Causes of Allais common consequence paradoxes: an...

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Journal of Mathematical Psychology ] (]]]]) ]]]]]]

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doi:10.1016/j.jm

Causes of Allais common consequence paradoxes: an experimentaldissection$

Michael H. Birnbaum*

Department of Psychology, CSUF H-830M, P.O. Box 6846, Fullerton, CA 92834-6846, USA

Received 20 September 2002; revised 12 January 2004

OOF

Abstract

The common consequence paradox of Allais can be decomposed into three simpler principles: transitivity, coalescing, and

restricted branch independence. Different theories attribute such paradoxes to violations of restricted branch independence only, to

coalescing only, or to both. This study separates tests of these two properties in order to compare these theories. Although rank-

dependent utility (RDU) theories, including cumulative prospect theory (CPT), violate branch independence, the empirical pattern

of violations is opposite that required by RDU theories to account for Allais paradoxes. Data also show systematic violations of

coalescing, which refute RDU theories. The findings contradict both original and CPTs with or without their editing principles of

combination and cancellation. Modal choices were well predicted by Birnbaums RAM and TAX models with parameters estimated

from previous data. The effects of event framing on these tests were also assessed and found to be negligible.

r 2004 Published by Elsevier Inc.

Keywords: Allais paradox; Branch independence; Coalescing; Common consequence paradox; Configural weighting; Cumulative prospect theory;

Event-splitting; Expected utility; Framing; Nonexpected utility; Rank and sign dependent utility; Rank dependent expected utility

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TED1. Introduction

The paradoxes of Allais (1953, 1979) revealed thatpeople systematically violate implications of ExpectedUtility (EU) theory. Different explanations have beenproposed for these paradoxes, including SubjectivelyWeighted Utility (SWU) theory (Edwards, 1962; Kar-markar, 1979), Original Prospect (OP) theory (Kahne-man & Tversky, 1979), Rank-Dependent ExpectedUtility (RDU) theory (Diecidue & Wakker, 2001;Quiggin, 1985, 1993), Rank- and Sign-DependentUtility (RSDU) theory (Luce & Fishburn, 1991, 1995;Luce, 2000), Cumulative Prospect (CPT) theory (Cha-teauneuf & Wakker, 1999; Starmer & Sugden, 1989;Tversky & Kahneman, 1992; Tversky & Wakker, 1995;Wakker & Tversky, 1993; Wu & Gonzalez, 1996, 1998),and Configural weight models, including the Rank-Affected Multiplicative Weights (RAM) and Transfer ofAttention Exchange (TAX) models (Birnbaum, 1997,

s received from National Science Foundation Grants

ES 99-86436, and BCS-0129453.

ing author. Fax: +1-714-278-7134.

ess: [email protected].

e front matter r 2004 Published by Elsevier Inc.

p.2004.01.001

1999a, b). The purpose of this paper is to test amongthese rival theories, which give different explanations forthe constant consequence paradoxes of Allais.

1.1. Constant consequence paradox and expected utility

The constant consequence paradox of Allais (1953,1979) can be illustrated with the following choices:

A:

$1M for sure

B:

0.10 to win $2M

0.89 to win $1M

0.01 to win $0

C:

0.11 to win $1M

D:

0.10 to win $2M

0.89 to win $0
0.90 to win $0
77

79

Expected Utility theory assumes that Gamble A ispreferred to B if and only if the EU of A exceeds that ofB: This assumption is written, AgB3EUA4EUB;where the EU of a gamble, G x1; p1; x2; p2;yxi; pi;y; xn; pn can be expressed asfollows:

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1The Ellsberg (1961) paradox can also be analyzed as a failure of at

least one of these same three premises; in particular, this paradox may

also result from violation of event coalescing. See Luce (submitted).2These models do, however, imply a still weaker form of

independence known as comonotonic branch independence, which is

the assumption that Expression 2 holds when corresponding con-

sequences (x and x0; y and y0; z and z0) retain the same rank orders(cumulative probabilities) in all comparisons

M.H. Birnbaum / Journal of Mathematical Psychology ] (]]]]) ]]]]]]2

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EUG Xni1

piuxi 1

According to EU, A is preferred to B iffu$1M40:10u$2M 0:89u$1M 0:01u$0: Sub-tracting 0:89u$1M from each side, it follows that0:11u$1M40:10u$2M 0:01u$0: Adding 0:89u0to both sides, we have 0:11u$1M 0:89u$040:10u$2M 0:90u$0; which holds ifand only if CgD: Thus, from EU theory, one candeduce that AgB3CgD: However, many peoplechoose A over B and prefer D over C: This pattern ofempirical choices violates the implication of EU theory,so such results were termed paradoxical.

1.2. Dissection of the Allais paradox

It is useful to decompose this type of paradox intothree simpler premises that can be used to deduce Allaisindependence (Birnbaum, 1999a), the property that isviolated in the paradox of Allais (1953, 1979). If peoplesatisfy transitivity, restricted branch independence, andcoalescing, then they will not violate Allais independence.

Transitivity, assumed in all of the models reviewedhere, is the premise that AgB and B gC ) AgC:

Coalescing is the assumption that if a gamble has two(probabilityconsequence) branches yielding identicalconsequence, those branches can be combined by addingtheir probabilities, without affecting the utility. Forexample, if G $100; 0:2; $100; 0:2; $0:6; thenGBG0 $100; 0:4; $0; 0:6; where B denotes indiffer-ence. Violations of coalescing combined with transitivityare termed event-splitting effects (Humphrey, 1995;Starmer & Sugden, 1993; Birnbaum, 1999a, b). Forexample, if GgA and G0!A; we say there is an event-splitting effect. Assuming transitivity, event-splittingeffects are violations of coalescing.

Restricted Branch independence is weaker than Sava-ges (1954) sure thing axiom. It is restricted togambles that have the same number of distinct branchesand the same probability distributions over thosebranches (same events produce those branches). Withthese restrictions, if two gambles have a commonprobabilityconsequence (or eventconsequence)branch, one can change the value of the commonconsequence on that branch without affecting thepreference induced by the other components.

For the case of three-branch gambles with nonzeroprobabilities p q r 1; restricted branch indepen-dence can be written as follows:

S x; p; y; q; z; rgR x0; p; y0; q; z; r3

S0 x; p; y; q; z0; rgR0 x0; p; y0; q; z0; r2

Transitivity, coalescing, and restricted branch inde-

pendence imply Allais independence, as illustratedbelow:

A:

$1M for sure

g

B:

0.10 to win $2M

0.89 to win $1M

0.01 to win $0

3 (coalescing & transitivity)

A0:
0.10 to win $1M g B: 0.10 to win $2M
0.89 to win $1M

0.89 to win $1M

0.01 to win $1M
0.01 to win $0
3 (restricted branch independence)

A00:
0.10 to win $1M g B0: 0.10 to win $2M
0.89 to win $0

0.89 to win $0

0.01 to win $1M
0.01 to win $0
3 (coalescing & transitivity)

C:
0.11 to win $1M g D: 0.10 to win $2M
0.89 to win $0
0.90 to win $0
PROO

F

The first step converts A to its split form, A0; A0 shouldbe indifferent to A by coalescing, and by transitivity, A0

should be preferred to B: From the third step, theconsequence on the common branch (0.89 to win $1M)has been changed to $0 on both sides, so by restrictedbranch independence, A00 should be preferred to B0: Bycoalescing branches with the same consequences on bothsides, we see that C should be preferred to D:

This derivation shows that if people obeyed thesethree principles, they would not show this paradox,except by chance or error. Because people showsystematic paradoxes, at least one of these assumptionsmust be false. By Allais independence, I mean to includeall such derivations with arbitrary values for probabil-ities and consequences that can be deduced from thepremises of transitivity, coalescing, and restrictedbranch independence. Similarly, the term Allais paradoxis used to designate a systematic pattern of violations ofAllais Independence.1

Different theories attribute Allais paradoxes todifferent causes (Birnbaum, 1999a). SWU (includingthe equation of OP) attributes the Allais paradox toviolations of coalescing. In contrast, the class of RDU,RSDU, and CPT explain the paradox by violations ofrestricted branch independence.2 It is important to keepin mind that restricted branch independence is not anaxiom of either the class of RDU/RSDU/CPT modelsor the class of TAX and RAM models; indeed, all ofthese models can violate this property. Similarly,coalescing was not initially stated as an axiom of the

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RDU/RSDU/CPT theories, but it can be deduced fromthem.

The configural weight, RAM and TAX models implythat both coalescing and restricted branch independenceare systematically violated (except in special cases wherethese models reduce to EU); like the rank-dependentmodels, these models satisfy restricted comonotonicbranch independence.

This paper will compare the theories in Table 1 byseparating tests of branch independence from those ofcoalescing in Allais common consequence paradoxes.

1.3. SWU and OP models

One way to describe Allais paradoxes is to replaceobjective probabilities with subjective weights (Edwards,1962) as follows:

SWUG Xni1

wpiuxi; 3

where SWUG is the subjectively weighted utility(SWU) of gamble G: In this model, the weight of agiven objective probability is a function of its prob-ability. In this model, there is no contradiction inchoosing A over B and D over C: In particular, if wp isan inverse-S function of p; paradoxes of Allais can bedescribed by Eq. (3).

However, Eq. (3) implies that people will violatetransparent dominance in ways that few humans woulddo (Fishburn, 1978). For example, with parameterschosen to fit the Allais paradoxes, Eq. (3) implies thatpeople should prefer E $103; 0:98; $102; 0:01;$101; 0:01 over F $120; 0:5; $110; 0:5; despite thefact that the lowest consequence of F is better than thebest consequence of E (see Birnbaum, 1999a).

In their Original Prospect (OP) model, Kahnemanand Tversky (1979) restricted Eq. (3) to gambles with no

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Table 1

Comparison of decision theories

Branch Independence

Coalescing Satisfied Violated

Satisfied EU (OP/CPT) RDU/RSDU/CPT

Violated SWU/OP RAM/TAX

Notes: Expected Utility (EU) theory satisfies both properties. OP Original Prospect theory and CPT Cumulative Prospect theory;these theories make different predictions with and without their editing

rules. The editing rule of combination produces satisfaction of

coalescing and the editing rule of cancellation implies branch

independence. CPT has the same representation as Rank Dependent

Expected Utility (RDU). With or without the editing rule of

combination, CPT satisfies coalescing. The Rank Affected Multi-

plicative (RAM) and Transfer of Attention Exchange (TAX) models

are configural weight models that violate both branch independence

and coalescing.

PROO

F

more than two nonzero consequences (which putsGamble E and the four gambles of Expression 2 outsidethe domain of OP). They also added editing rules to OPin order to avoid certain implausible implications ofEq. (3). For example, people are assumed to detect andconform to transparent dominance. Three other editingprinciples of OP are relevant to this paper: (1)Combination assumes that people combine brancheswith identical consequences by adding their probabilities(which implies coalescing). (2) Cancellation postulatesthat people will cancel elements that are identical in twogambles of a choice, implying restricted branch inde-pendence. (3) Simplification is an editing rule wherepeople round off and ignore small differences, whichfacilitates cancellation or combination of nearly equalbranches. Starmer and Sugden (1993) refer to strippedprospect theory as Eq. (3) without the editing rules ofOP, which can be extrapolated to include three branchgambles.

1.4. Rank-dependent expected utility models

Quiggin (1985, 1993) proposed RDU theory, whichaccounts for the Allais paradoxes without violatingstochastic dominance. Luce and Narens (1985) devel-oped a dual bilinear representation for two branchgambles that generalizes the original form of Quiggin,which required the weight of 1/2 to be 1/2. RSDU waslater proposed, which generalized the rank-dependentapproach to allow different weightings for positive andnegative consequences (Luce & Fishburn, 1991, 1995;Luce, 2000). Tversky and Kahneman (1992); Tverskyand Wakker (1995); Wakker and Tversky (1993)proposed CPT, which combined rank- and sign-depen-dent weighting with the editing principles of OP (see alsoStarmer & Sugden, 1989). All of these rank-dependenttheories have the same representation for gamblescomposed of strictly positive consequences:

RDUG Xni1

WPi WQiuxi; 4

where the consequences are ranked, such that

x14x24?4xi4?4xn40; Pi Pi

j1 pj; the (decu-

mulative) probability that a consequence is greater than

or equal to xi; and Qi Pi1

j1 pj is the probability that a

consequence is strictly greater than xi:Eq. (4) satisfies stochastic dominance, avoiding the

need for the editing principle of dominance detection(Tversky & Kahneman, 1992). It also automaticallysatisfies coalescing, eliminating the need for the editingrule of combination (Birnbaum, 1999a; Birnbaum &Navarrete, 1998; Luce, 1998). CPT generalizes OP togambles with more than two nonzero consequences. TheSP/A theory of Lopes and Oden (1999) also satisfiescoalescing and stochastic dominance.

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The rank-dependent theories attribute the Allaisparadox to violations of restricted branch independence.Eq. (4) links the pattern of violation of branchindependence to the Allais paradox, because bothphenomena are (in theory) produced by the sameweighting function (Birnbaum & McIntosh, 1996;Birnbaum & Chavez, 1997).

Wu and Gonzalez (1998) presented an illuminatinganalysis of three distinct types of common consequenceparadoxes, among which the original versions of Allaisrepresent only one type. These correspond to changingthe consequence on the common branch from the lowestto middle, from middle to highest, and from lowest tohighest consequence in the choice. They showed that ifthe weighting function has an inverse-S shape, theobserved paradoxical choices in these three types ofcommon consequence paradoxes can be fit by Eq. (4).Birnbaum (2001b) replicated all three types of commonconsequence paradoxes with chances at real but modestmonetary prizes.

The studies of Wu and Gonzalez (1998) and ofBirnbaum (2001b) investigated violations of Allaisindependence, which confounds branch independenceand coalescing. All three types of constant consequenceeffects can be predicted equally well by the CPT model(which attributes the paradoxes to violations of branchindependence) and by Birnbaums configural weightmodels (which attribute constant consequence para-doxes mainly to violations of coalescing). The presentpaper will dissect these two properties in order todistinguish the models in Table 1; therefore, the designallows a comparison of these four classes of rivaltheories.

In this paper, CPT will be tested both with andwithout its editing principle of cancellation, whichimplies branch independence. Allowing CPT both itsequation and its contradictory editing principles is a verylenient standard, since it allows CPT to handle two ofthree possible outcomes of a test of branch indepen-dence, including mixtures of those two. The standard isas follows: Either the Allais paradox and violations ofbranch independence will be linked by the sameweighting function in Eq. (4), or branch independencewill hold in transparent tests, or the data will beintermediate between these two patterns.

Note that OP and CPT with or without these editingrules lay claim to three of the four cells in Table 1. Thereis only one possible outcome of the experiment thatwould refute both prospect theories with or withouttheir editing rules.

This experiment is designed to test predictions ofprospect theories against the configural weight modelsof Birnbaum (1997, 1999a). The configural models werefit to data of Birnbaum and McIntosh (1996) forviolations of restricted branch independence in gambleswith three equally likely branches, and to Tversky and

PROO

F

Kahnemans (1992) data for certainty equivalents ofbinary gambles with nonnegative consequences. Calcu-lations from those parameter estimates are termed herethe prior predictions, and should not be confusedwith predictions based on a post hoc fit of a model tothe same data being predicted.

1.5. Configural weight, RAM and TAX models

Birnbaum (1974); Birnbaum and Stegner (1979)proposed configural, branch weighted averaging modelsin which the weight of a branch depends in part on itsrank within the set. Birnbaum employed this configuralweighting to explain interactions in judgment data(Birnbaum, 1973, 1974), risk aversion and risk seekingin buying and selling prices (Birnbaum & Sutton, 1992),and violations of branch independence (Birnbaum &McIntosh, 1996). Although these models have somesimilarities to rank-dependent utility models, it isimportant to keep in mind that the definition of rankin Birnbaums models applies to consequences ondiscrete branches, and not to cumulative probability.In the class of models that have subsequently come to beknown as rank-dependent models (including RDU,RSDU, CPT, and SP/A), rank refers to cumulativeprobability.

In a risky gamble, the term branch refers to eachprobabilityconsequence pair that is distinct in thegambles presentation. In this notation, gambles thatrepresent the same prospect may be subjectively distinct.Event-splitting produces extra branches, whereas coa-lescing reduces the number of branches in a gamble. Forexample, G $98; 0:8; $2; 0:2 is a two branch gamblethat is distinct from the three branch gamble, H $98; 0:4; $98; 0:4; $2; 0:2; even though they are thesame objectively, and represent the same prospect.

These two classes of representations (configural versusrank-dependent) cannot be distinguished when appliedseparately to certain types of experiments, such asexperiments of Birnbaum and McIntosh (1996), Tverskyand Kahneman (1992), or Wu and Gonzalez (1998).However, these two classes of models can be distin-guished by other tests (Birnbaum, 1997), including newtests used in this paper (Table 1).

The Rank-Affected Multiplicative Weights Model(RAM) and Transfer of Attention Exchange (TAX)models (Birnbaum, 1999a, b; Birnbaum & Navarrete,1998) are two configural weight models that makeidentical predictions for modal choices in the presentstudy, but which can be distinguished by other tests(Birnbaum, 1997; Birnbaum & Chavez, 1997). BothRAM and TAX models are special cases of branch-weighted configural expected utility models in which theweight of each distinct branch of a gamble gets a weightthat is affected by its probability, the rank of itsconsequence, and the weights of other branches, as

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RAM Model

0

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100

0 0.2 0.4 0.6 0.8 1

Probability to win y

100(

CE

(y, p

; x)

x

)/(y

x

)

Fig. 1. Predictions of the RAM model with prior parameters for

certainty equivalents (CE) of two-branch gambles of the form, F x; p; y; plotted as a function of p: Each CE has been linearlytransformed to a 0100 scale. The height of the curve at p 1=2 can beused to solve for branch rank weights, and the value of g can beestimated from the curvature.


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follows:

CWUG Pn

i1 wpi;GuxiPni1 wpi;G

5

where CWUG is the configurally weighted utility ofgamble G; and wpi;G is the configural weight of thebranch with consequence xi in Gamble G x1; p1; x2; p2;y; xi; pi;y; xn; pn; where the conse-quences are ranked such that x14x24?4xn: Thisexpression is quite general and includes CPT, RAM andTAX as special cases.

RAM model. In the RAM model, each configuralweight is a product of a function of branch probabilityand a function of the rank and augmented sign of thebranchs consequence (Birnbaum, 1997). For gambleswith strictly positive consequences, the weight of eachbranch is assumed to be a product as follows: wpi;G ai; nspi; where spi is a function of branch prob-ability and ai; n is the (positive) weight of the branchhaving the ith ranked consequence in gamble with ndiscrete branches. Substituting this assumption for theweights in Eq. (5) yields the RAM model:

RAMG Pn

i1 ai; nspiuxiPni1 ai; nspi

: 6

The rank weights describe how much weight isapplied to each discrete branch depending on the rankof the consequence of that branch in the gamble. Inpractice, for n 2; 3, and 4 branches, the estimated rankcoefficients in the RAM model are approximately equalto their ranks; i.e., ai; n i; with 1 highest; 2 second highest, 3 third highest. In practice, sp andux are approximated by power functions, spi pgiwith 0ogo1; and ux xb where 0obo1: It has beenfound that for positive cash prizes in the domain ofpocket money $1oxo$150; the approximations,ux x; and sp p0:6 gives a good fit to choicesmade by undergraduates. These will be termed theprior parameters of RAM, as they were selected toapproximate previous data of Birnbaum and McIntosh(1996) and Tversky and Kahneman (1992). The use of alinear utility function reveals that risk aversion in thisRAM model is ascribed entirely to the rank weights,with lower ranked branches receiving more weight.

The Certainty Equivalent (CE) of Gamble G is thevalue of cash for which a person would be equally happyto accept the cash or Gamble G; i.e., CEGBG; whereB represents indifference. Predicted certainty equiva-lents of gambles of the form, G x; p; y; 1 p; basedon the prior parameters of RAM are shown in Fig. 1.

These are an inverse-S function of probability: CEG a1;2tp

a1;2tpa2;2t1p: The height of the curve at p 1=2 canbe used to estimate the ratio of rank weights; forexample, in Fig. 1, a2; 2=a1; 2 2 : 1: If go1; thecurve will have an inverse-S shape in which people are

PROO

Frisk-seeking for small p; and with g41; it will have an S-shape in which people can be risk averse for small p: Theprior RAM model, illustrated in Fig. 1, agrees with dataof Tversky and Kahneman (1992) and with the Tverskyand Wakker (1995) model of CPT for two branchgambles.

Like CPT, the RAM model violates restricted branchindependence and satisfies comonotonic restrictedbranch independence. Unlike CPT, which violatesdistribution independence, RAM satisfies distributionindependence (Birnbaum & Chavez, 1997). If all of therank weights were equal to each other (e.g.,ai; n 18i; j), the RAM model would imply noviolations of restricted branch independence, but wouldstill violate coalescing and properties derived fromcoalescing, such as stochastic dominance. When thebranch rank weights are all equal and sp p; thenRAM reduces to EU.

It is important to keep in mind that although CPTand RAM can both account for violations of restrictedbranch independence, they make opposite predictionsfor the types of violation, given their prior parameters(which as we will see below, are needed to account forAllais paradoxes).

TAX model. The TAX model is also a special case ofExpression 5. In the TAX model, a portion of theprobability weight of each branch is transferred amongbranches according to the ranks of the consequences onthe branches. To explain risk aversion in the TAXmodel, it is assumed that weight is taken from brancheswith higher valued consequence and given to brancheswith lower valued consequences.

Birnbaum and Chavez (1997) represented branchweights in the TAX model for a risk-averse subject by

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TAX Model

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0 0.2 0.4 0.6 0.8 1

Probability to win y

100/2(1+ /3)100(

CE

(y, p

; x)

x

)/(y

x

)

Fig. 2. Predictions of the prior TAX model for binary gambles,

plotted as in Fig. 1. Height of the curve at p 1=2 can be used toestimate d; the curvature can be used to estimate g:


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a model that can be rewritten as follows:

wn tn On; 7a

where wn is a 1 n vector containing the relative weightsof branches 1n; respectively, tn is a 1 n vectorcontaining transformed probabilities normalized tosum to 1 in each gamble. In practice, the normalized

probability weighting is approximated as follows: ti pgiPn

j1 pgj

; where g is the parameter of the probability

transformation. Each entry of the n n matrix, On; oijnrepresents the proportion of weight transferred frombranch i to branch j in a gamble with n branches. Thesum of each row of this matrix is 1. Birnbaum andChavez (1997) further simplified this model by theassumption that in choice, weight transfers are given byoijn on d=n 1 for j4i; and on 0 for joi:For three branch gambles, this model can be written asfollows:

w1 w2 w3 tp1 tp2 tp3

1 2o o o

0 1 o o0 0 1

264

375; 7b

where o d=4; for three branch gambles.If o 0; then all weight transfers are zero and there

would be no violations of restricted branch indepen-dence. In this case, each branchs weight would be asimple function of branch probability. If d 0 andtp p; this TAX model reduces to EU. When d0; theproportion of weight transferred is assumed to be a fixedproportion d=n 1 of the branch giving up theweight. Therefore, the sum of the weights is constant, sothat weight is neither created nor destroyed, but onlytransferred from one branch to another.

When do0; weight is transferred from branches withhigher ranked consequences to branches with lowerranked consequences. Intuitively, the transfer of weightsin the TAX model represents a transfer of attentionfrom branches with higher valued consequences tobranches with lower valued consequences. The TAXmodel, like RAM, can imply risk aversion withoutpostulating a nonlinear utility function. With theassumption that utility is linearly related to cash value,risk averse behavior for binary gambles with p 1=2holds iff do0 in this model.

Birnbaum and Chavez noted that if d 1; thismodel would reproduce violations of branch indepen-dence reported by Birnbaum and McIntosh (1996).Birnbaum and Navarrete (1998), who tested three newproperties not examined in the previous work, estimatedthe median value of d for 100 undergraduates to be1:09; close to this prior value of 1: With d 1; thetransfers are as follows: In two branch gambles, one-

PROO

F

third of the probability weight of the branch with thehigher consequence is transferred to the lower valuedbranch. In three branch gambles, one-fourth of theweight of each higher branch is transferred to each lowervalued branch. Birnbaum and Navarrete (1998) re-ported that the median estimate of g was 0.74, close tothe prior value of 0.7 that was chosen to mimic data ofTversky and Kahneman (1992). Birnbaum (1999b)reported median estimates for a more highly educatedsample of people tested via the Internet to be d 0:33and g 0:79:

Predictions for the prior TAX model in this paper arecomputed with the following parameters: ux x;tpi p

0:7iPn

j1 p0:7j

; and d 1: These parameters were

chosen to approximate data of Tversky and Kahneman(1992) and Birnbaum and McIntosh (1996), but theyalso work well for predicting other phenomena (Birn-baum, 1999a). Predictions for two branch gambles forthis TAX model are shown in Fig. 2, plotted as in Fig. 1.The value of the configural parameter, d; can beestimated from the relative height of the curve at p 1=2; and the curve will be inverse-S or S-shaped when gis less than or greater than 1, respectively.

When there is a fixed number of positive-valuedbranches with a fixed probability distribution, RDU/RSDU/CPT, RAM and TAX are all equivalent to eachother and reduce to what Birnbaum and McIntosh(1996) called the generic rank-dependent configuralweight model, which Luce (2000) calls the rankweighted additive model.

In both RAM and TAX models, the approximationux x; gives a good fit to data in studies involvingrisky decisions with monetary consequences in thedomain of pocket money, i.e., when $1oxo$150: Withthis assumption, RAM and TAX have fewer parametersthan the comparable CPT model for strictly positiveconsequences. But in this paper, RAM and TAX will use

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= 2

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Fig. 3. Analysis of violation of stochastic dominance in the TAX

model. The values calculated are CEG CEG for Choice 5 inTable 4. Negative values show violations of stochastic dominance,

which occur for do0 (risk averse for p 1=2) and go1 (inverse-S inFig. 2). Violations of stochastic dominance can also occur for risk-

neutral d 0 and even risk-seeking d40 people if g is smallenough.


YJMPS : 1517

prior parameters to predict the new results, requiringnothing to be estimated from the present data. Predic-tions of CPT with parameters of Tversky and Kahne-man (1992) will be calculated for comparison. Thus,both models are tested on equal footing.3

RAM and TAX models violate both restricted branchindependence and coalescing, except when they reduceto EU. Because they violate coalescing, these modelsviolate properties that can be derived from coalescingincluding stochastic dominance and Allais indepen-dence.

Unlike stripped prospect theory, RAM and TAXmodels do not, however, violate transparent dominance(Birnbaum, 1999a): improving the consequence of agiven branch (holding everything else constant) im-proves a gamble. Similarly, moving probability from abranch with a lower valued consequence to a branchwith a higher valued consequence (holding everythingelse constant) improves the gamble.

Although they satisfy transparent dominance, bothRAM and TAX violate first order stochastic dominancefor specially constructed choices (Birnbaum, 1997).

1.6. Violations of stochastic dominance

Because RAM and TAX models violate coalescing,they violate stochastic dominance. Birnbaums (1997)recipe for creating violations of stochastic dominance inconfigural weight models is based on splitting the loweror higher-valued branch of a root gamble. For example,let the root gamble be G $98; 0:9; $12; 0:1; andconstruct the following:

D

G:

3To

follow

lerton

0.85 to win $98

explore predictions

ing URL for links t

.edu/mbirnbaum/calc

G:

of TA

o free

ulator

0.90 to win $98

E 91

0.05 to win $90

0.05 to win $14

0.10 to win $12
0.05 to win $12
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UN

CORR

ECT

According to the configural weight RAM and TAXmodels, splitting the higher branch of G gives greatertotal weight to higher consequence(s). Thus, eventhough the consequence on the 0.05 splinter has beenreduced from $98 to $90 (G is dominated by G),TAX G4TAX G: Splitting the lower branch of Gcreates G; which dominates G; but in theory, now thelower consequences get greater weight, making Gworse than G; even though the 0.05 splinter has beenincreased in value from $12 to $14. So, according toRAM and TAX models, people should choose G overG; even though G dominates G :

An analysis of predicted violations of stochasticdominance in the TAX model is shown in Fig. 3. Theordinate plots the value of CEG CEG; as afunction of the value of d; the configural weighting

109

X, RAM, CPT, and EV, visit the

on-line calculators: http://psych.ful-

s/

PROO

Fparameter of TAX. Separate curves are used fordifferent values of g: Negative values on the ordinaterepresent violations of stochastic dominance.

Fig. 3 shows that the TAX model always violatesstochastic dominance in this case if do0 and go1:Given the data of Tversky and Kahneman (1992), oneconcludes that d o 0 because people are risk-averse fortwo-branch, 5050 gambles to win positive conse-quences. In addition, one concludes that g o 1 becausepeople are simultaneously risk-seeking for 2-branchgambles with very small probabilities to win (for a fit ofthis model to their data, see Birnbaum, 1997, Fig. 9).Therefore, given the data of Tversky and Kahneman(1992), the TAX model of Eq. (7b) is forced to predictviolations of stochastic dominance in this recipe.Birnbaum (1997, p. 94) put this prediction in print,specifying both G and G and stating (p. 94), Itseems worthwhile to test such predictions...

Birnbaum and Navarrete (1998) tested this interestingprediction and found that about 70% of 100 under-graduates tested violated stochastic dominance in thischoice and three others like it. Birnbaum, Patton, andLott (1999) found similar results with a new group of110 subjects and five new choices constructed from thesame recipe.

In a subsequent study, Birnbaum (1999b) found that72% of a new group of 124 undergraduates violateddominance on the above choice, but only 15% of thesame people violated stochastic dominance when thesame (objective) choices are presented in split form, asfollows:

GS:

0.85 to win $98

GS:

0.85 to win $98

111
0.05 to win $90 0.05 to win $98
0.05 to win $12

0.05 to win $14

http://psych.fullerton.edu/mbirnbaum/calculators/http://psych.fullerton.edu/mbirnbaum/calculators/

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YJMPS : 1517

Table 2

Dissecti

No R

p

6

9 S

12 R

16 R

19 C

Notes:

RBI Rcolor of

calculat

Tversky

0.05 to win $12

UNC

on of Allais paradox

elation to

revious row

Choic

FU

First g

10 bla

90 pur

plit # 6 10 red

10 blu

80 wh

BI #9 10 red

80 blu

10 wh

BI #9, 12 80 red

10 blu

10 wh

oalesce #16 90 red

10 wh

The common branch

estricted Branch Ind

marbles on corresp

ed certainty equivale

and Kahneman (19

OR

(Series

e as in c

amble,

ck marb

ple mar

marble

e marbl

ite marb

marble

e marbl

ite marb

marble

e marbl

ite marb

marble

ite marb

was 8

ependen

onding

nt value

92).

0.05 to win $12

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Because the choice between G and G is the same asthat between GS and GS; except for coalescing,people should make the same choices, if coalescingholds. Approximately 62% of undergraduates tested(significantly more than half!), however, switched fromG to GS; whereas fewer than 5% switched in theopposite direction; these systematic preference reversalsindicate that coalescing is not descriptive of humanchoice (Birnbaum, 1999b, 2000; Birnbaum & Martin,2003). The prior RAM and TAX models predicted bothresults.

One might express reservations about these previoustests of event-splitting, however, based on the followingargument. The choice between GS and GS mightinvoke the editing mechanism of dominance detection,so the apparent violation of coalescing might beproduced by comparison processes such as editing,rather than by the evaluation function. The presentstudy provides new tests of coalescing that do notinvolve dominance.

1.7. Event framing versus coalescing

Tversky and Kahneman (1986) presented a case inwhich more violations of stochastic dominance wereobserved in a framed and coalesced choice than in adifferently framed and split form of the same choice.They noted that their theory assumed coalescing (seealso Kahneman, 2003, p. 727), and they emphasizedinstead the importance of the event framing used tomask the dominance relationship. In their framing,

RECT

EDA) (each entry is the percentage in each

ondition

R Second gamble, S

les to win $98 20 black marbles to w

bles to win $2 80 purple marbles to w

s to win $98 10 red marbles to win

es to win $2 10 blue marbles to win

les to win $2 80 white marbles to w









0 marbles to win $2 in Choices 6 and

ce, R Risky Gamble, S Safe Gambbranches; in condition UF, these were

s of each gamble with prior parameter

PROO

F

the dominated gamble was made to appear as if for anynamed event (for any color of marble drawn from anurn) the dominated gamble gave either an equal orbetter consequence. Because there were (slightly) differ-ent numbers of marbles of each color in the two urns,however, the so-called events were not really thesame. Because the numbers of marbles were nearlyequal, Tversky and Kahneman theorized that judgeswould simplify the choice by canceling the nearly equalbranches produced by the same named events. Theyconceded, however, that in their test, several differentinterpretations were confounded, including a compar-ison of the number of branches with positive or negativeconsequences in each gamble.

A second purpose of the present study is therefore toassess the importance of event framing (as opposed toevent-splitting/coalescing) in tests of branch indepen-dence and Allais independence as well as in tests ofstochastic dominance. Event framing would be expectedto reduce violations of branch independence in the splitforms. Such choices might be termed transparent testsof branch independence in the framed form, becauseboth gambles would clearly share a common eventconsequence branch. In such a framed format, adecision-maker should find it easy to cancel branchesthat are identical in two choices and to make a choicebased strictly on what is left.

The event framing manipulation is illustrated inChoice 16 of Table 2, which is framed as opposedto Choice 14 in Table 3, which is unframed. If peopleattend to framing and cancel common branches, theywould presumably show greater conformance to branch

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condition choosing the safe gamble, S)

Condition Prior TAX

model

Prior CPT

model

FU UF Rep R S R S

in $40 41 37 36 13:34 9.0 16:94 10.7in $2

$40 69 66 60 9.6 o11.1 16.94 10.7$40

in $2

$40 62 55 47 30.6 o40.0 38.0 o40.0$40

in $40

$98 47 45 39 62:64 59.8 67.6 o74.5$40

in $40

$98 81 82 72 54.7 o68.0 67.6 o74.5in $40

9, $40, in Choice 12, and $98 in Choices 16 and 19, respectively.

le, In condition FU, these choices were all framed by having the same

unframed; Rep replication study with n 150: TAX model showss. CPT model shows calculated certainty equivalents under model of

NCOR

RECT

ED P

ROOF

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Table 3

Dissection of Allais paradox into branch independence and coalescing (Series B)

No Relation to

previous row

Choice as in condition FU Condition Prior TAX

model

Prior CPT

model

First gamble, S Second gamble, R FU UF Rep S R S R

10 15 red marbles to win $50 10 blue marbles to win $100 86 74 82 13.6 o18.0 15.9 o22.185 black marbles to win $7 90 white marbles to win $7

17 Split #10 10 red marbles to win $50 10 black marbles to win $100 53 45 46 15:64 14.6 15.9 o22.105 blue marbles to win $50 05 purple marbles to win $7

85 white marbles to win $7 85 green marbles to win $7

20 RBI #17 10 red marbles to win $50 10 black marbles to win $100 44 49 52 50:04 40.1 504 49.285 white marbles to win $50 85 purple marbles to win $50

05 blue marbles to win $50 05 green marbles to win $7

14 RBI # 17, 20 85 red marbles to win $100 85 black marbles to win $100 63 63 60 68.4 o69.7 82:24 79.010 white marbles to win $50 10 yellow marbles to win $100

05 blue marbles to win $50 05 purple marbles to win $7

8 Coalesce #14 85 black marbles to win $100 95 red marbles to win $100 25 16 34 75:74 62.0 82:24 79.015 yellow marbles to win $50 05 white marbles to win $7

Notes: Gambles unframed, as in condition FU; in condition UF, each choice was framed. Each entry is the percentage of people in each condition

who chose the second, risky gamble, R: The common branch is 85 marbles to win either $7 (in Choices 10 and 17), $50 (Choice 20), or $100

(Choices 14 and 8), respectively). RBI restricted branch independence.

Table 4

Violations of stochastic dominance and coalescing linked to event framing and event-splitting

No. Choice (as in condition FU) Condition Prior TAX Prior CPT

G G FU UF Rep G G G G

5 90 red marbles to win $96 85 red marbles to win $96 73 85 76 45.8 o63.1 70:34 69.705 blue marbles to win $14 05 blue marbles to win $90

05 white marbles to win $12 10 white marbles to win $12

11 85 red marbles to win $96 85 red marbles to win $96 15 11 11 53:14 51.4 70:34 69.705 blue marbles to win $96 05 blue marbles to win $90

05 green marbles to win $14 05 green marbles to win $12


15 90 red marbles to win $96 85 black marbles to win $96 77 74 78 45.8 o63.1 70:34 69.705 yellow marbles to win $14 05 blue marbles to win $90

05 pink marbles to win $12 10 white marbles to win $12

7a 94 black marbles to win $99 91 red marbles to win $99 78 74 70 46.0 o66.6 76:24 75.903 yellow marbles to win $8 03 blue marbles to win $96

03 purple marbles to win $6 06 white marbles to win $6

13a 91 black marbles to win $99 91 red marbles to win $99 10 16 13 54:24 53.2 76:24 75.903 pink marbles to win $99 03 blue marbles to win $96

03 yellow marbles to win $8 03 green marbles to win $6

03 purple marbles to win $6 03 white marbles to win $6

18a 94 red marbles to win $99 91 red marbles to win $99 75 72 70 46.0 o66.6 76:24 75.903 blue marbles to win $8 03 blue marbles to win $96


Notes: Choices 5, 11, and 18 were framed. In Choices 5, 11, and 15, the dominant gamble was presented first and in Choices 7, 13, and 18, G waspresented second. Each entry is the percentage of people in each condition who violated stochastic dominance (bold type shows violations in framed

choices.


YJMPS : 1517

Uindependence in framed than unframed tests. Choices 9and 16 (Table 2) should more likely yield the samedecisions in the FU condition, where the commonbranch has the same color than they would in conditionUF, where the colors of marbles on correspondingbranches are different.

Similarly, people should be more likely to violatestochastic dominance on Choice 5 in Table 4 in the FUcondition, with common color framing, than in the UFcondition where it is unframed. The reasoning here usesthe editing principles (Kahneman & Tversky, 1979) ofsimplification and cancellation: the common colorbranches to win $96 and $12 in Choice 5 are nearly

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equal (in probability), which if cancelled from bothsides, leaves a branch that favors the (dominated)gamble.

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2. Method

Deciders made 20 choices between pairs of gambles.They viewed the materials on-line via the Internet,clicking the button beside the gamble they would ratherplay in each choice. They were informed that 3 luckyparticipants would be selected at random to play one oftheir chosen gambles for money, with prizes as high as$110, so they should choose carefully. Prizes wereawarded as promised. Each choice appeared as in thefollowing example:

1.

Which do you choose?

75
A: 50 red marbles to win $100
50 white marbles to win $0

77
OR
B:

50 blue marbles to win $35

79
50 green marbles to win $25
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UNCO

RREC

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Instructions read (in part) as follows:Think of probability as the number of marbles in

one color in an urn (container) containing 100 otherwiseidentical marbles, divided by 100. Gamble A has 50 redmarbles and 50 white marbles; if a marble drawn atrandom from urn A is red, you win $100. If a whitemarble is drawn, you win $0. So, the probability to drawa red marble and win $100 is 0.50 and the probability todraw a white marble and get $0 is 0.50. If someonereaches in urn A, half the time they draw red and win$100 and half the time they draw white and win $0. Butin this study, you only get to play a gamble once, so theprize will be either $0 or $100. Gamble Bs urn has 100marbles also, but 50 of them are blue, winning $35, and50 of them are green and win $25. Urn B thusguarantees at least $25, but the most you can win is$35. Some will prefer A and others will prefer B. Tomark your choice, click the button next to A or B...

2.1. Allais paradoxes: coalescing and branch

independence

Choices for Series A and B of Allais paradoxes areshown in Tables 2 and 3, respectively. Each choice iscreated from the choice in the row above by eithercoalescing/splitting or by restricted branch indepen-dence. Within each series, choices should be the same inevery row, according to EU, except for random error. InSeries A, the common branch is 80 marbles to win $2(first two rows), $40 (middle row), or $98 (last tworows). In Series B, the common branch is 85 marbles towin $7 (first two rows), $50 (third row), or $100 (fourthand fifth rows). Note that the positions (First or Second)

PROO

F

of the S or safe gamble with higher probability to wina smaller prize and the R; or risky gamble arecounterbalanced between Series A and B.

2.2. Framing manipulation

Each choice was either framed or unframed. In theframed version, the same marble colors are used for eachordered branch. A framed and coalesced test ofstochastic dominance is shown in Choice 5 of Table 4,and the unframed version of the same (objective) choiceis shown in Choice 15. The framed and split form of thischoice is shown as Choice 11 of Table 4.

There were two conditions to which participants wererandomly assigned by means of a JavaScript routine(Birnbaum, 2001a, p. 211). In the FU condition (shownin Tables 2, 3, and 4), all choices in Series A (Table 2)were framed, all in Series B (Table 3) were unframed;Choices 5, 11, and 18 of Table 4 were framed andChoices 7, 13, and 15 were not. In the UF condition,framing was reversed from that of FU.

The first four choices, which served as a warm-up,were the same as those of Birnbaum (1999b), formattedin terms of the marbles. Complete materials can beviewed at URLs: http://psych.fullerton.edu/mbirnbaum/Exp2 urnsUF A.htm http://psych.fullerton.edu/mbirn-baum/Exp2 urnsFU A.htm

2.3. Participants

Participants were 200 people recruited by links on theWeb and from the usual subject pool in thepsychology department of California State University,Fullerton. When each condition had 100 participants,the study was deemed complete.

2.4. Replication study

An additional 150 participants, recruited entirelyfrom the Web, were randomly assigned to ConditionsFU or UF, and were tested in a simple replication of theentire study.

3. Results

3.1. Allais paradoxes

Tables 2 and 3 show the percentage of participants ineach condition who chose the second gamble in eachchoice of Series A and B. Separate columns show choicepercentages for each framing condition of the mainstudy and for the combined results of the replicationstudy. According to EU, choices should be the same inevery row within Table 2 and within Table 3, except forerror; therefore, the choice percentages should not

http://psych.fullerton.edu/mbirnbaum/Exp2_urnsUF_A.htmhttp://psych.fullerton.edu/mbirnbaum/Exp2_urnsUF_A.htmhttp://psych.fullerton.edu/mbirnbaum/Exp2_urnsUF_A.htmhttp://psych.fullerton.edu/mbirnbaum/Exp2_urnsUF_A.htmhttp://psych.fullerton.edu/mbirnbaum/Exp2_urnsFU_A.htmhttp://psych.fullerton.edu/mbirnbaum/Exp2_urnsFU_A.htmhttp://psych.fullerton.edu/mbirnbaum/Exp2_urnsFU_A.htmhttp://psych.fullerton.edu/mbirnbaum/Exp2_urnsFU_A.htm

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YJMPS : 1517

UNCO

RREC

TED

change systematically from row to row. The originaltype of the Allais paradox involves comparison ofChoices 6 and 12 in Table 2, but any systematic changein preference from row to row in Tables 2 and 3 wouldbe a violation of Allais independence. The data showsystematic reversals, demonstrated by the finding thatchoice percentages change significantly from row to rowin both tables.

For example, Choice 6 in Table 2 (averaged overframing) shows that only 39% chose the safe gamble(with 20 marbles to win $40) over the risky gamble(with 10 marbles to win $98 and 10 to win $2) when thecommon branch was 80 marbles to win $2. However, inChoice 19 (where the common branch was 80 marbles towin $98), 82% chose the safe gamble. Similarly, Table3 shows that the percentage choice changed from 20%(for the safe gamble in Choice 10) to 80% (for thesafe gamble in Choice 8), as the common consequenceon the branch with 85 marbles was increased from $7 to$100. These are large violations of Allais independence.The replication study (column labeled Rep) yieldedvery similar results, which are averaged over twoframing conditions.

These violations of Allais independence are statisti-cally significant, even by the conservative standard thatthe modal choice had to be significantly reversed. For asample of n 200; the binomial distribution with p 1=2 has a mean of 50% and a standard deviation of3.5%; therefore, observed percentages outside theinterval from 43% to 57% deviate significantly from50% by a two-tailed test with a 0:05: Combining overconditions, the percentage choosing the risky gamblechanged from significantly less than 50% to significantlygreater than 50% as the common consequence wasincreased from the lowest value to the highest value inboth Tables 2 and 3. The replication study n 150also shows significant reversals by the same test in bothtables.

The binomial test of correlated proportions is a moresensitive test of significance of within-subject changes inchoice proportions. This test checks for equality ofchoice proportions, rather than requiring a significantreversal of the mode. Comparing Choices 619, forexample, this test compares the number who switchedfrom choosing the (risky) first gamble in Choice 6 tothe (safe) second gamble in Choice 19 against thenumber who switched preferences in the oppositedirection. In the FU condition of Series A, 47 switchedfrom risky to safe, against only 7 who switched inthe opposite direction z 5:44; in the UF condition,the numbers were 50 versus 5 z 6:07: The criticalvalue of jzj for a two-tailed test with a 0:05 is 1.96.The term significant and asterisks are usedthroughout this paper to denote significant differencesby this test.

PROO

F

For Series B (Table 3), the first gamble was the safeone; hence, the same directional change is observed inboth tables, counterbalanced for position. The numberswho switched from the risky second gamble in Choice10 to the safe first gamble in Choice 8 were 64 and 60in the FU and UF conditions, respectively, against only4 and 4 who switched in the opposite directions(z 7:28 and 7:00), respectively.

3.2. Tests of coalescing in Allais paradoxes

Tables 2 and 3 separate coalescing/splitting frombranch independence. Note that in Table 2, Choice 9 isthe same as Choice 6, except for coalescing. The firstgamble in Choice 9 is the same as the first gamble inChoice 6, except the lower branch of 90 marbles to win$2 has been split into 80 marbles to win $2 and 10marbles to win $2. According to the class of RDU/RSDU/CPT models, this manipulation should have noeffect; however, the TAX model (with its priorparameters) implies that splitting the lower branchincreased the relative weight of winning only $2 andthus made the first gamble worse.

Similarly, the second gamble of Choice 9 is the sameas in Choice 6, except that the 20 marble branch to win$40 was split, which should have no effect according tothe RDU/RSDU/CPT models, but makes the secondgamble better according to RAM or TAX. If peopleobeyed coalescing, they should make the same decisionsin Choices #6 and #9. However, according to the priorTAX model, this split makes the second gamble inChoice 9 better because more weight is transferred to thehigher consequence.

Choice 19 in Table 2 is the result of coalescing theupper branches (to win $98) in the first gamble of Choice16 and coalescing the lower branches (to win $40) in thesecond Gamble. Table 3 (Series B) is based on the sameplan.

According to the class of RDU/RSDU/CPT models(apart from editing), a person should make the samedecisions in Choices 6 and 9, 16 and 19, 10 and 17, and14 and 8, since each of these comparisons involves onlyevent coalescing/splitting (and transitivity). The valueslabeled CPT in the tables show calculated certaintyequivalent (cash value) of each gamble, based on themodel and parameters of Tversky and Kahneman(1992). Although a particular model was used tocalculate these predictions, the invariance with respectto coalescing/splitting holds for any functions with anyparameters in CPT.

According to the TAX model, however, coalescing/splitting affects the weights of the branches and hencethe values of the gambles. The three tables showcalculated cash equivalents for each gamble based theTAX model with prior parameters (see Introduction).From Choice 9 to 12, the lower branch of the first

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gamble has been split. In theory, this split increases theweight of $2, thereby making the risky first gambleR seem worse (in Choice 9) than it does in thecoalesced form of Choice 6 (the calculated certaintyequivalent of R drops from $13.3 to $9.6). In addition,the higher-valued branch of the safe second gambleS has been split, which increases the weight of $40,making the Second gamble seem relatively better in #9compared to #6 (its CE increases from $9 to $11.1).Therefore, the prior TAX model correctly predicts thatthere will be a reversal in the modal choice from R inChoice 6 to S in Choice 9.

Similar predictions in Tables 2 and 3 can be under-stood from these implications of branch weighting inRAM and TAX: splitting the lower branch makes agamble worse and splitting the higher branch makes agamble better.

Testing separately in each framing condition, all eighttests of coalescing in the main design are significant bythe test of correlated proportions, with all eight shifts inthe direction predicted by the configural weight TAXmodel with its prior parameters. For example, fromChoice 6 to 9, 68 people switched from picking R inChoice 6 to S in Choice 9, compared with only 6 whoreversed preferences in the opposite direction. FromChoices 16 to 19, the higher valued branches of R havebeen coalesced, making R seem worse in Choice 19 thanit did in Choice 16, whereas the lower consequences of Shave been coalesced, making S seem better in Choice 19.In this case, 77 switched from R to S compared withonly 8 who switched in the opposite direction. ForChoices 10 and 17 in Series B (Table 3), the results aresimilar: 73 reversed preferences in the directionpredicted by the TAX model compared to only 12who switched in the opposite direction. For Choices 14and 8, 96 reversed preferences in the predicted directionagainst only 13 who switched in the opposite direction.

All eight significant changes due to coalescing/splitting are predicted by the configural weight modelwith its prior parameters, and all eight results areinconsistent with the class of RDU/RSDU/CPT modelswith any set of parameters, because those models requireno systematic effects of coalescing or splitting. Thereplication study confirms the same conclusions: in allfour cases, the shifts are significant by the test ofcorrelated proportions. In three of the four cases (allexcept Choice 10 versus 17), the shift is even significantby the conservative standard that coalescing signifi-cantly reverses the mode.

3.3. Tests of branch independence in Allais paradoxes

Choices 9, 12, and 16 in Table 2 differ only in that thecommon branch of 80 marbles to win $2 in Choice 9 hasbeen changed to 80 marbles to win $40 in Choice 12, andto 80 marbles to win $98 in Choice 16. Therefore, if

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choices obeyed restricted branch independence, deci-sions in Choices 9, 12, and 16 would be the same.Similarly, Choices 17, 20, and 14 in Table 3 are thesame, except that the consequence on the commonbranch of 85 marbles was either $7, $50, or $100.

Consider, the following defense of CPT for theviolations of coalescing in the previous section. Supposethat in transparent tests of branch independence, peopleused the editing rule of cancellation at least part of thetime. Such a strategy would produce greater satisfactionof restricted branch independence. For example,Choices 9 versus 16 might be called a transparenttest of branch independence because in each case, thejudge could simply cancel the common branch (80marbles to win $2 or $98), which leaves the sameremainder in Choices 9 and 16. Similarly, Choices 17and 14 would be the same if one canceled the commonbranch of 85 marbles to win $7 or $100. Such editingmight give an explanation for significant shifts (betweenthe coalesced and split forms in Tables 2 and 3) observedwith the tests of correlated proportions.

The problem with this argument, however, is that theviolations of branch independence are statisticallysignificant and in the opposite direction from thatobserved in the coalesced versions of the same choices.Summed over framing conditions, 57 switched from thesafe to the risky gamble from Choice 9 to Choice 16compared to only 17 who switched in the oppositedirection z 4:65: Similarly, 55 reversed from safeto risky from Choices 17 to 14, compared with only28 who made the opposite switch z 2:96:

In the replication study, significantly more than halfchose S in Choice 9 and significantly less than half choseS in Choice 16. Similarly, significantly less than halfchose S in Choice 6 but significantly more than halfchose S in Choice 9. In addition, there was a significantreversal in the majority choice between 16 and 19. Thesesignificant reversals, even by the conservative standard,show that CPT, even with its editing principle ofcancellation, can be rejected. If we theorize thatcancellation is used some of the time by some of thejudges, we must conclude that the results would havebeen even more devastating for CPT.

According to CPT, RDU, and RSDU models, Choice6 is the same as Choice 9 and Choices 16 and 19 are thesame. Similarly, Choice 10 is the same as 17 and 14 is thesame as Choice 8 in Table 3. Assuming people some-times do and sometimes do not use cancellation, theresults might have been intermediate between noviolations of branch independence and the patternneeded to explain the Allais paradox. For example,had the percentages of choosing the second gamble inChoices 9 and 16 been 47% and 69%, respectively,instead of 69% and 47%, respectively, it might havebeen argued that the apparent violations of coalescingare caused by partial used of cancellation. However, the

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results in Tables 2 and 3 are opposite those required tosave the CPT/RDU/RSDU models even with thisediting argument.

Even by the conservative standard of significantlyreversing the mode (but lenient to CPT), there is at leastone significant switch in each series of the modal choicefrom significantly less than 50% to significantly greaterthan 50%. (These significant switches are observed inChoices 6 versus 9 in Series A, and Choices 14 versus 8in Series B). In the replication study, there are threereversals that significantly reverse by this conservativetest (Choices # 6 versus 9, 16 versus 19, and 14 versus 8).It is hard to see how to reconcile such results with CPT,even with the editing principle of cancellation, becausethe violations are not merely reduced by splitting (asmight be expected from the editing principle ofcancellation) but significantly reversed.

3.4. Tests of stochastic dominance and coalescing

Table 4 summarizes tests of stochastic dominance andcoalescing. In Choices 5, 11, and 15, the first gambledominates the second. The second gamble dominates theFirst in Choices 7, 13, and 18. In Table 4, all percentagesrepresent violations of stochastic dominance. Violationsof stochastic dominance are significantly greater than50% in all 12 tests of coalesced choices (values exceeding60% are significant). There are 12 values in the tableexceeding 70%, with an average of 75%.

In contrast, violations of stochastic dominance aresignificantly less than 50% in all 6 tests of theappropriately split versions of the same choices (averageof 13%). Each of the 12 tests of event-splitting/coalescing is also significant by the test of correlatedproportions. The replication study again repeats thepattern of significant violations observed in the mainstudy. These results reinforce those of previous tests ofstochastic dominance and coalescing in this recipe.

3.5. Event framing

The framing effect of marble color had very smalleffects. For framed and coalesced choices, violations ofstochastic dominance had an average of 74% comparedto 78% for coalesced and unframed. In the split forms,there were 15.5% violations in the framed cases and10.5% in the unframed cases. These are small effects,and they go opposite the directions anticipated by theediting notion, which predicted more violations whenframed and coalesced and fewer violations in the framedand split conditions.

For tests of branch independence, summing overSeries (Choices 9 versus 16 and 17 versus 14), there were65 choices with the SR switch compared to 26 whoshowed the opposite in framed choices. With unframedchoices, there were 47 showing the SR switch compared

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to 19 showing the opposite switch. The editing notionheld that there should be fewer violations of branchindependence in the framed cases, so the data againshow small effects that are opposite those predicted. Insum, event framing had minimal effects and did notshow the patterns expected from the idea that framingwould increase the use of cancellation. The replicationstudy also found that event framing effects wereminimal (not shown).

3.6. Prior predictions

This experiment was designed based on calculationsunder CPT and TAX models. As noted by Birnbaumand McIntosh (1996), the prediction of violations ofbranch independence requires a careful fishnet designunless parameters are known in advance. Based on theprevious parameters, it was possible to design anexperiment that should distinguish these theories, ifthe prior parameters also work for new choices and newparticipants. Naturally, fitting models to the same datawould give a better fit than using previous data topredict new results. However, because post hoc predic-tions can take advantage of lack of constraint in anexperimental design, I think results are more impressiveif one can use a model and its parameters to predictfrom one study to new properties tested in anotherstudy.

Tables 24 show calculated certainty equivalents fromthe TAX model with prior parameters (Birnbaum,1999a). The RAM model with its prior parametersmakes the same (directional) predictions as TAX in thisstudy (not shown). TAX and RAM models correctlypredict the majority choice in 31 of 32 percentages of themain study listed in Tables 24 (all except Choice 17 inthe FU condition, where the 53% should have been lessthan 50%). It also predicts the majority choice in all buttwo of 16 choices in the Replication study (in Choice 12,the 47% should have been greater than 50% and inChoice 20, the 52% should have been less than 50%). Innone of these three cases where prior TAX was wrongwere the percentages significantly different from 50%.

There are eight choice sets where CPT and TAX makedifferent predictions (Choices 9, 16, 17, 14, 5, 15, 7, and18). Of the 24 empirical choice percentages for theseeight cases, the modal choice agreed with TAX in 23cases and with CPT in only one choice (Choice 17 in theFU condition, 53%). In 19 of 24 cases, empirical choiceproportions were significantly different from 50%; in allof these cases, CPT predicted the wrong choice. If thetwo models were equally good, one would expect half ofthese 19 significant cases to favor either model.

The prior TAX and RAM models agree with CPT forChoices 6, 12, 19, 10, 20, 8, 11, and 13. That is, theseconfigural models make the same predictions as CPTfor coalesced tests of Allais independence and for split

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tests of stochastic dominance. RAM and TAX modelsdiffer from any member of the rank-dependent modelsin that they predict violations of stochastic dominancein Choices 5, 15, 7, and 18 of Table 4. They also differfrom any of the rank-dependent models in predictingreversals in Tables 24 due to coalescing/splitting.

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4. Discussion

Table 1 shows how the separation of restricted branchindependence and coalescing allows one to distinguishtheories of decision making. The results are inconsistentwith both original and cumulative prospect theoriesbecause the data show significant violations of bothproperties. In addition, the type of violation of restrictedbranch independence is opposite what is needed by CPTto account for the Allais paradoxes.

These results are instead consistent with the conclu-sion that the primary cause of Allais common con-sequence paradoxes is violation of coalescing. In everytest of coalescing/splitting in Tables 2 and 3, there arelarge and significant changes that agree in direction withthe Allais paradoxes.

Although pure tests of restricted branch indepen-dence show systematic violations in Table 2 (Choice 9versus 16) and Table 3 (Choice 17 versus 14), thesepure violations go in the opposite direction from whatis needed by rank-dependent models RDU/RSDU/CPTto explain the Allais paradoxes (first to last row inTables 2 and 3). Put another way, the inverse-S functionwith CPT correctly predicts the results of Choices 6, 19,10 and 8 in Tables 2 and 3. However, that sameweighting function fails to predict the results of Choices9, 16, 17, and 14, which are identical prospectsaccording to CPT. Both sets of results are consistentwith the pattern predicted by RAM and TAX.

All systematic reversals of preference due to splitting/coalescing reported in Tables 24 are in the directionone expects from the RAM or TAX models if splitting abranch gives those splinters greater total weight thanthey would receive when coalesced. Splitting the highervalued branch of a gamble should make the gamblerelatively better and splitting the lower valued branchshould make it relatively worse. Note that the configuralmodels imply that each splitting and coalescing opera-tion (from top to bottom rows of Tables 2 and 3) shouldimprove S and diminish R:

Therefore, even if there were no violations of branchindependence [i.e., even with d 0 in TAX or withai; n 18i; n in RAM], TAX and RAM models wouldstill imply Allais paradoxes from violations of coalescingproduced by the curvature of tp (in TAX) or sp (inRAM) alone.

Violations of coalescing also explain violations ofstochastic dominance. Splitting or coalescing (in Table

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4) changes the percentage of violations from 76% to13%, significantly reversing the preference between twoequivalent prospects.

The RDU/RSDU/CPT models imply coalescing;therefore, these models cannot account for any of theeffects in Tables 24 produced by coalescing (orsplitting).

The class of RDU/RSDU/CPT models easily violaterestricted branch independence. However, the particularobserved pattern in previous studies as well as this one isopposite that predicted by the inverse-S form (Birn-baum & Chavez, 1997; Birnbaum & McIntosh, 1996;Birnbaum & Navarrete, 1998). A new feature of thepresent study is that the experimental tests of branchindependence are elegantly coupled precisely with Allaiscommon consequence paradoxes to disentangle branchindependence and coalescing in the Allais paradoxes. Inthis design, CPT must violate branch independence inorder to explain the Allais paradox, but it does so in theopposite way from what is observed.

4.1. The case against CPT

The cumulative array of data that violate the class ofRDU/RSDU/CPT models has now reached a criticalthreshold where those theories must be questioned asdescriptive of human decision making. The weight ofevidence against CPT now exceeds the case against EUtheory reviewed by Kahneman and Tversky (1979).

The RDU/RSDU/CPT models can be replaced by amore accurate model that uses no more parameters, butwhich accounts for seven different results that refute thisclass of theories. CPT (with any choice of functions andparameters) cannot account for violations of coalescing(event-splitting effects), violations of stochastic dom-inance, violations of lower cumulative independence,violations of upper cumulative independence, or viola-tions of 3-branch tail independence. In addition, theCPT model, in order to account for Allais paradoxes,violates both branch independence and distributionindependence in the opposite direction from thatobserved. Each of these seven phenomena has now beenwell established by systematic experiments, and mosthave been replicated in more than one study orconfirmed in the same study with multiple variations.Each of these seven phenomena are consistent with theTAX model, which predicted five of them in advance,including the dissection of the Allais paradox in thepresent paper.

Birnbaum (1997) derived the properties of lowercumulative independence and upper cumulative inde-pendence to clarify the contradiction between empiricalviolations of branch independence and empirical evi-dence that implied the inverse-S weighting function inRDU/RSDU/CPT models. These two theorems can bededuced from transitivity, monotonicity, coalescing, and

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comonotonic restricted branch independence (Birn-baum, 1997). They were originally deduced directlyfrom the RDU representation (Birnbaum et al., 1999,Appendix). Based on RAM and TAX models, Birn-baum (1997) predicted violations of these properties,which were subsequently confirmed in several studies(Birnbaum, 1999b, 2000; Birnbaum & Navarrete, 1998).

Without modification, the class of rank dependentmodels does not account for violations of 3-branchupper tail independence reported by Wu (1994) andreplicated by Birnbaum (2001b), a property that can bededuced from transitivity, coalescing, and restrictedcomonotonic branch independence. These results are,however, consistent with RAM and TAX.

4.2. RAM and TAX models

Both RAM and TAX models predicted the phenom-ena that violate the class of CPT and RDU models.Indeed, the TAX model was used to design the empiricaltests in this paper. Birnbaums (1999a) review showedthat with the same set of parameters, RAM and TAXmodels predict results of tests of stochastic dominance,event-splitting effects, lower and upper cumulativeindependence, and branch independence.

TAX also implies violations of distribution indepen-dence (Birnbaum & Chavez, 1997), which violate RAM.The TAX model, with the same prior parameters,explains both classic and modern variations of theAllais paradoxes (Birnbaum, 2000, 2001a,b; Birnbaum& Martin, 2003; Wu & Gonzalez, 1996, 1998), as well asother data that can be fit with the inverse-S weightingfunction (Abdellaoui, 2000; Bleichrodt & Pinto, 2000;Gonzalez & Wu, 1999; Tversky & Wakker, 1995;Quiggin, 1993; Luce, 2000; Starmer, 2000). The presentstudy completes the picture by showing that the TAXmodel with the same (prior) parameters correctlypredicts the effects of branch independence and coales-cing on new variations of the common consequenceparadoxes of Allais.

4.3. Can CPT be saved by changing procedure?

It has been found that violations of CPT are obtainedwhen choices are presented in many different forms andformats (Birnbaum & Yeary submitted; Birnbaum,Yeary, Luce & Zhou submitted). Majority violationsof stochastic dominance and coalescing have beenobserved whether branches are presented in juxtaposedformat or with two other arrangements and whetherbranches are presented in increasing or decreasing orderof consequences (Birnbaum & Martin, 2003). They havebeen observed whether probabilities are presentednumerically or accompanied by pie charts that see-mingly reveal dominance. They are found both with andwithout financial incentives. They are observed with

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highly educated people as well as students (Birnbaum,1999b, 2000). They are found with students tested inclass, in the lab, or via the Web (Birnbaum & Martin,2003). They are observed whether probabilities arepresented as decimal fractions, as natural frequencies,or as lists of equally likely consequences (Birnbaum &Yeary submitted; Birnbaum et al. submitted). They areeven found when gambles are presented with decumu-lative probabilities, which should help people seestochastic dominance, because they do not need to useaddition to compute the probability of getting a certainprize or higher. The present data show that violations ofstochastic dominance are observed whether choices areframed by the same marble colors or not. This growingcollection of null findings represents a waste basket offailed attempts to explain violations of CPT bymechanisms other than coalescing, which has substan-tial effects in all of these different studies despite surfacedifferences in how splitting/coalescing appears indifferent formats.

Similar results are also obtained whether people makechoices between gambles or judge buying and sellingprices of G and G on different trials (Birnbaum &Beeghley, 1997; Birnbaum & Yeary submitted). Viola-tions of stochastic dominance in judgments of valuesuggest that one should look to theories of theevaluation of gambles, rather than models that focuson contrasts or comparisons between gambles. In otherwords, if judgments, which (necessarily) satisfy transi-tivity, also show violations of CPT, one infers thatcomparison processes or other mechanisms specific tochoice seem less plausible than theories of the evaluationof the gambles.

4.4. Predicting choice percentages

This study used the prior TAX model to accuratelypredict majority choices; however, one might want topredict exact (numerical) choice percentages. Birnbaumand Chavez (1997) used an approximate model forpredicting choice percentages. Their model, like themost restricted case of Thurstones law of comparativejudgment or Luces choice model, assumed that choicepercentages are a function of utility differences betweengambles. However, such models are oversimplified, sincethey do not distinguish utility difference from ease ofcomparison.

For example, in Choice 5 of Table 4, the TAX modelimplies a utility difference of 63:1 45:8 17:3; and theempirical choice percentages range from 73% to 85%.However, in Choice 11, the utility difference is muchsmaller, 51:4 53:1 1:7; but the choice percentagesare more extreme, ranging from 11% to 15%. Here thesmaller absolute utility difference produced the moreextreme choice percentage. Similar results can be foundin other cases in the tables. To account for such data,

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one can use a choice model in which the predicted choicepercentage is a function of the difference in utilitydivided by a parameter representing the difficulty ofdiscrimination (see Diederich & Busemeyer, 1999).Despite the small difference, it is easy to see that thefirst gamble is better in Choice 11. The parameterrepresenting the difficulty of discrimination can bethought of as the standard deviation of the difference.Jerome Busemeyer (pers. comm., 2003) is currentlyworking on ideas for calculating this standard deviationfor the case of independent gambles.

4.5. Are tests of CPT unfair?

It is sometimes claimed that because CPT has beenaxiomatized, it must satisfy axioms like branchindependence. Although branch independence is a clearprinciple, it is certainly not an axiom of either CPT orTAX. Branch independence should be violated accord-ing to both CPT and TAX (given their prior para-meters), so it can hardly be an axiom of either class oftheories. These prior CPT and TAX models predictopposite types of violations of this property, however,when their parameters have been chosen to explain theAllais paradoxes.

In this paper, CPT was granted its equation (whichimplies violation of branch independence) and theoption of invoking the editing rule of cancellation(which satisfies branch independence). The present data,however, show that even with this extra flexibility CPTcan still be rejected. Because CPT was allowed two ofthree possible outcomes, whereas TAX was granted onlyone, the test was not equally fair to both models. Thepoint is, however, that the theory that was granted thelarger space of compatible outcomes was the onerejected by the data.

Prospect theories were also granted a larger space ofpossibilities in Table 1, since they were allowed toinvoke or not invoke two editing principles and twoequations. It is difficult to calculate how many freeparameters are consumed by an editing rule that may ormay not be exercised, but it should be clear that theediting rules permit more flexibility to prospect theoriesthan to TAX or RAM.

If we treat the editing rules of combination andcancellation as free-standing scientific hypotheses(rather than as post hoc excuses or as partial tendenciesthat operate only in some of the people some of thetime), then they can be rejected by the present results inTables 2 and 3. Both of these editing rules aresystematically violated by significant departures.

Coalescing was not initially stated as an axiom ofRDU, RSDU, or CPT (Tversky & Kahneman, 1992;Luce & Fishburn, 1991, 1995; Wakker & Tversky, 1993).Birnbaum (1997) used coalescing to simplify his proofsof lower and upper cumulative independence and

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considered coalescing as a testable theorem that can bederived from the RDU/RSDU/CPT representation.Luce (1998) subsequently showed that this simpleproperty can be treated as an axiom, and showed thatit forces RDU in the context of a fairly general class ofrank-weighted utility models. The point, however, isthat the property was derived as a theorem implied bythe RDU representation before it was used as an axiomto derive the representation. Thu

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