Auction Theory

Class 5 – single-parameter implementation and risk aversion

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Outline

• What objective function can be implemented in equilibrium?

– Characterization result for single-parameter environments.

• Revenue effect of risk aversion.– Comparison of 1st and 2nd price auctions.

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Implementation• Many possible objective functions:

– Maximizing efficiency; minimizing gaps in the society; maximizing revenue; fairness; etc.

– Many exogenous constraints imply non-standard objectives.

• Problem: private information

• Which objectives can be implemented in equilibrium?– We saw that one can maximize efficiency in equilibrium.

What about other objectives?

• We will show an exact characterization of implementable objectives.

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Reminder: our setting• Let v1,…,vn be the private values (“types”) of the

players (drawn from the interval [a,b])

• Each player can eventually win or lose.– Winning gains the player a value of vi, losing gains her 0.– (More general than single-item auction.)

• An allocation function: Q:[a,b]nq1,…,qn

– qi = the probability that player i wins.

• Given an allocation function Q, let Qi(vi) be the probability that player i wins.

– In average, over all other values. 5

CharacterizationRecall that an auction consists of an allocation function

Q and a payment function p.

Theorem: An auction (Q,p) is truthful if and only if1. (Monotonicity) Qi() is non-decreasing for every i.

2. (Unique payments) Pi(vi)= vi·Qi(vi) – avi Qi(x)dx

Conclusion: only monotone objective functions are implementable.

– Indeed, the efficient allocation is monotone (check!).

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Theorem: An auction (Q,p) is truthful if and only if1. (Monotonicity.) Qi() is non-decreasing for every i.

2. (Unique payments.) Pi(vi)= vi·Qi(vi) – avi Qi(x)dx

Reminder: our settingProof:

We actually already proved: truthfulness (monotonicity) + (unique payments)

Let’s see where we proved monotonicity:

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Proof• Consider some auction protocol A, and a bidder i.• Notations: in the auction A,

– Qi(v) = the probability that bidder i wins when he bids v.

– pi(v) = the expected payment of bidder i when he bids v.

– ui(v) = the expected surplus (utility) of player i when he bids v and his true value is v.

ui(v) = Qi(v) v - pi(v)

• In a truthful equilibrium: i gains higher surplus when bidding his true value v than some value v’.– Qi(v) v - pi(v) ≥ Qi(v’) v - pi(v’)

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=ui(v’)+ ( v – v’) Qi(v’)=ui(v)

We get: truthfulness ui(v) ≥ ui(v’)+ ( v – v’) Qi(v’)

Proof• We get: truthfulness

ui(v) ≥ ui(v’)+ ( v – v’) Qi(v’) or

• Similarly, since a bidder with true value v’ will not prefer bidding v and thus

ui(v’) ≥ ui(v)+ ( v’ – v) Qi(v) or

Let dv = v-v’

Taking dv 0 we get:

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)'(– v’ v

)(v’u- (v)u ii vQi

)(– v’ v

)(v’u- (v)u ii vQi

)'(dv

)(v’u- dv)(v'u)'( ii dvvQvQ ii

)'(dv'

)(v'du i vQi

Given that v>v’

Rest of the proofWe will now prove the other direction:

if a mechanism satisfies (monotonicity)+(unique payments) then it is truthful.

In other words: if the allocation is monotone, there is a payment scheme that defines a mechanism that implements this allocation function in equilibrium.

Let’s see graphically what happens when a bidder with value v’ bids v>v’.

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Proof: monotonicity truthfulnessProof: We saw that truthfulness is equivalent to:

for every v,v’ : ui(v) - ui(v’) ≤ ( v – v’) Qi(v)

We will show that (monotonicity)+(unique payments) implies the above inequality for all v,v’. (Assume w.l.o.g. that v>v’)

We first show:

Now,

11

)'()'(')'( vpvQvvu iii

dvvQdvvQvQvvQvv

ai

v

aiii

''

)()()'(')'('

dxxQdxxQvuvuv

ai

v

aiii

'

)()()'()( dvvQv

vi

'

)(

Due to the unique-payment

assumption

Due to monotonicity

)(' vQvv i

Single vs. multi parameter

A comment: this characterization holds for general single-parameter domains

– Not only for auction settings.

• Single parameter domains: a private value is one number.

– Or alternatively, an ordered set.

• Multi-dimensional setting are less well understood.– Goal for extensive recent research.– We will discuss it soon.

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Theorem: An auction (Q,p) is truthful if and only if1. (Monotonicity.) Qi() is non-decreasing for every i.

2. (Unique payments.) Pi(vi)= vi·Qi(vi) – avi Qi(x)dx

Outline• What objective function can be implemented in

equilibrium?– Characterization result for single-parameter environments.

• Revenue effect of risk aversion.– Comparison of 1st and 2nd price auctions.

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Risk AversionWe assumed so far that the bidders are risk-neutral.

– Utility is separable (quasi linear), vi-pi

Now: bidders are risk averse (שונאי סיכון).– All other assumptions still hold.

We assume each bidder has a (von-Neumann-Morgenstern) utility function u( ).∙

– u is an increasing function (u’>0): u($10)>u(5$)– Risk aversion: u’’ < 0

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Risk Aversion – reminder.

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$5 $10

u($5)

u($10)

½*u($10) + ½*u($5)

7.5

u( ½* $10 + ½*$5 )

A risk averse bidder prefers the expected value over a lottery

with the same expected value.

Auctions with Risk Averse Bidders• The revenue equivalence theorem does not hold

when bidders are risk averse.

• We would like to check: with risk-averse bidders, should a profit maximizing seller use 1st-price or 2nd-price auction?

• Observation: 2nd-price auctions achieve the same revenue for risk-neutral and risk-averse bidders.

– Bids are dominant-strategy, no uncertainty.

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Auctions with Risk Averse Bidders

• Intuition: – risk-averse bidders hate losing. – Increasing the bid slightly increases their potential

payment, but reduces uncertainty. • Gain ε when you win, but risk losing vi-bi

The equilibrium bid is higher than in the risk-neutral case.17

Theorem: Assume that

1. Private values, distributed i.i.d.2. All bidders have the same risk-averse utility u( )∙

Then, E[1st price revenue] ≥ E[2nd price revenue]

1st price + risk aversion: proof • Let β(v) be a symmetric and monotone equilibrium

strategy in a 1st-price auction.

• Notation: let the probability that n-1 bidders have values of at most z be G(z), and G’(z)=g(z).

– That is, G(z)=F(z)n-1

• Bidder i has value vi and needs to decide what bid to make (denoted by β(z) ).

– Will then win with probability G(z).

• Maximization problem: 18

)()(max zxuzGz

1st price + risk aversion: proof Proof:FOC:

Or:

But, since β(v) is best response of bidder 1, he must choose z=x:

We didn’t use risk aversion yet…

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)()(max 1 zvuzGz

0)(')(')()()( 11 zzvuzGzvuzg

)(

)()(')()('

1

1

zGzg

zvuzvuz

)(

)()(')()('

1

1

11

11

vGvg

vvuvvuz

1st price + risk aversion: proof Fact: if u is concave, for all x we have (when u(0)=0) :

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xxuxu

'

u(x)

x

xu’(x)

xuxxuor '

1st price + risk aversion: proof For risk-averse bidders:

With risk-neutral bidders (u(x)=x, u’(x)=1 for all x).

Therefore, for every v1

Since β(0) = b(0) =0, we have that for all v121

)(

)()(')()('

1

1

11

11

vGvg

vvuvvuz

)()()(

1

111 vG

vgvv

)()()()('

1

111 vG

vgvvzb

)(')(' 11 vbv

)()( 11 vbv

xxuxu

'

1st price + risk aversion: proof Summary of proof:

in 1st-price auctions, risk-averse bidders bid higher than risk-neutral bidders.

Revenue with risk-averse bidders is greater.

Another conclusion: with risk averse bidders,

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Revenue in 1st-price auctions

Revenue in 2nd-price auctions

>

Summary• We saw today:

– Monotone objectives can be implemented (and only them)– Risk aversion makes sellers prefer 1st-price auctions to 2nd-

price auctions.

• So far we discusses single-item auction in private value settings.

– Next: common-value auctions, interdependent values, affiliated values.

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