therefore social welfare is not the aggregation of

individual utilities. Marxist studies contrast sharply with the liberalist view about the

political and social structure.

23 ? For example, to construct a park on a track of land owned by many owners, individual bargaining will make it impossible for construction because of the hold-out problem. Society will decide the value of land collectively, usually the market value (if such value exists) and impose this value on the individuals.

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In legal decision-making, the society values the commodity by its exchange

value (market value), and not by the value derived from individual utilities.

Therefore it is natural to consider social decision-making on the base of the market

value of resources owned by individuals, or on the resource allocation among

individuals. This contrasts to the public choice theory, in which the society is merely

a mechanism for combining private preferences into a social decision-making. Even

though many researches in public choice reach results against the rationality of social

decision-making, the incoherence predicted by public choice theory is not reflected

in the actual behavior of the society. Legislature is not chaotic; outcomes in legal

decision-making are predictable and stable; and law must treat equals equally.

Especially, consistency is a basic requirement for legal decision-making, as required

by justice and fairness.

Therefore, the rationality in social decision-making can be better described

by the consistency in resource allocations among social members. We assume very

intuitive assumptions about social rationality about resource allocation among

individuals: a social preference is transitive and complete; the society prefers more

than less; mixing two resource allocations with any third one in a specific way

preserves the order of preference. Together with other technical conditions, we show

that any social decision-making is described by maximizing a subjective social

welfare function, with a social value judgment and a social attitude towards

distributional inequality as parameters of the social preferences. The result is

obtained by using the advance in Savage’s subjective utility theory in individual

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decision-making with uncertainty and finite states, in which an individual can assign

a subjective probability to the states of the world and a value to each outcome.

However, the states of the world in subjective utility theory are replaced by the set of

individuals in the society, and the outcome is replaced by the resource of each

individual. The subjective social welfare function represents the subjective

perception of the society; and it expresses the social ethics. When making social

decision, the society puts more weight on the interest of the party with more social

value; the attitude toward distributional inequality reflects the social ideology on

distributional inequality.

The value judgment and the attitude toward distributional inequality are

internal to a society. They are endogenous to the social life and can be changed by

social and political activities. Social preferences vary across different societies, so do

the value judgment and the notion of justice and fairness. If lump sum transfers are

possible, a society always makes decisions that maximize total social resources and

then distributes them among all parties. Otherwise, either because of the property

law or other constraints, the bias to the interest of some individuals (or group of

individuals) will lead the social decision-making away from wealth maximization.

Therefore, the purpose of law can be different from the maximization of social

wealth.

As an application, we consider the determination of the due precaution level

and the award of damage compensation in tort law. In the economic analysis of law,

80

the due precaution level is determined by maximizing total social wealth, and the

award of damage compensation is always equal to the actual damage. The only

reason for punitive damage is that there is a probability of avoiding the compensation

by injurers (Polinsky and Shavell (1998)). Other theories for punitive damage

include that punitive damage serves as deterrence and (or) retribution24. Cooter and

Ulen (1988) argued that punitive damage is necessary because some illegal benefits

should not be counted in the social welfare function25. The view of retribution

suggests that the awards of punitive damage express social outrage at some form of

behavior (Chapman and Trebilcock (1989)), which implies that there is a moral

judgment consideration behind the awards of punitive damage.

From our results, in a rational legal decision-making setting, if the injurer in a

tort case is valued less than the victim by the society, the society imposes a due

precaution level that is stricter than the one maximizing total social wealth. To

induce such a precaution, punitive damage must be awarded. Similarly, if the victim

is valued less than the injurer by society, the society imposes a due precaution level

that is less strict than the one maximizing total social wealth. Correspondingly, the

compensation is less than the actual damage. When punitive damage awards become

necessary, the exact amount of the award of punitive damage award is very sensitive

to the estimation bias of social value judgment. This can explain why punitive

damage seems arbitrary, as observed in the empirical evidence (Daniels and Martin

(1990)).24 There is a large literature on the analysis of punitive damage. See Biggard (1995) for some of the citations.25 As pointed out in Klevorick (1985), it is not clear why there is such a domain, as illicit, and what activity is illicit.

81

Section 2 uses a result analogous to the existence of subjective expected

utility theory on a finite state space (Nakamura (1990), Chew and Karni (1994)) to

discuss rational social decision-making for resource allocation among individuals (or

among groups of individuals). The social welfare function and the value judgment in

legal decision, as well as the efficiency in legal decision-making are discussed in this

section. Section 3 discusses the due precaution level and the award of damage

compensation in tort law. We also provide another explanation for the award of

punitive damage. Section 4 concludes the paper.

2. Rationality, subjective social welfare function and value

judgment

Social decision-makings involving the exchange of commodities are not

based on individual utilities of the commodity; they are based on the exchange

values (or market values) of these commodities. For a given kind of activity, each

social decision leads to a resource allocation among the involved parties. The society

compares all possible resource allocations and makes decisions based on such

comparison. We use the similar notation as in subjective expected utility theory as in

Nakamura (1990), substituting the states of the world by parties involved in a social

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decision, and the set of consequences by the set of resources received by each

individual.

Assume that there are parties ( individuals or groups of individuals) in

an activity, . All subsets of , , are the subgroups of

these parties. The resource that party receives is an element in the space ,

which can be one dimensional (as profit or loss) or multidimensional. A resource

allocation is denoted by , in which is the resource of party .

Assume also that there is a nature ordering on (such as the nature ordering of

profit on ). The set of all possible allocations is denoted by .

Social preference is a binary relation on .

For two allocations in , means that allocation is at least as good as

allocation from the social point of view.

Several additional notations and definitions as defined in Nakamura (1990) will be

useful in stating the assumption about social rationality. For any resource allocations

, any individual resource allocation , and any subgroups ,

means that in allocation and allocation , all parties in the subgroup

have the same resource: for all ; means that in allocation , all

parties in subgroups get resource that is equal to . A binary allocation is an

allocation with and , in which each parties in subgroup get resource

, while each of the rest of the parties get resource . Such binary allocation is

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denoted by . An equal allocation is an allocation such that all parties have

the same resource , and every can be identified with an equal allocation.

A partition of is a sequence of non-empty subgroups that are mutually disjoint and

whose union equals ; it divides the parties into exclusive subgroups. For a k-

partition and , if for all , that is, in allocation , if

each parties in subgroup have resource for all , then the allocation is written

as .

A null subgroup is a subgroup such that for all , :

allocations for parties in a null subgroup’s have no effect on the social preference.

On the contrary, a universal subgroup is a subgroup such that for all

, : allocations for parties in a universal subgroup determine the

social preference. The social preference is bounded if, for each allocation , there are

such that : for any allocation, there is a more preferred equal

allocation, and there is a less preferred equal allocation. A standard sequence is a set

for which there exists such that , and either

and for all , or , and

for all .

Given the above notation, we now state the axioms concerning the rationality in

social decision over resource allocations among parties. These axioms apply to all

allocation , all individual resources , and all subgroups :

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A1. The binary relation on is bounded, complete and transitive.

A2. If , then for some .

A3. If is not null, then iff ; if is not universal, then iff

.

A4. If and , then .

A5. Every strictly bounded standard sequence is finite.

Since on is bounded, and by A2, for each allocation , for some 26: for

every allocation, there is an equivalent equal allocation. We denote the equal

allocation corresponding to as . For binary allocation , the

corresponding equal allocation is denoted by .

The following is the key axiom for the representation theorem:

A6. For any partition of , , ,

, , denote allocation as the allocation in which each

party in has resource , as the allocation in which each party in

has resource , and as the equal allocation for the binary allocation

, then:

26 Since on is bounded, for each allocation , there are such that . By definition of an allocation, and . Therefore, by A2, we have for some

.

85

A1 implies that the society has a complete ranking of all resource allocations; any

pair of allocations can be compared and such ranking does not lead to cyclical

results. A2 is a restricted solvability axiom, which implies that every allocation has

an equivalent equal allocation among all parties. A3 and A4 are monotonic axioms.

In A3, if equal allocation is preferred than equal allocation (for example, is

larger than in quantity), then for any equal allocation and any subgroup ,

suppose we change the equal allocations to such that those parties in have

resource , and we change the equal allocation to such that those parties in

have resource , then allocation is preferred to . More specifically, for any

equal allocation, if we increase the resource for some parties, then the society is

better off.

In A4, if equal allocation is preferred than equal allocation (for example, is less

than in quantity), and if we change the equal allocation to the allocation such

that those parties in have resource , which is more than , and we change the

equal allocation to the allocation such that those parties in a subset of have

resource , then the allocation is preferred to the allocation . For any equal

allocation, if we increase the resource for more parties, the society is better off.

A5 is an Archimedean axiom, which guarantees a good form of utility function in the representation.

A6 is the key axioms and there are different forms of this axiom as in Nakamura

(1990) and in Chew and Karni (1994). The form here is taken from Chew and Karni

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(1994). It is analog to the independent axiom of Savage. The analog to the

independence axioms can be seen as follows: Take arbitrary allocations and , any

subgroup , and any third allocation . Consider the allocation constructed from

, and by requiring that the resource of each individual to be the resource ,

such that the equal allocation is equivalent to the allocation in which each parties

in B has the resource that party has as in and all other parties has the resource

that party has as in (i.e., and A2 assure that such equivalent equal

resource allocation exists). Similarly, construct from , and as above. Then if

is preferred to , then is preferred to . As in the independent axiom, the

order of preference is preserved when and are “mixed” with the same allocation

in the sense above. However, because of the finiteness mixture and the specific way

of mixing, this axiom is weaker than the independent axiom in the original Savage

Axioms.

Using the result of Nakamura (1990), Chew and Karni (1994), we have the following

theorem:

Theorem 2.1. Suppose social preference relation satisfies A1-A6. Then there

exists a unique (relative) weight with , and a real valued

function , unique up to a positive linear transformation, such that for any

allocations , , we have:

(2.1)

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Therefore, a rational social preference can be represented by a subjective social

welfare function:

(2.2)

It is important to notice that the social welfare function and the parameters

and are all derived from the revealed social preference, and they are subjective to

the society, representing the internal belief and perception of the society. We call the

social welfare function as the subjective social welfare function.

This theorem is only a restatement of the finite state Savage's theorem of

individual decision-making with uncertainty in Nakamura (1990), Chew and Karni

(1994), where interested readers can find the detailed proof of the theorem. The

subjective social welfare function obtained has the same form as an individual

expected utility27. For individual expected utility, represents the belief over the

probability of the states of the world, while represents the individual attitude

towards risk. We will see that these parameters have different meanings in our

setting, as is illustrated by the following example of pure redistribute policy.

27 There are two different approaches for individual decision-making. One is Ascumbe-Aumann (A-A) approach, which is based on the objective lottery of some prospectus, where the probability is exogenously given. Another approach is Savage's subjective utility approach based on the preference of alternative actions, where the probability is endogenous to the preference.

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We consider a problem of pure social income redistribution among parties without

cost and distortion. The initial income is . Redistribution is implemented through

lump sum taxes and subsidies.

The society maximizes the social welfare function under the resource constraint:

(2.3)

Without loss of generality, we suppose . From the well-known results in

optimization problem, if is strictly convex, the optimal allocation for society is to

give all the resources to the first party and nothing to the rest of the parties. If is

strictly concave, the first order condition becomes

(2.4)

If , from condition (2.4), the society allocates more resource to party than to

party . Society emphasizes party 's interest more than that of party 's. Therefore

we can define as the social value of party . These social values can appear, for

instance, as the social perception that party is more important for the society or

more valuable for the future of the society.

Function represents the social attitude toward distributional inequality. If is

convex, the society is inequality loving and all resources are concentrated to the most

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influential party. If is concave, the society is inequality averse. As in individual

utility theory, the curvature of , as defined in the Arrow-Pratt coefficient ,

reflects the extent of distributional inequality averse.

Corollary 2.2. In the subjective social welfare function, the weight represents the

social value of party , and the function represents the social attitude toward

distributional inequality. Parties with larger social value get more resources. Arrow-

Pratt coefficient describes the society's extent of inequality aversion (or willingness

of wealth transfer).

Therefore, the subjective social welfare function includes two important

aspect of social ethics: the social values assigned to members of the society and the

social attitude toward distributional inequality, which are not any general principle

imposed on the society from outside, but internal and subjective to the society.

Contrast to standard social choice theory, in which the social welfare is a

function of individual utilities, this subjective social welfare function has no relation

with any specific individual utility. Both and are society level variables. Since

the social ethics is internal to the society, different society has different ethics; social

preference can be changed through social activities (the change of social preference

through interest group competition is out the scope of the present paper28), and ethics

in a society can be changed. Social preferences can change without any changes in 28 In Zheng (2001), we relate the social preference to democratic voting equilibrium, and discuss how interest groups can influence social preference.

90

individual preferences, and vice versa. The social values and the attitude toward

distributional inequality reflect the social and political power, social norms and

institutional constraints.

Social preference can be independent from each individual preference;

however, society uses its forces to impose its ethics on its members by changing the

incentive of individuals through legal and other social decision-makings. Inevitably,

social preferences often conflict with individual preferences and have to be imposed

by force, and collective decisions are used sometimes as instruments to keep some

groups’ advantageous social position.

We can explicitly include the product activities in social decision-making.

Each productive activity leads to a certain kind of resource distribution. For

production activities with significant externalities (either positive or negative) and

high transaction costs, some kinds of social decisions have to be made. In some

cases, some parties are benefited from the activity while other parties incur a loss.

For example, in tort cases, society has to choose the due precaution levels. Each

social decision leads to a resource allocation among all parties involved in the

production.

Suppose that the productive activity that the society can choose from is .

If the society chooses activity , then the resource obtained by group is . The

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total social wealth is . From the above analysis, a rational social

decision is determined by maximizing a certain social welfare function:

If lump sum transfers are available and there is no cost for wealth transfer, it

is easy to see that the society always chooses the activity that maximizes total social

wealth. Even with the consideration of the social ethics, economic efficiency is still

achieved.

In reality, lump sum transfers are not always possible and wealth

maximization is no longer the objective of the society. The society cannot transfer

freely the individual wealth, either because of property law or other considerations.

For simplicity, we assume that the society does not care about the distribution of

wealth for legal decisions, as distribution objective is not important for many legal

decisions and such objective should be left to other public policies, such as tax-

transfer policy. Under such simplification, function in the subjective social welfare

function is linear. Social welfare function in determining the optimal activity is

represented by:

.

The society does not choose the resource allocation; rather it only chooses an

optimal action . If all involving parties have the same social value, i.e., is

constant for all , then the social welfare function becomes total social wealth (up to

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a constant). The society chooses the action that maximizes total social wealth. This is

exactly the argument in law and economic analysis: the purpose of law is to

maximize the total social wealth.

However, if parties have different social values, i.e., the i ’s are not constant,

then maximizing the social welfare is different from maximizing total social wealth.

With different social values, the society chooses the actions that are biased to the

interests of its more valued parties; the more diverse of the social values, the less

likely total social wealth is maximized. As we will see in tort cases, economic

efficiency (social wealth maximization) can be achieved only if the injurers and the

victims have the same social value. Otherwise, the due precaution level is either too

strict (when the victim is more influential) or too loose (when the injurer is more

influential).

An activity that creates considerable wealth may still be socially undesirable

if it hurts a more valuable party. An activity that reduces the wealth could be socially

desirable if it benefits the party with more social value and hurts some other parties.

Different societies have their own social preferences, so the value judgment

in legal decision-making can be different, and the notion of justice and fairness can

be different. There is no universal concept of justice or fairness, as the critical legal

studies emphasizes, see Fitapatrick and al. (1987). Changing social environments,

93

such as technology progresses and interest group activities can change social

preference and thus social value judgments.

The lobbying activities in the process of law making are aimed at changing

social preferences, making the law decision beneficial to some parties. Judges and

juries, considered as “with characteristically judicial virtues of rationality, neutrality

and respect for the institution”, are trying to find the social preference in legal cases.

Activities of lawyers are also aimed at influencing social preferences.

Proposition 2.3. All legal decisions involve social value judgment. The

purpose of law is for social efficiency, which is the maximization of a social welfare

function, with the social value judgment and the attitude toward distributional

inequality as the parameters of social preference. When lump sum transfers are

always possible, legal efficiency leads to social wealth maximization. Otherwise,

legal efficiency is different from social wealth maximization if the involved parties

have different social values, and the interests of the parties with more social value

judgment are more emphasized by the society.

Posner's work (for example, Posner (1979)) claims that efficiency in

economic analysis of law is wealth maximizing. He suggests that wealth

maximization seems a more defendable principle than utilitarianism. Dworkin (1980)

criticizes that wealth is neither a value nor an instrument of social value. The above

94

result shows that the wealth maximization can be the purpose of law, but only in

some special cases.

Individuals only care about their own interests, trying to get more resources

from society and to maximize their own utilities. There are conflicts between social

preferences and individual preferences. Society has to use coercive power to impose

social values on individuals. The imposition of compensation in tort law is aimed at

inducing individuals to act in the social interest (to maximize the social welfare

function) and to take the social optimal level of precaution.

3. The determination of efficient precaution level and the

damage compensation in tort law

As an application of the above analysis, we consider the determination of due

precaution levels and the award of compensation in torts. For simplicity, we consider

circumstances involving only two parties: the injurer and the victim, and only the

injurer takes action, which is characterized by the level of precaution. The injurer

chooses precaution level , leading to a resource allocation between injurer and

victim: , with . Legal decisions cannot make

lump sum transfer because of the property law constraint, therefore, to achieve the

desired resource allocation, the society imposes a due precaution level by imposing

some compensation schemes.

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Legal decisions maximize a subjective social welfare function, with its social

values and attitude toward distributional inequality as parameters. We assume that

the society is inequality neutral in legal decision-making (i.e., is linear). This

assumption is reasonable if the society does not want to use tort law for

redistribution of wealth. The social welfare function associated with an action of

precaution level is29:

(3.1)

If both parties have the same social value, , maximizing social

welfare is equivalent to maximizing total social wealth. The social efficiency

becomes the wealth maximization, as claimed by Posner (1979). When ,

maximization of the social welfare function is different from the maximization of

social wealth. In the extreme case, when is approaching 1 or 0, the wealth of one

party is almost not counted by the society, which is the case described by Cooter and

Ulen (1988): some illegal benefits should not be counted in the social welfare

function.

It is easy to check that even when is not linear, if is concave and

, maximizing social welfare still leads to maximizing the total social

wealth.

29 We assume, without loss of generality, that the social value is constant for such behavior. The precaution level has not any effect on the social value. For cases that the social values are functions of the precaution level, the analysis is the same.

96

For most contract and tort cases, members of the society have equal chance to

be the injurer and the victim; the injurer and the victim have the same social value. In

these cases, wealth maximization is the motive of society. Legal decision tries to

impose the actions that maximize social wealth. This can explain why in most cases

of contract and tort law, standard economic analyses of law without considering

value judgment is very successful. Full compensation induces the socially optimal

precaution.

However, for some other cases, by history or interest group activities, the

group of injurer and the group of victims has very different social value. Wealth

maximization will be no longer true.

Suppose the social values of the injurer and the victim are and . For precaution

level , the revenue and cost function of the injurer are and , with

property . More precaution reduces profit of the action. At the

same time, the action causes a fixed damage to a potential injurer with probability

. More precaution will reduce the probability of an accident but at a decreasing

rate: . The action with precaution level results in an expected

allocation , with . Social welfare associated

with precaution level is:

.

(3.2)

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The efficient level of precaution for the society is determined by:

.

Therefore, the social efficient precaution level is determined by:

, (3.3)

The wealth maximizing precaution level is determined by:

(3.4)

Without the imposition of the compensation, the injurer does not consider the

externality that he imposes on the victim. The preferred precaution level of the

injurer is 0. Society must provide incentives for the injurer to increase the precaution

level by imposing an award of damage on the injurer, forcing him to internalize

the externality. The injurer's profit after the compensation is:

.

The injurer chooses a precaution level to maximize his utility. If the

injurer's utility is only a function of his profit from the activity, with a form as

, then injurer maximizes his profit. The first order

condition is:

(3.5)

To induce the socially optimal precaution , by comparing condition (3.3)

and condition (3.5), the compensation should be set to:

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(3.6)

Under a negligence rule, the injurer pays compensation only if his

precaution level is less than the due care level . Under a strict liability rule, injurer

pays compensation once damage occurs. As in standard economic analysis of law,

both a negligence rule and a strict liability rules induce the socially optimal

precaution.

There are three different cases:

1. . The group of injurer has higher social value. According to (3.3)

and (3.6), the due precaution level under a negligence rule is less strict than the

precaution level maximizing total social wealth, . The compensation to be paid

is also lower than full compensation, .

2. . The group of victim has higher social value. The due precaution

level under a negligence rule is more strict than the precaution level maximizing total

social wealth, . The compensation to be paid is greater than the actual damage,

. The injurer has to pay a punitive damage.

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3. . This is the case extensively studied in law and economics. The due

precaution level under a negligence rule is equal to the precaution level maximizing

total social wealth, . The compensation is equal to the actual damage, .

If the injurer has higher social value, the injurer is less likely to be found

negligent and he pays less for compensation when he is found negligent. The injurer

takes less precaution than the precaution level maximization total social wealth and

gets more profit than under a full compensation rule. If the victim has higher social

value, the society imposes a stricter due precaution level such that the victim has

more chance to get compensation given the injurer's action. The compensation is

greater than the actual damage. The injurer gets less profit from his action than under

the wealth maximization precaution level .

In standard law and economic analysis, punitive damages can be awarded

only if there is a possibility that injurer can avoid compensation with some

probability. The compensation should be the harm multiplied by the reciprocal of the

probability that the defendant can escape from compensation (see Polinsky and

Shavell (1998)). However, there are many cases in which the probability of detection

and full compensation is very high and punitive damages are still rewarded (for

instance in the case of certain assaults, pollution, etc). Our analysis shows that the

award of punitive damage is possible even without possibility of avoiding

compensation, and the award of punitive damage is rather a consequence of different

social value of the injurer and the victim. An injurer with very small social value is

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the one morally repugnant, thus the award of punitive damage always associates with

a moral judgment.

Most literature on tort compensation focuses on the award of full damage and

punitive damage, neglecting the case of awarding less than full damage30. In the case

of less than full damage, the action of the injurer is often regarded as accidental or

inevitable.

We now look at the expected post-compensation resource allocation. Under a

strict liability rule, the payoff of the injurer is proportional to the social welfare:

.

The payoff of the victim is proportional to the actual damage:

.

Under a negligence rule, the injurer's payoff is:

if ,

and

if .

The payoff of victim is

if ,

and30 Dobbs (1989) considered the case of under-compensation.

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if .

The payoff of the injurer is greater under a negligence rule than under a strict

liability rule and the payoff of the victim is less under a negligence rule than under a

strict liability rule.

The payoffs of the victim and of the injurer depend on their relative social

values. If victim has less social value than the injurer, , he always ends up

with a negative payoff, even under a strict liability rule. In case where both parties

have the same social value, only strict liability will make the victim whole; a

negligence rule will make the victim poorer than he would be. In both cases, the

society sacrifices the victim's interest to the injurer's benefit.

If the victim has higher social value than the injurer, , then under a

strict liability rule, the victim gets a positive payoff from the action. He may get

positive payoff even under a negligence rule. Therefore, the choice of compensation

schemes implicitly transfers wealth from the party with less social value to the party

with more social value. The choice of due care level also reflects “who we wish to

make richer or poorer” (Calabresi (1985)).

The common view is that it is sufficient for the injurer to make the victim

whole. The victim plays only a passive role. In some states in US, even in case where

the punitive damage is awarded, victims only get full compensation and the rest of

the compensation goes to the state treasury. However, under a negligence rule,

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without the award of punitive damage to the victim, the victim cannot be made

whole in general.

Proposition 3.2. Tort law maximizes a subjective social welfare and induces

a socially optimal precaution level by the imposition of compensation in case of

damage. The optimal precaution level is more or less strict than the precaution level

maximizing total social wealth and the compensation is greater or less than the actual

damage, depending on the relative social value of the injurer and the victim. Tort law

implicitly transfers wealth from the less socially valued party to the more socially

valued party.

One of the characteristics of punitive damage award is its arbitrariness.

Whatever the purpose of the punitive damage award, it is criticized of being

unpredictable, even out of control31. A recent paper of Sunstein et al. (1998) studies

the source of such arbitrariness. They choose 899 jury-eligible citizens and estimate

the results of deliberation consisting of different composition of juries. They find that

people's moral judgment are remarkably widely shared, (and punitive damages are

largely determined by value judgment), but people have a great deal of difficulty in

mapping their moral judgment on to an unbounded scale of dollars. They accredit

such difficulty to the human behavior factor.

31 One study of 47 counties in US over a several-year period, median verdicts ranged from less than $10,000 in some area to as much as $204,000 in San Diego. See Daniels and Martin (1990).

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Such arbitrariness can be explained by the excessive sensibility of the dollar

value of a punitive damage award to social value estimation bias in deciding the

punitive damage. When deliberating a case, the juries are attempting to assess the

social preference in that case, the “sense of community”. In our setting, juries are

trying to find the exact value of . The bias in the estimation is inevitable. From

expression (9), the award of damage is determined by: . For relative

large , a small estimation bias will not affect greatly the value of . However,

when is very small (thus punitive damage should be awarded), the value of

becomes extremely sensitive to the small bias of estimation of .

For a numerical example, we consider a case: the damage caused by the

injurer's activity is . The jury has to find the value of relative social value

judgment. Suppose there is an estimate bias . For any value of , the estimation

can be in the range , with . Therefore the estimation

of the award of damage lies in the range , with .

We take three values of : 0.9, 0.5, and 0.1. When =0.9, the victim is

under-compensated. When =0.1, punitive damage is awarded. We give a range of

the award of damage when there is a small estimation bias, =0.05.

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Table 3.1.

Sensitivity of the Award of the Punitive Damage to the Estimation Bias of

Range of award of damage

0.9 0.053D to 0.176D

0.5 0.818D to 1.220D

0.1 5.667D to 19.00D

The range of award of damage is from 0.053D to 0.176D when ;

when , the range of award of damage is from 0.818D to 1.220D; and when

, the range of award of damage is from 5.667D to 19.00D. When

and punitive damage is necessary, the possible value of the award of damage can

range from about 6 times to 19 times of the actual damage, demonstrating a

significant arbitrariness.

This kind of excessive sensibility of the dollar value of the punitive damage

award to the estimation of social value requires the procedure considerations of

punitive damage, as in the criminal law case. The award of punitive damage should

treat this problem seriously. A strict procedure and some very detailed judicial rules

should be proposed to solve the problem (Ellis (1989)).

4. Summary

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In this paper, we establish a subjective social welfare function in legal decision-

making, with a social value judgment and a social attitude toward distributional

inequality as parameters of the social preference. The starting point is the coherence

and the rationality of legal decision-making; and the social preference is not based on

individual utilities, but based on the resource allocation among social members.

Contrast to the view of a general concept of justice and morality, the social ethics is

subjective to the society.

The existence of a social value judgment also implies that social wealth

maximization is the purpose of law only if all parties involved in a legal decision-

making have the same social value. Otherwise, legal decisions emphasize the interest

of the parties with more social values, and total social wealth is not maximized.

As an application, in tort law, the determination of the due precaution level and the

award of damage compensation depend on the social value judgment. If the injurer

has a higher social value, the due precaution level is less strict than the precaution

level maximizing total social wealth, and the compensation to be paid is lower than

the full compensation. If the victim has a higher social value, the due precaution

level is stricter than the precaution level maximizing total social wealth, and the

compensation to be paid is greater than the actual damage: the injurer has to pay a

punitive damage. By comparing the after-compensation resource allocation, we find

that tort law implicitly transfers wealth from the less socially valued parties to the

more socially valued parties. Our paper provides another explanation of the award of

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punitive damage, and provides an explanation of the arbitrariness of the dollar value

of the punitive damage award.

References

Biggar D. (1995). “A model of punitive damage,” International Review of Law and

Economics 15: 1-24.

Calabresi G. and Melamed A. D. (1972). “Property rules, liability rules, and

inalienability: on view of the cathedral,” Harvard Law Review 88: 163-172.

Calabresi G. (1985). Ideals, Beliefs, Attitudes, and the Law. Syracuse, Syracuse

University Press, 1985.

Chapman B. and Trebilcock M. (1989). “Punitive damage, divergence in search of a

rational,” Alabama Law Review: 741-729.

Chew S. H. and Karni E. (1994), "Choque Expected Utility with a Finite State Space:

Commutativity and Act-Independence," Journal of Economic Theory 62, 469-479.

Cooter R. and Ulen T. (1988). Law and Economics. Scott, Foresman and Company,

Glenview, IL.

107

Daniels S. and Martin J. (1990), “Myth and reality in Punitive damage,” Minnesota

Law Review 75: 1.

Dobbs D. P. (1989). “Ending punishment in punitive damage: deterrence measured

remedies,” Alabama Law Review: 831-917.

Ellis Jr. D. D. (1989). “Punitive damage, due process, and the jury,” Alabama Law

Review, Spring 1989, Vol 40, Number 3, 975-1008.

Dworkin R. M. (1980), “Is wealth a value?” The Journal of Legal Studies. 9: 192-

226.

Fitapatrick, Peter and Hunt, Allen (eds.) 1987. Critical Legal Studies, New York:

Basil Blackwell.

Klevorick A. K. (1985). “On the economic theory of crime,” in J.Poland Pennock

and W. Chapman (eds), Criminal justice: Nomos XXVII, 289-309.

Mercuro, N and Samuels, W. J. (1986), “Wealth Maximization and Judicial

Decision-making: The Issue Further Clarified,” International Review of law and

Economics, Vol 6, No. 1 (June 1986), pp133-138.

108

Nakamura, Y. (1990), "Subjective Expected Utility with Non-additive Probabilities

on Finite State Spaces," Journal of Economic Theory 51, 346-366.

Polinsky and Shavell (1998). Punitive damages: an economic analysis, Harvard Law

Review 111 (4): 869-962.

Posner R. A. (1979). “Utilitarianism, economics and legal theory,” The Journal of

Legal Studies. 8:103-140.

Posner R. A. (1984), “Wealth Maximization and Judicial Decision-Making,”

International review of Law and economics, Vol. 4, No. 2 (December 1984), pp.131-

135.

Posner R. A. (1992). Economic Analysis of Law, 4th ed., Little, Brown.

R. P. Posner (1998). “The problematic of moral and legal theory,” Harvard

Law Review, Vol. III, May 1998, 1637-1717.

Sunstein C., Kahnemann D. and Shkade D. (1998). “Assessing punitive

damages (with notes on cognition and valuation in law),” Yale Law Journal 107:

2071.

109

Zheng M. (2001). Social welfare and competition among interest groups,

mimeo, University of Toronto

110