(Advanced) Decision Theory - UAB Barcelonapareto.uab.es/mballester/Teaching_files/DecisionTheory7.pdfDecision theory appears to be at a crossroad, in more sense than one. Over half

(Advanced) Decision Theory

Miguel A. Ballester

Universitat Autonoma de Barcelona (BGSE, MOVE)

2011-2012, IDEA program

Course Evaluation

I Problem Set

I Report

I Paper

I Take-Home Exam

I Participation, including paper presentation

Problem Set

Exercises appear in the slides. Some of them (if not all !!) requireconceptual discussion. Each exercise must be written in at mostone page (one side).

Due Time: December 7th, 8 pm (box or e-mailed)

Report

List of possible papers to report in the slides. There is a group ofsix papers with close discussions. The first five must be reportedby two students that can discuss the paper but have to reportindividually. The last paper will be reported by a single student(notice however that the author of this paper is available fordiscussion at the department). Use a maximum of three pages(one side) for your reports.

Paper Assignment: November 7th, 11.10 am (students willcome with a proposal to class. If the assignment fails to satisfy theconditions above, I will impose an assignment)

Due Time: November 28th, 8 pm (box or e-mailed)

Paper

Students will divide in groups of 3/4 to write a paper proposal onthe topics of the course. Group Formation: November 7th,11.10 am (students will come with a proposal to class. If theassignment fails to satisfy the conditions above, I will impose anassignment). Each group will use a maximum of four pages (oneside) for the paper. The paper has to contain:

I A novel question in decision theory.

I The model to address it.

I Conjectures or Hypothetical results (credible!).

I Interpretation and Conclusions.

First Proposal Limit: November 30th, during the day (office)

Due time: December 21st, before presentations that willstart at 10 am

Take-Home Exam

Students will be given few questions involving between half a dayand one day of work. They will be e-mailed December 19th, 8pm, and after solved individually, they must be handed back (boxor e-mailed) before December 21st, 9 am (please notice thiscoincides with the presentations day).

Participation

Participate, you are here because you want !

Introduction

What is Decision Theory? What is Behavioral Economics?

Introduction

I A theoretical/descriptive research field aiming to helpunderstanding and driving economic phenomena through theunderstanding of individual decisions?

I A theoretical/descriptive research field in the service of(individual and collective) normative economics by helping usto understand how individuals make decisions?

I A purely normative enterprise whose goal is to help individualdecision makers pursue their own goals?

How people make/should make decisions?

Introduction

Paraphrasing Gilboa:

Decision theory appears to be at a crossroad, in more sense thanone. Over half a century after the defining and, for many years,definitive works of the founding fathers, the field seems to beasking very basic questions regarding its goal and purpose, its mainquestions and their answers, as well as its methods of research.

This soul searching is partly due to empirical and experimentalfailures of the classical theory. Partly, it is the natural developmentof a successful discipline, which attempts to refine its answers andfinds that it also has to define its questions better. In any event,there is no denying that since the 1950s decision theory has notbeen as active as it is now.

Introduction

Homo Oeconomicus

I Rationality as Maximization: max U(·)I Consistency in dealing with uncertainty:

∑p(z)U(z)

I Consistency in dealing with time:∑ρtU(xt)

I Choice from Menus. maxx∈A U(x)

I Non-Altruistic Preferences: Ui (xi )

I Game Theoretical considerations

I Advanced Decision Theory and Economics: ExperimentalEconomics, Game Theory, I.O., Macroeconomics...

ADVANCED DECISION THEORY

I Riskless Choice**II Choice and Uncertainty*III Menu Choice*

Complement with Time Preferences (Temporal inconsistencies,hyperbolic discounting...)

I RISKLESS CHOICE

1 Standard Individual Decision-Making2 Basic Violations of Consistency3 Reference-dependent models4 Sequential models5 Other Multicriteria models6 Search models7 A classification of cyclical models8 Measuring Consistency9 Welfare analysis

1 STANDARD INDIVIDUAL DECISION-MAKING

1.1 Preference and Utility1.2 Choice Behavior1.3 Rationalizability in a consumer framework1.4 Rationalizability in a general framework

1.1 Preference and Utility

Preferences/Maximization

I Preference over alternatives, as binary relations R ⊆ X × X .

I Greatest Alternatives y ∈ A : yRx for all x ∈ AI Undominated Alternativesy ∈ A : for all x ∈ A, not (xPy) = max(A,R)

I Consistency on preferences: xRy , yRz ⇒ xRz (transitivity),absence of cycles with some strict relation (acyclicity),comparability (completeness), . . .


Utility/Maximization

I Utility Function u : X → RI Maximization: max u(·) and arg max u(·)


A utility function u : X → R represents the preference R if, for allx , y ∈ X : xRy ⇔ u(x) ≥ u(y)

Representation problem: Which preferences can be represented?

The finite and countable case. Transitivity (and completeness) asthe only requirement. Hint: measure the lower contour sets ofalternatives by counting or by weighting-counting each of thealternatives which is inferior

U(xj) =∑

k:xjPxk

1k

U(xj) =∑

k:xjPxk

(1

2)k


The general case: Transitivity (and completeness) is not enough.

Let X be the unit square, that is, X = [0, 1]× [0, 1]. Let x %k y ifxk ≥ yk . The lexicographic preferences %L induced from %1 and%2 are:

(a1, a2) %L (b1, b2) if a1 > b1 or both a1 = b1 and a2 ≥ b2


Proposition The lexicographic preference relation %L on[0, 1]× [0, 1] does not have a utility representation.

Proof: Assume, by contradiction, that u : X → R represents %L.For any a ∈ [0, 1], it is (a, 1) L (a, 0). Hence, it must beu(a, 1) > u(a, 0) and there exists a rational number q(a) in theinterval (u(a, 0), u(a, 1)). The function q : [0, 1]→ Q should be aone-to-one function because if b > a, then (b, 0) L (a, 1) andhence, u(b, 0) > u(a, 1) and q(b) > q(a). But the cardinality ofthe rational numbers is lower than that of the continuum, acontradiction.

The general case: A preference can be represented if and only ifthere exists a countable dense set

Continuous representations: Debreu (1954), Eilenberg (1941),Rader (1963)

1.2 Choice Behavior

Samuelson (1938): I propose, therefore, that we start anew indirect attack upon the problem, dropping off the last vestiges ofthe utility analysis. This does not preclude the introduction ofutility by any who may care to do so, nor will it contradict theresults attained by use of related constructs. It is merely that theanalysis can be carried on more directly, and from a different set ofpostulates. All that follows shall relate to an idealised individualnot necessarily, however, the rational homo-economicus.

I assume in the beginning as known, i.e., empirically determinableunder ideal conditions, the amounts of n economic goods whichwill be purchased per unit time by an individual faced with theprices of these goods and with a given total expenditure. It isassumed that prices are taken as given parameters not subject toinfluence by the individual.

1.2 Choice Behavior

Samuelson identifies choice as a more fundamental notion thanpreference mostly because the latter is not observable while theformer is. A natural question thus arises:

Rationalizability problem: Can we uncover the utility/preferencesof a decision maker by observing the choices she makes whenchoosing a commodity bundle from her budget set?

1.2 Choice Behavior

Choice Behavior Let X be a set of alternatives. A choice behavioris a mapping c : Ω ⊆ 2X \ ∅ → X such that c(A) ∈ A. Theelements of Ω are called menus.

I Consumer Choice: X ≡ Rn, Ω is a set of compact sets definedn∑

i=1

pixi ≤ m.

I Universal Domain: X is a general set of alternatives,Ω = 2X \ ∅

I Binary Domain: X is a general set of alternatives, Ω subsetsof two alternatives.

Choice correspondences versus choice functions.

1.3 Rationalizability in a consumer frameworkRationalizability: Existence of a preference relation over bundlessuch that xt is the maximal/greatest bundle achievable at prices pt .Key observation by Samuelson: . . . if an individual selects batchone over batch two, he does not at the same time select two overone.

A preference for the first batch is said to be revealed. The logicbehind this terminology is straightforward: A decision maker whoviews an alternative as superior to another will never choose thelatter when the former is available (Stochastic Choice !)

Revealed Preference Given some vectors of prices and chosenbundles (pt , x t) for t = 1, . . . ,T we say x t is directly revealedpreferred to a bundle x (written x tRDx) if ptx t ≥ ptx .

Weak Axiom of Revealed Preference If x tRDx s then it is notthe case that x sRDx t . Algebraically, ptx t ≥ ptx s impliespsx s < psx t .

1.3 Rationalizability in a consumer frameworkSamuelson (1948) offered a graphical proof of rationalizability withtwo commodities under the Weak Axiom. Houthakker (1950)offered a formal proof for any dimension by using the StrongAxiom.

Indirect Revealed Preference We say x t is (indirectly) revealedpreferred to x (written x tRI x) if there is some sequencer , s, t, . . . , u such that prx r ≥ prx s , psx s ≥ psx t , . . . , puxu ≥ puxThe relation RI is the transitive closure of the relation RD .

Strong Axiom of Revealed Preference If x tRI xs then it is not

the case that x sRI xt .

Generalized Axiom of Revealed Preference If x tRI xs then

psx s ≤ psx t

Exercise 1: We can just rewrite SARP in the form...if x tRI xs then

it is not the case that x sRDx t . Why? Can you think of otheralternative ways to write this axiom?

1.3 Rationalizability in a consumer framework

Rose (1958) offered a formal argument that the Strong Axiom andthe Weak Axiom were equivalent in two dimensions, providing arigorous proof for Samuelson’s earlier graphical exposition.

Afriat (1967) contributed by showing a general version admittingindifferences (GARP) and a constructive method for the utilityfunction.

1.3 Rationalizability in a consumer framework

I Varian (2005)

I Rubinstein (2007)

1.4 Rationalizability in a general framework

Rationalizability A choice function c is rationalizable if thereexists a well-behaved preference P such that c(A) = max(A,P).


Revealed Preference (choice functions) Alternative x isrevealed better than alternative y if y ∈ A and c(A) = x .

Weak Axiom of Revealed Preference (choice functions) Ifx , y ∈ A∩B and x = c(A) then it is not the case that y = c(B).

Chernoff Condition (α condition or IIA). If x ∈ S ⊆ T andx = c(T ) then it must be the case that x = c(S).

See Chernoff (1954) and Sen (1969, 1971). For a simple survey onthe classical ideas, see Moulin (1984).


Proposition: The following statements are equivalent (in theuniversal domain):

1. A choice behavior c is rationalizable

2. c satisfies WARP

3. c satisfies the α property.


Exercise 2: Check whether WARP is sufficient to guaranteerationalizability in a domain Ω which is not universal (a domainthat contains only some but not all menus). Formulate the StrongAxiom of Revealed Preference in this general framework and checkwhether it is sufficient to guarantee rationalizability.


Proof: 1⇒ 2. Let P the preference that rationalizes c . Considertwo menus A and B and two alternatives x , y ⊆ A ∩ B. Ifx = c(A) then, x = max(A,P). In particular, xPy . Hence,y 6= max(B,P) = c(B).

2⇒ 3. Let x ∈ S ⊆ T and x = c(T ). If y = c(S) with y 6= x , wewould have a violation of WARP involving menus S and T andalternatives x , y .

3⇒ 1. Define the binary revealed preference xPBy ⇔ x = c(xy).We show that PB is transitive. Suppose xPBy and yPBz andconsider the set x , y , z. It cannot be c(xyz) = y as this is acontradiction with c(xy) = x . It cannot be c(xyz) = z as this is acontradiction with c(yz) = y . Hence, c(xyz) = x . By the αproperty, c(xz) = x and thus xPBz . Clearly, PB rationalizes c.


Expansion x = c(A) and x = c(B) imply x = c(A ∪ B).

Always Chosen x = c(xy1), . . . , x = c(xyk) implyx = c(xy1 . . . yk).


Proposition: A choice behavior is rationalizable if and only if: (i)the Binary Revealed Preference is acyclical and (ii) it satisfies thealways chosen property.

Proof: We only need to prove the sufficiency part. Since thechoice behavior does not contain any binary cycle, finitenessguarantees that PB is well-behaved. Let A be a subset ofalternatives, and consider max(A,PB). This element dominatesany other alternative in A and hence, it is chosen in all binaryproblems. By the always chosen property, we are done.

See Tyson (2008) for extra discussions on the Binary RevealedPreference and refinements of rationalizability.

2. BASIC VIOLATIONS OF CONSISTENCY

2.1 Existence of Binary Cycles2.2 Status Quo and Endowment Effect2.3 Attraction Effect2.4 Compromise Effect2.5 Other Effects

2.1 Existence of Binary Cycles

The cycle of order three is well-known in voting, being calledCondorcet Cycle. In essence, majority voting over the pairs ofalternatives in a triple does not entail necessarily a transitiverelation.

This natural violation of transitivity may appear in many choicesituations in which aggregation of criteria sounds reasonable. Nocyclical choice is observed when choosing from amounts of cash !

Consider an agent that studies a collection of items by focusing ona group of characteristics/components common to all these items.Circularities thus may arise if items are ordered in conflicting waysaccording to each component.


May (Econometrica 1954). College students face the choice ofthree hypothetical marriage partners diferring in intelligence, looksand wealth.

x y z

Very Intelligent Intelligent Fairly Intelligent

Plain Looking Very Good Looking Good Looking

Well-off Poor Rich


In a very simple presentation, 27 percent of students presentedcyclical preferences of the type xPByPBzPBx .

The intransitive pattern is easily explained as the result of choosingthe alternative that is superior in two out of three criteria.

Other cyclical experiments: Tversky (1969), Loomes, Starmer andSugden (1991), Roelofsma and Read (2000), 20-50 percent ofcyclical choice.

I Waite (2001). Birds prefer to take 1 raisin from a 28cm tubeto 2 raisins from a 42cm tube, and the latter to 3 raisins froma 56cm tube. However, they choose the last confronted to thefirst.

I Female Lions always choose the biggest male lion (compatiblewith transitivity of choice in cash environments).

2.2 Status Quo and Endowment Effect

Experimental evidence has revealed that Individuals are reluctantto make changes in their current status (often called Status QuoBias). Consider the following hypothetical description of threeprofessional positions.

x y z

Very Stimulating Work Fairly Stimulating Work Stimulating Work

Pleasant Life Very Pleasant Life Fairly Pleasant Life


While having no particular work (studying), the agent declares aninformal preference for position y over position x .

The agent gets position z . After some time working, she receivesan offer to work in position y . It comprises a much better qualityof life but a less stimulating work. Doubting on it, the agentrejects the offer.

The agent receives an offer to work in position x . The offer, beingbetter than the original position in all dimensions, is accepted.

The previous example conforms a cycle among alternatives x , y , z .


Samuelson and Zeckhauser (1988) constitutes the first and widestanalysis of SQB.

Each question begins with a brief description of an individual, amanager, or a government policymaker, followed by a set ofmutually exclusive alternative actions from which to choose. Thesubject plays the role of the decision maker and in many of thedecisions, one alternative occupies the status quo position(externally framed or self-selected in a prior stage).

Presence of (statistically significant) status quo bias.


Knetsch (1989) provides the simplest experiment on SQB.

Mug Candy

Mug, 76 89 percent 11 percent

Candy, 87 10 percent 90 percent

choice, 55 56 percent 44 percent


See Masatlioglu and Uler (2011) for a modern experiment that alsocompares explanatory theories.

These phenomena are also observed in real markets by means offield studies:

I Insurance: Madrian and Shea (2003)

I Organ donations: Johnson and Goldstein (2003)

I Residential Electric Service: Hartman, Doane and Woo (1991)

I Religious Choices: Chaves and Montgomery (1996)


Individuals are likely to value an alternative they possess more thanone that they do not (often called the Endowment Effect). Theusual consequence of the endowment effect is the willingness toaccept (WTA)/willingness to pay (WTP) gap, i.e., the fact thatWTA is larger than WTP. Consider the following hypotheticalsituation

I A person moves to a new apartment and finds there an olddrawing. She is offered 100 euros for the drawing and rejectsthe offer, thus revealing that x = (1, I )PBy = (0, I + 100).

I If the person had moved to the new apartment and found 100euros, she would have wished to buy the drawing for no morethan 50 euros in the paintings shop, thus revealing thaty = (0, I + 100)PBz = (1, I + 30).

The previous example conforms a cycle among alternatives x , y , z .


Thaler (1980) provides evidence of the Endowment Effect, mainlyon lottery choices. Kahneman, Knetsch and Thaler (1990) showsimilar results for a trading experiment. For the most basicconsumption setup on bundles, see for instance Bateman et al.(1997, 2005).


Suggested explanations for these effects:

I Experimental Misconceptions: Plott and Zeiler (2005, 2007,2011).

I Rational explanations: Transaction costs, uncertainty.

I Cognitive Misperceptions: Perceived losses are more relevantthan perceived gains, switching from the SQ may generate aloss in some attribute/value, or a mere loss. Similar toanchoring process where we adapt views of the world relativeto the current position but in an insufficient amount.

I Psychological Commitment: sunk costs as relevant even if thisis not rational, to justify previous commitments orinvestments, i.e., people like to think their decisions werecorrect or avoid regret in future moments about past decisionsor feel in control of their actions and thus, avoid difficultchoices of any kind.


Exercise 3: Read in detail one of the recent suggestedempirical/experimental papers (last 20 years) and describe in detailone of the cycles that could appear as a consequence of theirrational behavior analyzed there. Might you suggest a rationalexplanation for the cycle? Alternatively, might you suggest aplausible cognitive failure that explains the cycle?

Exercise 4: Think of other real life example fitting these effectsand look for empirical evidence or literature supporting thealternative example.

2.3 Attraction EffectThe attraction effect refers to the ability of an asymmetricallydominated or relatively inferior alternative, when added to a set, toincrease the choice probability of the dominating alternative.

For a first observation, see Huber, Payne and Puto (1983).Students are asked to choose products (different categories), insets of two or three. The alternatives in each category represent atarget, a competitor and a decoy, that differ in several dimensions.

The target and the competitor do not dominate each other, butthe target dominates the decoy, and thus, a stimulus for the target.

The effect of the decoy is tested by checking the percentages inwhich target and competitor are chosen, with and without decoy.A greater percentage of target under the decoy treatment wouldsuggest that some students fall in the attraction effect. Theywould choose the competitor in the two alternative case but thetarget in the three alternative case.

2.3 Attraction Effect

It is observed an increase in Target Demand from 50 percent to 59percent.

The effect is stronger when the decoy shares with the target thedimension in which the target is better, and less important if thedecoy is mainly dominated in terms of the dimension in which thetarget is worse.

2.3 Attraction Effect

The attraction effect constitutes a violation of always chosen. Letx , y and z , where the target y clearly dominates the decoy z andhence yPBz . The competitor x is perceived as better than thetarget and the decoy, xPBy and xPBz . Hence, no cyclical choice ispresent.

However, the target is chosen when both the competitor and thedecoy are present, i.e., c(xyz) = y .

See also Shafir, Simonson and Tversky (1993).

2.4 Compromise EffectThe compromise effect refers to the ability of an extreme (but notinferior) alternative, when added to a set, to increase the choiceprobability of an intermediate alternative.

For a first observation, see Simonson (1989). The following is anexample from Herne (1997).

The alternatives were imaginary policy proposals concerningvarious topics from everyday politics (such as economic conditions,social security, taxation, environmental questions). Respondentswere asked to make their choices as a member of parliament, as amember of the local council, as a voter in a referendum, etc.

The amount of people who made a choice for the compromiseoption is smaller in the 3 alternative set versus the 2 alternativeset. However, the percentage of people who selected thecompromise alternative among those who did not select the newalternative increased from 64 to 75 percent.

2.4 Compromise Effect

The compromise effect can operate as the attraction effect or in amore crude way. Consider x , y and z , where the intermediatealternative y is never chosen against the extreme alternatives xand z . Hence, no cyclical choice is present. However, theintermediate alternative is chosen when both extremes are present,i.e., c(xyz) = y . This is also called a difficult choice.

2.5 Other Effects

Several empirical papers have reported on order effects in paneldecisions in contests such as the World Figure Skating Competition(Bruine de Bruin 2005), the International Synchronized SwimmingCompetition (Wilson 1977), the Eurovision Song Contest (Bruinede Bruin 2005) and the Queen Elisabeth Contest for violin andpiano (Glejser and Heyndels 2001).

Other empirical observations on how the alternatives areconsidered in an structured way, and the models that explain thiscan be seen in Caplin, Dean and Martin (Forthcoming). Considerthat, in general, Presentation/Description of the alternatives(framing effect) has been deeply analyzed.

3. REFERENCE DEPENDENT MODELS

3.1 Loss Aversion and Constant Loss Aversion models3.2 Anchored Preferences3.3 Status Quo Bias3.4 Endogenous version: Default Choice3.5 Other Models

3.1 Loss Aversion

Tversy and Kahneman (1991) introduce the firstReference-dependent model where losses and gains are definedrelative to a reference-point.

I Primitives are preferences over consumption bundles with ndimensions.

I For each reference point r, which is a consumption bundle,the decision-maker has preferences %r, and they assume theexistence of Ur

I Ur (x) =∑

i gi (ui (xi )− ui (ri )).

I Concavity of gi for a > 0. Diminishing sensitivity for gains.

I Convexity of gi for a < 0. Diminishing sensitivity for losses.

I gi (0) = 0 and gi (a) < gi (−a) captures loss aversion.

3.1 Loss Aversion

The theory is useful in a context of uncertainty (Prospect Theory),where there is one dimension, money. In a multidimensionalgeneral setting, however, it allows for very controversial behaviors.In particular, Reference-dependent choice cycles are permitted.

(Munro and Sugden, 2003): An agent may have preferences of theform y %x x , z %y y and x %z z , allowing for money pumps.

3.1 Constant Loss Aversion

Munro and Sugden (2003): Constant Loss-Aversion model

I gi is linear both on gains and losses.

I gi (a) < gi (−a) captures loss aversion.

Exercise 5: Exemplify with a two-dimensional setup why lossaversion models may present reference-dependent cycles. Showthat, if g1 and g2 are linear, this is not possible.

3.2 Anchored preferences

Sagi (2006): A model of Reference-dependent preferences forlotteries, alternative to prospect theory. It can be adapted toriskless choice.

I Primitives are preferences.

I Ur (x) = infu∈∆ u(x)− u(r), where ∆ is a set of real,continuous and bounded utility functions on X .

3.2 Anchored preferences

A graphical comparison of models.

3.3 Status Quo Bias

Masatlioglu and Ok (2005) present a model of choice with SQB.They differentiate decisions where the SQ is present and decisionswhere there is not a SQ.

I The latter are solved in a standard way, by maximizing astandard preference/utility over the set of alternatives.

I The former are solved by considering the set of alternativesthat dominate the SQ in all attributes. If this set is empty, theagent sticks at the SQ. If the set is non-empty, the agentmaximizes the standard preference/utility over such a set.

The model assumes the possibility of observing the Status Quo ofthe agent, and thus, deals with data of the form c(A, x) where A isthe menu, x is the status quo (if exists) and c(A, x) is the choice.

3.3 Status Quo Bias

SQ WARP If x , y ∈ A ∩ B and x = c(A, z) then it is not thecase that y = c(B, z).

SQ Independence If x 6= c(T , x) for any x ⊂ T ⊆ S , thenc(S , x) = c(S , ).

SQ Bias If y = c(S , x) then y = c(S , y).

3.3 Status Quo Bias

Theorem. A Choice Model with Status Quo satisfies WARP, SQIand SQB if and only if there exists a positive integer q, an injectivefunction u : X → Rq and a strictly increasing map f : u(X )→ Rsuch that:

c(S , x) =

x if y ∈ S : u(y) > u(x) = ∅,arg max

z∈y∈S :u(y)>u(x)f (u(z)) otherwise

3.3 Status Quo Bias

Proof:Step 1: Define the binary revealed preference when the status quois discarded (thus, clearly inferior even with the status quo plus).That is, x % y ⇔ x = c(xy , y). This relation is transitive.

Let x % y and y % z . Then x = c(xy , y) and y = c(yz , z). ByWARP, the second equality guarantees that z cannot be thechosen alternative in (xyz , z). By WARP, the first equalityguarantees that y cannot be the chosen alternative in (xyz , y). BySQB, y cannot be the chosen alternative in (xyz , z). Hence,c(xyz , z) = x and by WARP, c(xz , z) = z . Thus, x % z .

Step 2. Consider all linear orders extending %, P1, . . . ,Pn. We canrepresent them by a vector of utility functions u = (u1, . . . , un).

3.3 Status Quo Bias

Step 3. Define a transitive and complete preference P over all thealternatives by using the choices without status quo, c(·, ).Represent them by a utility function v .

Step 4. v is monotone on u.

Suppose u(x) > u(y). Then, it must be x % y and hence,x = c(xy , y). By SQB, it must be x = c(xy , ). Hencev(x) > v(y). We can thus write v = f (u) in a monotone way.

3.3 Status Quo Bias

Step 5. Final proof.

Suppose first that y ∈ S : u(y) > u(x) = ∅. Then, by definition,x = c(xy , x) for all y ∈ S , and by WARP, it must be x = c(S , x),as desired.

Suppose now that y ∈ S : u(y) > u(x) = A 6= ∅. Consider theset A ∪ x. For any subset T , it must be c(T , x) 6= x since x isdominated by all elements in A. Hence, by SQI, it must bec(A∪x, x) = c(A∪x, ) = arg maxz∈y∈S :u(y)>u(x) f (u(z)),as desired.

3.4 Endogenous version: Default Choice

Reference-Dependent models are usually exogenous, in the sensethat the reference point is an exogenous parameter.

Testing these models require thus to obtain data both on the set ofavailable options and the reference point at the moment of choice.

The validity of this assumption depends on the particularinterpretation of the reference point (status quo, default option,etc)


The simplest endogenous version of Masatlioglu and Ok’s modelcan be seen in Apesteguia and Ballester (2011a):

Default Choice: A choice function c is a default choice, DC, ifthere exists an element d ∈ X , a positive integer q, an injectivefunction u : X → Rq and a strictly increasing map f : u(X )→ Rsuch that:

I For all S ⊆ X with d 6∈ S , it is c(S) = argmaxx∈S f (u(x)).

I For all S ⊆ X with d ∈ S it is:

c(S) =

d if S ∩ X d = ∅,arg maxy∈S∩X d f (u(y)) if S ∩ X d 6= ∅.

with X d = x ∈ X : u(x) > u(d)


DC satisfy Expansion

Proof: Let x be chosen in sets S and T . If x = d , then itdominates each of the alternatives in at least one attribute. Hence,it would be chosen in S ∪ T . If x is not the Default, andd 6∈ S ∪ T , standard maximization satisfies Expansion. Finally, if xis not the Default but d ∈ S ∪ T , then the chosen element inS ∪ T is the alternative maximizing f (u) in (S ∪ T ) ∩ X d , call it,z . Since x ∈ (S ∪ T ) ∩ X d it must be f (u(z)) ≥ f (u(x)). Letwlog, z ∈ S . Then x , z ∈ S ∩ X d and hence, f (u(x)) ≥ f (u(z).It can only be z = x , as desired.


DC satisfy Cyclical Choice Consistency If we observe twocyclical chains xPByPBzPBx and aPBbPBcPBa, then there mustexist t ∈ x , y , z ∩ a, b, c such that c(xyz) and c(abc) are thealternatives that dominate t.

Proposition A choice function c is a DC if and only if it satisfiesExpansion and Cyclical Choice Consistency.


Proof: If there are no binary cycles, since c satisfies AC, it mustbe rationalizable. Hence, we can simply find a preference P and autility representation u for which the result holds.

If there are binary cycles, cyclical consistency guarantees that thereexists one alternative d present in all of them and such that thechoice in the triple is the alternative dominating d .

Consider the sets A = x ∈ X : xPBd, B = x ∈ X : dPBx.


Proof: Define P1 on X \ d as xPiy ⇔ xPBy . Complete it byplacing d on the bottom of the preference.

Place AP2dP2B, with P2 on A as xPiy ⇔ xPBy and P2 on B asxPiy ⇔ xPBy .

Represent Pi with the lower contour sets utility function ui . Finallydefine f (u(x)) = u1(x) + εu2(x). Also notice that A = X d for thegiven representation u.


If d 6∈ S , S presents no cycle and AC guarantees that thealternative selected is the maximal one according to PB or P1.Given the structure of f , this is the element that maximizes f , asdesired.

If d ∈ S and dPBx for all x ∈ S , then by AC, d = c(S). This ismerely what we desire as A ∩ S = ∅.

If d ∈ S and A∩ S 6= ∅, we need to prove that c(A) is the maximalalternative in A = X d according to f (u). To prove it, consider thatmaximal alternative z and any other y ∈ S . If zPBy , given that wealso have zPBd we can use AC to derive z = c(zyd). If yPBz itmust be y 6∈ A, because in the set A, f (u) is merely equivalent tou1 or u2 and z was the maximizer. Hence, we know thatzPBdPByPBz and by cyclical consistency, it must be z = c(zyd).The union of all triples zyd is merely S and using expansion, wehave c(S) = z , as desired.


SQB

Cycles NO

SQ cyles YES

Attraction NO

Difficult NO

Exercise 6: Consider three alternatives x , y and z , and a StatusQuo Model with two utility functions u1 and u2 (two attributes ortwo selves). Prove formally that the attraction and the difficultchoice effects cannot happen. Hint: Notice that this has beenalready proved in a more general form before, I am just asking toprove it in a different way using the model directly!

3.5 Other models

I A Model of Reference-Dependent Preferences (Koszegi andRabin, 2006): lotteries and expectations.

I A theory of reference-dependent behavior (Apesteguia andBallester, 2009): Bridge between choice and preference.Rationalizability.

I Money matters: an axiomatic theory of the endowment effect(Giraud, 2011): A monetary model to analyze axiomaticallythe endowment effect (WTA-WTP).

I Extended SQB: Extension of Rational SQB paper, not onlyPareto domination (Masatlioglu and Ok, 2011). Endogenousextended version in Ok, Ortoleva and Riella (2011).

I Aspiration levels: External Reference-Dependent behavior.Guney, Richter and Tsur (2011).

4. SEQUENTIAL MODELS

4.1 Elimination by Aspects and Lexicographic Semiorders4.2 Sequential Rationalizability4.3 Consideration Sets4.4 Rationalization and Categorize Then Choose4.5 Other Models

4.1 Elimination by Aspects and Lexicographic Semiorders

I Elimination by Aspects (Tversky 1972): The set ofalternatives is narrowed down sequentially by dropping inferioralternatives with respect to some aspect/attribute. Randommodel of choice.

I Lexicographic Semiorders (Tversky 1969): Preference isgenerated by the sequential application of numerical criteria,by declaring an alternative x better than an alternative y ifthe first criterion that distinguishes between x and y ranks xhigher than y by an amount exceeding a fixed threshold.Model of preferences based on the semiorder concept (seeLuce 1956).

4.2 Sequential Rationalizability

I The DM applies a number of criteria (incomplete binaryrelations) in a fixed order of priority, gradually narrowing downthe set of alternatives, until one is identified as the choice

I Same criteria applied in the same fixed order to every choiceproblem

I Individual choice: multiple criteria or selves, orderly applied

I Collective choice: refining the set of efficient allocations witha fairness notion


Manzini and Mariotti (2007):

Sequential Rationalizability by Asymmetric Rationales(SR(As)): A choice function c is sequentially rationalizable byasymmetric rationales or simply SR(As) whenever there exists anon-empty ordered list P1, . . . ,PK of asymmetric rationales onX such that c(A) = MK

1 (A) for all A ∈ P(X ), whereMK

1 (A) = M(M(. . .M(M(A,P1),P2), . . . ,PK−1),PK )


SR satisfy Always Chosen

Proof: Let x be chosen among alternatives y1, . . . , yk . We provethat x ∈ M j

1(x , y1, . . . , yk) for every j inductively. It is obviousfor j = 1, since otherwise yiP1x for some i and hence, x 6= c(xyi ).If x ∈ Mt

1(x , y1, . . . , yk), we prove thatx ∈ Mt+1

1 (x , y1, . . . , yk). Suppose that there existsyi ∈ x , y1, . . . , yk such that yiPt+1x . Then, since x = c(xyi ),there must exist p < t + 1 such that xPpyi and hence,yi 6∈ Mt

1(x , y1, . . . , yk). This proves thatx ∈ Mt+1

1 (x , y1, . . . , yk).


The case of two rationales.

Weak WARP If x , y ∈ B ⊆ A and x = c(xy) = c(A) then it isnot the case that y = c(B).

Proposition A choice function c is a 2-SR if and only if it satisfiesExpansion and Weak WARP.


Proof: xP1y if not yPDx and xP2y if xPBy .

Let c(S) = x . Clearly, there does not exist w ∈ S such that wP1x ,since xPDw thanks to set S . Then x survives the first round.

We now prove that for any y ∈ S such that yP2x , y is eliminatedin the first round. Otherwise, for all z ∈ S \ y , there exists Tyz

such that c(Tyz) = y . By Expansion, y = c(∪Tyz). Sincex , y ⊂ S ⊂ ∪Tyz , Weak WARP guarantees that x 6= c(S),absurd. Hence, y survives the second round too.

If some alternative y survives the first round, we know thatx = c(xy) and thus, xP2y . Hence, x is the only alternative inM2

1 (S), as desired.


SQB SR

Cycles NO YES

SQ cyles YES YES

Attraction NO NO

Difficult NO NO

Exercise 7: Consider three alternatives x , y and z , and a RationalShortlist Method (two attributes or two selves). Prove formallythat the attraction and the difficult choice effects cannot happen.Hint: Notice that, again, this has been already proved in a moregeneral form before !


I Choice by Sequential Procedures (Apesteguia and Ballester2011a). All Sequential procedures are equivalent if theyimpose acyclicity to each attribute. Manzini and Mariotti’sassymetric version is more general. Is it credible?

I Lexicographic compositions of two criteria for decision making(Houy and Tadenuma, 2009). Other Sequential Compositions.

I Manipulation of Choice Behavior (Manzini, Mariotti andTyson, 2011). How can this behavior be manipulated?

4.3 Consideration SetsChoice with Limited Attention (Masatlioglu, Nakajima and Ozbay,forthcoming)

I The revealed argument relies on the implicit assumption thata DM considers all feasible alternatives. Without the fullconsideration assumption, the standard revealed preferencecan be misleading. It is possible that the DM prefers x to ybut she chooses y when x is present simply because she doesnot realize that x is also available.

I The marketing literature calls the set of alternatives to whicha DM pays attention in her choice process consideration set.Due to cognitive limitations, DMs cannot pay attention to allthe available alternatives.

I The common property in the formation of consideration sets isthat it is unaffected when an alternative she does not payattention to becomes unavailable (attention filters).

I The agent proceeds in two-stages, by maximizing awell-behaved preference over the alternatives in herconsideration set, and only these.

4.3 Consideration Sets

Attention Filter A consideration set mapping Γ : 2X \ ∅ → 2X \ ∅,with Γ(A) ⊆ A is an attention filter if for any menu S ,Γ(S) = Γ(S \ x) whenever x 6∈ Γ(S).

Choice with Limited Attention A choice function c is a choicewith limited attention if there exists a preference P and anattention filter Γ such that c(A) = M(P, Γ(A)).


Choice with limited attention are not necessarily SR choices. Thereare attention filters that do not correspond to a maximizingprocess.

I Elements that are the best for one particular characteristic.

I Elements that are the most popular in the set according tosome characteristics.

Exercise 8: Consider a set of four alternatives a, b, c , d. Define achoice function in the universal domain that: (1) it is not arational shortlist method, (2) it is a Limited Attention model.Which of the properties of a Rational Shortlist Method is notsatisfied? Give an intuitive interpretation to the Attention Filterthat you have defined.


I Limited Attention model allows for cycles. Let xPyPz andΓ(xz) = z is the only problem with partial attention. ClearlyxPByPBzPBx .

I Limited Attention model allows for the attraction effect. LetxPyPz and Γ(xyz) = yz the only problem with partialattention. Clearly c(xyz) = y and the binary revealedpreference is consistent with P.

I Limited Attention model allows for Difficult Choices. LetxPyPz and Γ(xy) = y , Γ(xz) = z the only problems withpartial attention. Clearly, c(xyz) = x and the binary revealedpreference is yPBzPBx as in the compromise effect.


SQB SR LAt

Cycles NO YES YES

SQ cyles YES YES YES

Attraction NO NO YES

Difficult NO NO YES


I When More is Less: Choice by Limited Consideration (J.Lleras, Y. Masatlioglu, D. Nakajima and E. Ozbay, 2011).Variant.

I Limited information and advertising in the US personalcomputer industry, Goeree (2008). Awareness of a fraction ofproducts.

I Consideration sets and competitive Marketing (K. Eliaz andR. Spiegler, forthcoming). I.O. market implications, series ofpapers.

I A good application for Neuroeconomics ?! Neuroeconomics:How neuroscience can inform economics (Camerer,Loewenstein and Prelec, 2011)

4.4 Rationalization and Categorize Then Choose

Rationalization (Cherepanov, Feddersen and Sandroni, 2011)

I In a first stage, a decision maker uses a set ofrationales/norms to determine a subset of alternatives, theones she can rationalize i.e., those that are optimal accordingto at least one of her rationales.

I Among the alternatives that she can justify according to somerationale/norm, the agent maximizes a well-behavedpreference.


Categorize Then Choose (Manzini and Mariotti, forthcoming)

I The first stage involves a coarse form of maximization, using abinary relation (interpreted as a psychological shadingrelation) defined on categories, namely sets of alternatives.For instance, the presence of salad dishes in the menu shadespasta dishes, or the presence of hamburgers shades othertypes of sandwiches.

I Then in the second stage the agent picks an alternative whichis preferred to all surviving alternatives.


I Both models are equivalent in terms of explained behavior(choice functions).

I They can explain cycles and the attraction effect.

I They cannot explain difficult choices. Suppose x = c(xz) andy = c(yz) and reason according to rationalization. If zPx ,then z is dominated by x according to allnorms/rationalizations. Hence, it cannot be maximal in xyzfor any norm and z 6= c(xyz). It must be then xPz . Using thesame reasoning, yPz . Thus, in order to choose z from xyz weshould discard x and y . This means x and y are not maximalfor any norm and thus, z is maximal for all the norms. Butthen z would be maximal for all the norms in xz andz = c(xz) absurd.


SQB SR LAt R-CTC

Cycles NO YES YES YES

SQ cyles YES YES YES YES

Attraction NO NO YES YES

Difficult NO NO YES NO

5. OTHER MULTICRITERIA MODELS

5.1 Additive Difference Model5.2 Rationalization by Multiple Rationales5.3 Rationalization by Game Trees5.4 Aggregation and Dual Aggregation5.5 Other Models

5.1 Additive Difference Model

The Additive Model:

I Preferences. Alternatives can be represented in an attributespace Rn and there exist real functions f1, . . . , fn such thatx % y ⇔

∑i fi (xi ) ≥

∑i fi (yi ).

I Choices. Alternatives can be represented in an attribute space[0, 1]n and there exist real functions f1, . . . , fn such thatc(A) = arg max

∑i fi (x).

That model is equivalent to a rational individual.


The Additive Difference Model: Alternatives can be represented inan attribute space [0, 1]n and there exist functions g1, . . . , gn withgi (−a) = −gi (a) such that x % y ⇔

∑i gi (xi − yi ) ≥ 0.

This model guarantees transitivity only if gi are linear. To see this,just notice that∑

i

gi (xi − yi ) ≥ 0⇔∑i

λi (xi − yi ) ≥ 0⇔

⇔∑i

λixi ≥∑i

λiyi

which clearly leads to transitivity.


The general version may explain violations of rationality. However,it can only be defined for pairs of alternatives. How to write areasonable choice version of this model to apply revealed analysis?

I Reference-Dependence version. All alternatives are comparedto some alternative r in the set, receiving value

∑i gi (xi − ri )

I Aspiration level. All alternatives are compared to somenon-feasible ideal. For instance, ai = maxx∈A xi . Thatresembles a regret model, though regret is a concept mostlyanalyzed in decisions under uncertainty like Sugden (1993),Sarver (2008) or Hayashi (2008).

5.2 Rationalization by Multiple Rationales

Kalai, Rubinstein and Spiegler (2002)

I The choice set conveys information about its constituentelements and given this information, the DM chooses what hethinks is the best alternative.

I The DM has in mind a partition of the set of menus and sheapplies one ordering to each cell in the partition.

I A cell is like a state of the world. The DM’s behavior isrationalized after the state of the world is added to thedescription of the alternatives.


Rationalization by Multiple Rationales: A choice function c isRationalizable by Multiple Rationales whenever there exists anon-empty collection P1, . . . ,PK of linear orders X such that forall A ∈ P(X ), there exists i with c(A) = M(A,Pi ).


Every Choice Function is Rationalizable by MultipleRationales

Define P1, . . . ,Pn as a collection of |X | = n linear orders, eachof them placing an alternative x ∈ X on top.

The authors propose to focus on the minimal explanation in termsof number of rationales. The larger such a number, the lessmeaningful is the rationalization by multiple rationales that can begiven to the choice behavior. For an application, see Crawford andPendakur (2010).


Every Choice Function is Rationalizable by at most |X | − 1Rationales

Let x an alternative with c(X ) 6= x and define P1, . . . ,Pn−1 as acollection of |X | − 1 linear orders, each of them placing analternative y ∈ X \ x on top, and x as the second bestalternative. Let A be any menu. If c(A) 6= x , assign A to thecorresponding rationale with c(A) on top. If c(A) = x , it must beA 6= X and hence, there exists y ∈ X \ A. Assign A to therationale with y on top.

The proportion of choice functions that can be rationalizedby less than |X | − 1 orderings tends to 0 as |X | tends toinfinity.


SQB SR LAt R-CTC RMR

Cycles NO YES YES YES YES

SQ cyles YES YES YES YES YES

Attraction NO NO YES YES YES

Difficult NO NO YES NO YES

Exercise 9: Consider the set of alternatives X = 1, 2, . . . , n anddefine a choice function implicitly by considering one of theprevious models of choice (please define the specification of themodel formally!). Construct the explanation of this choice functionwith a minimal number of rationales using the idea of Kalai,Rubinstein and Spiegler.


Computational difficulties in explaining behavior. RMR model iscomputationally complex in general.

I Apesteguia and Ballester (2010)

I DeMuynck (Forthcoming)

5.3 Rationalization by Game Trees

Xu and Zhou (2007). For an extension, see Horan (2011a).

I The choices of the DM are the equilibrium outcome of anextensive game with perfect information. The tree hasalternatives of X as terminal nodes, each alternativeappearing once and only once. Every node of the treerepresents one criterion.

I Multiple selves competing between them

I Collective choice; hierarchical decision-making


Rationalizability by Game Trees: A choice function c isrationalizable by game trees whenever there exists an extensivegame with perfect information (G ,P) where alternatives of X areterminal nodes (each alternative appearing once and only once),every node of the tree represents the decision of some agent, Pi ,and c(A) = SPNE (G |A; P) for all A ⊆ X


I RGT allows for cycles. Let agent 1 choose between x or yz ,while agent 2 chooses for yz . If zP1xP1y and yP2z , we havexPByPBzPBx .

I RGT does not explain the attraction effect or the compromiseeffect. Suppose PB is well-behaved on a triple, with x over yover z and consider c(xyz). If one agent is decisive over theentire triple, then this agent has preferences xPiyPiz andc(xyz) = x . Hence, there must exist one agent i splittingx , y , z into a single option or a pair (for which anotheragent will decide later). If the alternative splitted is x , weknow that xPiy and xPiz and hence this agent would choosex as equilibrium. If the alternative splitted is z , we know thatxPiz and yPiz and hence this agent will choose xy . Thedecision on xy we know is x and hence, x will be theequilibrium. If the alternative splitted is y , we know that theagent deciding on xz prefers x . Hence i will go for xz for aresulting equilibrium of x .


SQB SR LAt R-CTC RMR RGT

Cycles NO YES YES YES YES YES

SQ cyles YES YES YES YES YES YES

Attraction NO NO YES YES YES NO

Difficult NO NO YES NO YES NO

Exercise 10: Prove formally (Please be very precise in defining thegame theoretical tools that you need) that any individual orcollectivity following an RGT model satisfies Always Chosen.

5.4 Aggregation Models

Green and Hojman (2011) formulate an additive model with avoting procedure.

I The choices of the DM are the result of a voting/aggregationof multiple criteria.

I Multiple selves aggregated.

I Collective choice; voting.


An aggregation-explanation of a choice function c is a pair (λ, v)consisting of a distribution of probabilities of the differentpreferences λ and a aggregation rule v such that c(A) = v(A, λ).

Any choice function can be explained using a monotonicaggregator.

Dual Aggregation: A Dual aggregation/explanation of a choicefunction c is a pair (λ = (λ1, λ2), v) consisting of a distribution ofprobabilities of two possibly different preferences and anaggregation rule v such that c(A) = v(A, λ). This can explain theattraction effect but not the other violations.


SQB SR LAt R-CTC RMR RGT DAGG

Cycles NO YES YES YES YES YES NO

SQ cyles YES YES YES YES YES YES NO

Attraction NO NO YES YES YES NO YES

Difficult NO NO YES NO YES NO NO

5.5 Other Models

Ambrus and Rozen (2011): An aggregation model, based onpreferences/utilities.

DeClipel and Eliaz (forthcoming): A bargaining model of twoselves.

6. SEARCH MODELS

6.1 Search and Report papers6.2 Choice by sequential elimination

6.1 Search and Report papers

Simon (1955) considered a model, satisficing, having the followingtwo characteristics:

I The examination of alternatives comes in an ordered,structured way.

I The individual chooses the first alternative over certainthreshold.

Papers to report analyze or extend the structure of the choice setor the existence of a satisficing idea in models similar to the onesanalyzed in this course.

6.1 Search and Report papers

1. Search, Choice and Revealed Preference (Caplin and Dean2011)

2. Choice by Iterative Search (Masatlioglu and Nakajima, 2011)

3. The satisficing Choice Procedure (Papi 2011)

4. Sequential Choice and Choice from Lists (Horan, 2011)

5. Choice over Lists (Rubinstein and Salant, 2006)

6. Choice in Ordered-Tree-Based Decision Problems (Mukherjee,2011)

6.2 Choice by sequential elimination

I The final choice is the surviving alternative of an eliminationprocess. The supermarket example, google lists, etc.

I Individual choice: models of choice by ordered elimination.The election in any menu of alternatives is determined, as inthe classical model, by binary comparisons. Instead of amaximization process, there is an elimination heuristics.Salant (2003), Rubinstein and Salant (2006), Apesteguia andBallester (2011a).

I Collective choice and Political Environment: Voting bysuccessive elimination as in Dutta, Jackson and LeBreton(2001, 2002) or Apesteguia, Ballester and Masatlioglu (2011).Elimination of alternatives is done by majority voting.

6.2 Choice by Sequential Elimination

Apesteguia and Ballester (2011a), Apesteguia, Ballester andMasatlioglu (2011)

Choice by sequential elimination: A choice function c is Choiceby Sequential Elimination whenever there exists a profile ofpreferences P and a linear order < over the set of alternatives (anagenda-list) such that for every A ∈ P(X ), c(A) is the alternativesurviving the majority process over the agenda-list <.


I Choice by sequential elimination satisfies Always Chosen

I Choice by sequential elimination allows cycles (consider theclassical Condorcet Cycle).


SQB SR LAt R-CTC RMR RGT DAGG CSE

Cycles NO YES YES YES YES YES NO YES

SQ cyles YES YES YES YES YES YES NO YES

Attraction NO NO YES YES YES NO YES NO

Difficult NO NO YES NO YES NO NO NO

7. Classification of Cyclical Models

7.1 Examples7.2 A Nested Classification of Cyclical Models

7.1 Examples

We have seen that the following models explain some sort ofcyclical behavior, but satisfy Always Chosen (and thus, do notexplain Attraction Effect or Compromise Effect).

I Reference Dependent Model: DC

I Sequential Model: SR

I Multicriteria Model: RGT

I Search Model: CSE

7.1 Examples

I Default: Alternative x

I Attribute 1: u1 = (1, 2, 4, 9)

I Attribute 2: u2 = (7, 8, 8, 5)I Average: u = (4, 5, 6, 7)

I Base Relationx

yoo

w

>>

// z

OO``

I c1(x , y ,w) = y , c1(x , z ,w) = c1(x , y , z ,w) = z

7.1 Examples

I Criterion 1: wP1z and zP1y

I Criterion 2: zP2yP2xI Criterion 3: xP3wP3y

I Base Relationx

yoo

w

>>

// z

OO``

I c4(x , y ,w) = w , c4(x , z ,w) = c4(x , y , z ,w) = x

7.1 Examples

P1

P2

P3

''x y z w

I Agent 1: zP1xP1wP1y

I Agent 2: yP2xP2wP2z

I Agent 3: yP3xP3wP3z

I Base Relationx

yoo

w

>>

// z

OO``

I c3(x , y ,w) = c3(x , y , z ,w) = w , c3(x , z ,w) = x

7.1 Examples

I Supermarket Products Ordered:

x // y // z // w

I Criterion 1: wP1yP1zP1x

I Criterion 2: xP2wP2zP2yI Criterion 3: zP3yP3xP3w

I Base Relationx

yoo

w

>>

// z

OO``

I c2(x , y ,w) = c2(x , z ,w) = c2(x , y , z ,w) = w

7.2 A Nested Classification of Cyclical Models

Proposition 1

CDC ⊂ CCSE


Proof (inclusion):

I Default: x

I Base Relationx

yoo

w

>>

// z

OO``

I c1(x , y ,w) = y , c1(x , z ,w) = c1(x , y , z ,w) = z

I Construct P to explain the Base Relation (McKelvey 195..)I To explain the first choice, we need to put alternative y after

alternatives x and w . Similarly, for the second choice we needto put alternative z after alternatives x and w . For instance,agenda x < w < y < z . General Agenda: default < Worsethan default < Better than default


Proof (strict):

I Base Relationx

yoo

w

>>

// z

OO``

I c2(x , y ,w) = c2(x , z ,w) = c2(x , y , z ,w) = wI Cycle x , y ,w determines as default alternative y . However,

this is incompatible with the existence of the Cycle x , z ,w .


Proposition 2

CCSE ⊂ CGT


Proof (inclusion):

I Political Agenda:

x // y // z // w

I Base Relationx

yoo

w

>>

// z

OO``

I c2(x , y ,w) = c2(x , z ,w) = c2(x , y , z ,w) = w


Proof (inclusion):

I

P1

P2

w

P3

z

x y

General Game with the same structure

I Pi consistent with the base relation. In the example, forinstance xP1wP2(y , z)


Proof (strict):

I Base Relationx

yoo

w

>>

// z

OO``

I c3(x , y ,w) = c3(x , y , z ,w) = w , c3(x , z ,w) = xI Cycle x , y ,w determines that alternative w comes after x in

the agenda. However, Cycle x , z ,w determines that alternativex comes after w in the agenda.


Proposition 3

CGT ⊂ CSR


Proof (inclusion):

I

P1

P2

P3

''x y z w

I Base Relationx

yoo

w

>>

// z

OO``

I c3(x , y ,w) = c3(x , y , z ,w) = w , c3(x , z ,w) = x


Proof (inclusion):First, decisions where players 2 and 3 will be decisive: yP1x , wP2z .Then, decisions where player 1 is decisive in any order, since onlytwo alternatives will remain at this point.

I Order linearly the players in the game, starting from the lowerpart of the hierarchy

I Associate to each player all binary comparisons for which theyare decisive

I Order linearly the binary comparisons respecting the order ofplayers and respecting some specific order regarding thepreference of the player involved

I Construct a rationale where only one binary comparison isexecuted, according to the base relation

7.2 A Nested Classification of Cyclical ModelsProof (strict):

I Base Relationx

yoo

w

>>

// z

OO``

I c4(x , y ,w) = w , c4(x , z ,w) = c4(x , y , z ,w) = xI Cycle x , y ,w determines that alternative w appears in a

different subgame than x , y . Cycle x , z ,w determines thatalternative x appears in a different subgame than x , y . Then,it must be

P1

P2

P3

''x y z w

However, since yP2x , c4(x , y , z ,w) cannot be explained


CDC ⊂ CCSE ⊂ CGT ⊂ CSR

Exercise 11: Prove formally that every choice function generatedby a Rational Shortlist Method is indeed a Default Choice. Find aDefault Choice function that cannot be explained sequentially byonly two rationales. Given that CDC ⊂ CSR , you can explain it withmore. Can you do it with only three? Can you explain any DefaultChoice function with a maximum of three rationales? How?

Exercise 12: Pick up one of the other models and analyze howgeneral it is in relation to the cyclical models. Which of thecyclical models are just particular cases of the one you haveselected? Why?

8. MEASUREMENT OF CONSISTENCY

8.1 Classical Measures8.2 Minimal Indices8.3 A characterization of minimal indices

8.1 Classical Measures

Two main features of the maximization principle in the classicaltheory of choice are:

I it provides a simple and versatile account of individualbehavior, and

I a tool for individual welfare analysis

However, how severe the observed deviations are from that simpledescription?


Applied work has tried to applied the idea of rationalizability sincelong ago. See Koo (1963) for an interesting analysis of householdconsumption. To just mention some experimental works that usethese classical measures, see

I Sippel (1997): Laboratory experiment.

I Harbaugh, Krase and Berry (2001): Rationality of children

I Andreoni and Miller (2002, 2008): Consistency on Altruistbehavior.

I Chen, Lakshminarayanan, and Santos (2006): Rationality oftufted capuchin monkeys.

I Choi et al. (2007): Portfolio Decisions

I Choi et al. (2011): Linking Rationality and Sociodemographicconditions.


A common framework for all measures by just allowing the samemenu observed several times.

I An observation (A, a) consists of a non-empty menu ofalternatives A ⊆ X and an element a that is chosen from A.Observations: O

I A collection of observations is a mapping f : O → Z+.Data: F

I An inconsistency index is a mapping I : F → R+ thatmeasures how inconsistent any collection of observations is.


Afriat (1973): To measure the amount of relative wealthadjustment required in each budget constraint to avoid violationsof the maximization principle.

Revealed preference at e: xRφf ,ey ⇔ ∃A : f (A, x) >

0 and y can be bought with a proportion of income e


Afriat can be reinterpreted as an attention concept, in a generalframework:

I An attention mapping φ assigns to every menu A thealternatives in A that are considered at the attention levele ∈ [0, 1].

1. φ(A, 0) = ∅2. φ(A, 1) = A, and3. φ(A, e) is increasing in e.

I Revealed preference at e:

xRφf ,ey ⇔ ∃A : f (A, x) > 0 and y ∈ φ(A, e)


IAf (f ) = infe:Rφf ,e is acyclic

(1− e).

1. Allows for exogenous information on the plausibility ofviolations.

2. Does not consider number of violations.

3. Does not allow for endogenous information (preferences) onthe plausibility of violations.


I Varian (1982) proposes a nonparametric approach for theestimation of demand functions and later in this line ofresearch, Varian (1990) refines Afriat’s measure ofinconsistency by considering potentially different levels ofattention in the different observations e = e(A,a)

xRφf ,ey ⇔ ∃A : f (A, x) > 0 and y ∈ φ(A, e(A,x))

I Varian is interested in the vector of attention levels e that isclosest to 1 and respects rationalizability

IV (f ) = infe:Rφf ,e is acyclic

∑(A,a)

f (A, a)(1− e(A,a)).

8.1 Classical MeasuresHoutman and Maks (1985) suggest to compute the maximalsubset of observations that is consistent with revealed preference.In other words, inconsistencies are measured by the minimalsubset of observations that needs to be eliminated from the datain order to make the remainder rationalizable.

IM(f ) = ming≤f :f−g is rationalizable

∑(A,a)

g(A, a).

1. Does not allow for exogenous information on the plausibilityof violations.

2. Considers the number of violations.

3. Does not allow for endogenous information (preferences) onthe plausibility of violations.

Banker and Maindiratta (1988) extend on this approach and Deanand Caplin (2011) discuss a quicker algorithm for computing thatmaximal subset.


I Rationality has also been measured by counting the number oftimes in the data a consistency property is violated (see, e.g.,Swofford and Whitney, 1985 or Famulari 1995).

I Echenique, Lee and Shum (2010) make use of the monetarystructure of budget sets to suggest a version of this notion,the money pump index, that captures also the severity of eachviolation.

8.2 Minimal Indices

Apesteguia and Ballester (2011b)A weighting function is a mapping w : P ×O → R+ such thatw(P,A, a) = 0 if and only if a = m(P,A). That is, observationsthat are explained by the preference P receive a nullweight/inconsistency, while any other observation receives apositive one.

Iw (f ) = minP∈P

∑(A,a)

f (A, a)w(P,A, a).

8.2 Minimal Indices

I Proposition 1: IM is a minimal index.

I Proposition 2: IV is a minimal index.

I Minimal Swaps Index:IS(f ) = minP∈P

∑(A,a) f (A, a)|x ∈ A : xPa|.

I Minimal Loss Index:IL(f ) = minP∈P

∑(A,a) f (A, a)(um(P,A) − ua(P)).

Exercise 13: Can you describe which are the weights that we needto consider for understanding Houtman-Maks and Varian asminimal indices? Can you explain with a simple example whycounting the number of cycles is NOT a minimal index?

8.2 Minimal Indices

Exercise 14: Consider a set X = 1, 2, . . . , n of cake pieces,ordered from biggest to smallest, where n is an even positiveinteger. Define, on the universal domain, the glutton-educatedchoice behavior as pick up the second largest piece of cake. Is thisrationalizable? Which of the effects described in chapter 2 arepresent in this behavior? Analyze whether this choice behaviorsatisfies each of the properties defined in these notes.

Exercise 15: Compute the inconsistency of the glutton-educatedchoice behavior for Houtman-Maks, the number of violations ofWARP and the swaps index.

8.3 A characterization of minimal indices

Denote by r ∈ R a rationalizable collection of observations whereall binary menus are observed. Given f and r , we say that f isr -invariant if I (f ) = I (f + r).

1. Rationality (RAT): I (f ) = 0 if and only if f is rationalizable.

2. Invariance (INV): For every f , there exists r such that f isr -invariant.

3. Attraction (ATTR): For every f and every r , there exists apositive integer z such that f + zr is r -invariant.

4. Separability (SEP): For any two collections of observations fand g , I (f + g) ≥ I (f ) + I (g), with equality if and only f andg are r -invariant for some r .


























Theorem 1: An inconsistency index I satisfies (RAT), (INV),(ATT) and (SEP) if and only if it is a minimal index.


1. Categorization: Relate f with the corresponding r (and P r ).

2. Separation: For certain constant z rn that forces all collections

with at most n observations to be r -invariant collections,proceed to I (f ) = I (f + nz r

nr) =∑

(A,a) f (A, a)I (1(A,a) + z rnr)

3. Representation: Let w(P r ,A, a) = 0 whenever P r explains(A, a), and w(P r ,A, a) = I (1(A,a) + z r

1r) otherwise. Thenshow that I (1(A,a) + z r

1r) = I (1(A,a) + z rnr).

4. Minimality: For any P ′ and r ′ consider g = f + zr ′ with zlarge enough. We know that

∑(A,a) g(A, a)w(P ′,A, a) =

I (g) ≥ I (f ) + 0 =∑

(A,a) f (A, a)w(P r ,A, a).


Extra properties characterize the specific families/indices describedabove.

I Corollary 1: I satisfies (RAT), (INV), (ATTR), (SEP), (UC)and (PI) if and only if I is a scalar transformation of aVarian’s index.

I Corollary 2: I satisfies (RAT), (INV), (ATTR), (SEP), (UC),(PI) and (NEU) if and only if I is a scalar transformation ofthe minimal inconsistent subset index.

I Corollary 3: I satisfies (RAT), (INV), (ATTR), (SEP), (UC),(NEU) and (DC) if and only if it is a scalar transformation ofthe minimal swaps index.

I Corollary 4: I satisfies (RAT), (INV), (ATTR), (SEP),(NEU) and (COM) if and only if it is a minimal loss index.

9. WELFARE ANALYSIS

9.1 A neutral approach using Pareto9.2 The non-neutral approach9.3 A neutral approach using measures of rationality

9.1 A neutral approach using Pareto

Two main features of the maximization principle in the classicaltheory of choice are:

I it provides a simple and versatile account of individualbehavior, and

I a tool for individual welfare analysis

how to extract relevant information from the choices of theindividual to do welfare analysis?


Accepting the welfare information involved in individualpreferences, we may approach the second question in two differentways:

I The neutral approach: We have limited information about thebehavioral violations of the consistency conditions and hence,we cannot adopt one specific model as reflecting individualbehavior. We can only assume there are severalframings/ancillary conditions that determine differentpreferences/choices of the individual, or the preferences of theindividual are subject to some variability across problems.From that, we can try to extract welfare implications.

I The non-neutral approach: We have specific information thatthe agent proceeds in a specific behavioral way. Under somebehavioral processes, there is an identifiable preference thatwe can intuitively use as the preference of the individual.


Bernheim and Rangel (2010): Choice with ancillary conditions.

Individuals can be inconsistent across choice problems and also inthe same problem if operating under different framings/ancillaryconditions.

We can declare x better than y if for every choice problem and forevery ancillary condition, y is not directly revealed preferred to x .


Limitation of the approach:

We make very limited judgments on individual welfare, and rejectmany information available (as the likelihood of different ancillaryconditions or the consistency of the individual within ancillaryconditions, which is treated as a yes/no binary issue merely.

9.2 The non-neutral approach

Koszegi and Rabin (2007), Rubinstein and Salant (2011): The useof the underlying behavioral process is inexcusable.

The real preference of the agent can contradict the revealedpreference.


An example of non-neutral analysis. Limited Attention.

Remember that Limited Attention model allows for cycles. LetxPyPz and Γ(xz) = z is the only problem with partial attention.Clearly xPByPBzPBx .

However, this another agent zP ′xP ′y overlooking z in xyz and yzmakes the same choices.

But these are the only two possible cases. Thus, we know for surethat x is preferred to y !


Proposition Under the limited attention model, we can claim xPyif and only if there exists T such that c(T ) = x 6= c(T \ y).

Proof: Since c(T ) 6= c(T \ y), we know for sure that y wasconsidered in T . Since c(T ) = x , it must be xPy .

Now we show that we can complete such P in any way. Let beany completion of P. Define Γ(S) = x ∈ S : c(S) x ∪ c(S).Clearly, c(S) is the maximal element according to in Γ(S). Also,Γ is an attention filter, since y 6 inΓ(S) means y x and byconstruction, not xPy . Hence, c(S) = c(S \ y) and henceΓ(S) = Γ(S \ y).

Exercise 16: Check whether the educated-glutton behavior can beexplained by some of the models in these notes. If it can be,describe the welfare implications of accepting the correspondingmodel (which would be the real preference of the agent).

9.3 A neutral (?!) approach using measures of rationality

Apesteguia and Ballester (2011b), minimal indices.

Wonder about which preference relation approximates best thechoices, and the extent of such an approximation:

I the latter provides a measure of inconsistency/rationality asdescribed before, and

I the former a tool for the welfare analysis of possiblyinconsistent individuals


Computation of P∗ is a complex matter in general. Strategies:

1. Identification of the optimal solution through well-knownquick algorithms/techniques. Dynamic programming, branchand bound methods, etc.

2. Identification of a suboptimal solution.

3. Specific domains that allow easy computations.


Algorithms and Suboptimality:

I Dean and Martin (2011) connect Houtman-Maks to theMinimum Set Covering Problem, and using this relation, theyprovide a quick algorithm for computing the optimalpreference.

I Apesteguia and Ballester (2011b) connect the Minimal Swapsindex to the (integer) Linear Order Problem. For a study anda series of applications of the LOP problem, see Brusco, Kohnand Stahl (2008). The most relevant application to economicsis related to the triangularization of input-output matrices(see for instance Fukui, 1986). Connecting LOP and theswaps index allows for quick algorithms and also, identificationof suboptimal solutions.


The integer LOP problem over the set of vertices X and directedweighted edges that connect all vertices x and y with (integer)cost cxy consists of finding the linear order relation over the set ofvertices that maximizes or minimizes the total aggregated cost.

That is, if we denote by Π the set of all mappings from X to1, 2, . . . , k, the LOP involves solving

arg minπ∈Π

∑π(x)<π(y)

cxy


Theorem:

1. For any collection of observations f one can define a LOPwith vertices in X , the solution of which provides the optimalpreference for the minimal swaps index.

2. For any LOP with vertices in X one can define a collection ofobservations f , its optimal preference being the solution tothe LOP.

Exercise 17: Consider again the glutton-educated behavior.Imagine that each menu has a different ancilliary condition. Whichis the Bernheim-Rangel incomplete preference over thealternatives? Which is the optimal preference for theHoutman-Maks and the Swaps indices?


Specific Domains:

We say that a collection of observations f is balanced if all themenus of alternatives of the same cardinality are observed thesame number of times. That is, |A| = |B| ⇒∑

x∈A f (A, x) =∑

y∈B f (B, y) (universal domain, binarydomain...)

Given a collection f , a basic preference relation PB(f ) is anypreference relation such that∑

(A,a):a=x f (A, a) >∑

(A,a):a=y f (A, a)⇒ xPB(f )y . Clearly, sucha preference relation is extremely easy to compute. Hence, thefollowing result is of substantial interest.


Theorem: For any balanced collection of observations f , and forany vector u, P is an optimal preference relation for the minimalloss index if and only if P is a basic preference relation for f .

II Choice under Uncertainty

10. Objective and Subjective Uncertainty11. Experimental Observations12. Objective Uncertainty and Sophisticated Behavior13. Choquet EU Model14. Multiple Priors

10. Objective and Subjective Uncertainty

10.1 VonNeumann and Morgenstern10.2 Anscombe and Aumann10.3 Savage

10.1 VonNeumann and Morgenstern

Von Neumann and Morgenstern (1947)

Let L the set of all probability measures on a (finite) prize set Z . Abinary relation % on L satisfies

I independence/substitution, i.e., for any p, q, r ∈ L and anyα ∈ (0, 1), p % q ⇔ αp ⊕ (1− α)r % αq ⊕ (1− α)r .

I archimedean/continuity, i.e., if p q then there areneighborhoods B(p),B(q) such that for allp′ ∈ B(p), q′ ∈ B(q), we have p′ q′.

if and only if there are numbers v(z)z∈Z such that

p % q ⇔ U(p)∑z∈Z

p(z)v(z) ≥ U(q) =∑z∈Z

q(z)v(z)

See for instance Rubinstein (2007) or any basic textbook!

10.2 Anscombe and Aumann

Anscombe and Aumann (1963)

I When the preference is defined over roulettes, P, preferencesatisfies the VNM axioms.

I Horse lotteries (H) are defined by [p1, . . . , ps ], with each stateproviding a roulette lottery.

I Let P∗ be the set of roulette lotteries with prizes being horselotteries in H.

I When the preference is defined over these compoundedroulette lotteries, the preference satisfies the VNM axioms.

I Monotonicity and Reversal connect the two systems ofpreferences.

10.2 Anscombe and Aumann

THEOREM is a binary relation on P∗ satisfying the axioms ifand only if there exists a unique set of s non-negative numbersπ1, . . . , πs summing to 1, such that

u∗[p1, ., ps ] = π1u(p1) + . . . πsu(ps)

10.3 Savage

Savage (1954)

Let S be the set of states and X be the set of consequences. Actsare the collection F of mappings from S to X and is a binaryrelation on F .

Ordering is a preference relation.

Sure-Thing Principle For all f , f ′, g , g ′ and A ⊆ S , if f = f ′,g = g ′ on A, and f = g , f ′ = g ′ on Ac , then f g if and only iff ′ g ′.

Conditional Preferences make sense thanks to Axiom 2, byequating the outcomes in the complementary event. An event A isnull if f ∼ g whenever f = g on Ac .

10.3 Savage

Eventwise Monotonicity For all f , g ∈ F , x , y ∈ X and non-nullA ⊆ S , if f = x and g = y on A, then f g given A if and only ifx y .

Conditional Preferences respect preferences over constant acts.

Weak Comparative Probability For all f , f ′, g , g ′ ∈ F ,x , y , x ′, y ′ ∈ X , A,B ⊆ X , if

I x y , x ′ y ′

I f = x on A, f = y on Ac , g = x on B, g = y on Bc , and

I f ′ = x ′ on A, f ′ = y ′ on Ac , g ′ = x ′ on B, g ′ = y ′ on Bc

then f g if and only if f ′ g ′.

Thanks to this axiom, we are able to define a qualitativeprobability relation on events. An event A is more likely than anevent B if we prefer the good prize on A.

10.3 Savage

Nondegeneracy There exist x , y ∈ X such that x y .

Small Event Continuity For all f ∈ F , x ∈ X , there exists a finiteset of events A1, . . . ,An forming a partition of S such that: (1)for all i , f g ′ with g ′ = x on Ai and g ′ = g otherwise, and (2)for all j , f ′ g , with f ′ = x on Aj and f ′ = f otherwise.

10.3 Savage

THEOREM A binary relation on F satisfies (Ord), (STP), (EM),(WCP), (Non) and (SEC) if and only if there exists a unique,finitely additive (non-atomic) probability measure µ on S and astate independent utility u on X such that finite outcome acts areranked according to:

v(f ) =

∫u(f (s))dµ(s) =

n∑i=1

u(xi )µ(f −1(xi ))

This result characterizes probabilistically sophisticatednon-expected utility maximizers. These preferences are oftenreferred to as yielding a separation of preferences from beliefs.

11. Experimental Observations 11.1 Allais’ Paradox11.2 Ellsberg’s Paradox

11.1 Allais’ Paradox

1 2− 11 12− 100

f1 1M 1M 1M

f2 0 5M 1M

f3 1M 1M 0

f4 0 5M 0

For most decision makers, f1 f2 but f4 f3.


The paradox can be easily discussed for any model of uncertainty.Under any reasonable assumption, Allais is describing roulettes. Nosensible individual would derive subjective probabilities other thanthe objective ones. Hence, the paradox seems to rely on thenon-expected utility maximizer nature of agents.This paradox would be resolved just by considering individuals who:

I Face objective uncertainty and then rank the lotteries not inan expected utility form, or

I Even if they face subjective uncertainty, form sophisticatedbeliefs (assign probabilities to states) and then rank thelotteries induced from the acts, but not in an expected utilityform.

Exercise 18. Prove formally that a VNM individual facing theobjective urn, or any Savage type of individual cannot behaveaccording to the Allais paradox. In the latter case, discuss whichaxioms in Savage’s theory are key for this observation.


I Camerer (1995): Survey on Experimental Observations.

I Starmer (2000): Survey on Experimental/Theoretical research.

11.2 Ellsberg’s Paradox

Red(1/3) Black Yellow

g1 100 0 0

g2 0 100 0

g3 100 0 100

g4 0 100 100

For most decision makers, g1 g2 but g4 g3.


Difference with Allais.

I Comparisons between the acts in the Ellsberg example involveonly exchanging a pair of outcomes, thus concern only beliefsand not the risk preferences !

Comparisons between acts in the Allais paradox concern alsothe risk preferences of the agent. Specically, it requires theagent to compare getting 1M with sometimes getting 0 andsometimes getting 5M. Probabilistic SophisticatedNon-Maximizers may therefore incur in Allais paradox, but notin Ellsberg, as they build well-behaved beliefs.

Exercise 19. Prove formally that a Savage type of individual withany beliefs cannot behave according to the Ellsberg paradox.Discuss which axioms in Savage’s theory are key for thisobservation.


I Camerer (1995): Survey on Experimental Observations.

I Halevy (2007): Comparisons of Models explaining Ellsberg’sparadox.

I Eliaz and Ortoleva (2011): Uncertainty on probabilities andprizes.

12. Objective Uncertainty and Sophisticated Behavior

12.1 Prospect Theory12.2 Sophistication

12.1 Prospect Theory

Prospect Theory (Kahneman and Tversky, 1979)

I Value is assigned to gains and losses instead of final assets, ina reference-dependent sort of way. Concave for wins, convexfor losses.

I Probabilities are transformed. Overweighting of smallprobabilities.

12.2 Sophistication

Probabilistic Sophistication (Machina and Schmeidler, 1992)

It contains a general behavior regarding subjective probability inwhich agents are fully sophisticated (they form probabilities overstates in a consistent way) but are not utility maximizers. That is,the agent transforms acts into roulettes(u(x1), µ(f −1(x1), . . . , u(xn), µ(f −1(xn)) with a subjectiveprobability µ but does not apply EU.

A different functional w on roulettes is usedv(f ) = w(u(x1), µ(f −1(x1), . . . , u(xn), µ(f −1(xn)). Linearity onprobabilities is discarded. The authors preserve some mixturecontinuity properties and monotonicity properties (related tostochastic dominance) in the functional w .

12.2 Sophistication

Probabilistically Sophisticated Non-Maximizers satisfy (EM) and(WCP). To see the first, consider that the individual formswell-behaved probabilities and thus we can define conditionalprobabilities. In this case, a dominating act will define stochasticdominant outcomes and thus (EM) will hold thanks to themonotonicity condition. To see the second, f g impliesw(x , µ(A), y , 1− µ(A)) w(x , µ(B), y , 1− µ(B)). Given thedominance in between x and y , it must be µ(A)µ(B) and then wecan apply stochastic dominance onw(x ′, µ(A), y ′, 1− µ(A)) w(x ′, µ(B), y ′, 1− µ(B)) to concludef ′ g ′.

The key difference can be seen to be (STP). However, we cannotjust dispense completely with this axiom or the individual may failto be sophisticated.

12.2 Sophistication

Strong Comparative Property.For all f , f ′, g , g ′ ∈ F , x , y , x ′, y ′ ∈ X , A,B ⊆ X A ∩ B = ∅, if

I x y , x ′ y ′

I f = x on A, f = y on B, g = y on A, g = x on B, and f = gon (A ∪ B)c .

I f ′ = x ′ on A, f ′ = y ′ on B, g ′ = y ′ on A, g ′ = x ′ on B andf ′ = g ′ on (A ∪ B)c .

then f g if and only if f ′ g ′.Exercise 20. Prove formally that SCP implies SWP but they arenot equivalent. Prove also that any Probabilistically SophisticatedNon-Maximizer can neither behave according to the paradox, evenif we have eliminated the Sure-Thing Principle. How is thispossible?

13. Choquet EU Model

13. Choquet EU model

Ellsbergs examples capture the idea that ones condence in theprobability assignment is relevant. Following Schmeidler (1989)

and Gilboa (1992) we discuss Choquet Expected Utility (CEU).They need of subjectivity and hence, the former is in theAnscombe-Aumann framework, and the second in the Savage’sframework, but share the same idea. They capture the notion ofuncertainty or ambiguity aversion. The model combines a capacity

(instead of an additive probability) and a utility function.


I A real valued set function v on events (subsets of S) is acapacity if v(∅) = 0, v(S) = 1 and v is monotonic, i.e.,v(E ) ≤ v(F ) if E ⊆ F .

I Let a be a finite step function where a =∑k

i=1 αiEi and

α1 > α2 > · · · > αk and Eiki=1 is a partition of S and υ bea capacity. The Choquet integral of a with respect to υ isdenoted by

∫adυ and defined by∫

adυ =k∑

i=1

(αi − αi+1) υ(∪ij=1Ej

)where αk+1 = 0.


If the capacity is additive, it allows to define a probability overstates by considering the value of v in each state. In this case, theintegral coincides with the classical notion.

Axioms in the two setups characterizing behaviors of the form

f g ⇔∫

u (f (s)) dυ ≥∫

u (g (s)) dυ

Exercise 21. Describe the condition for the capacity v to beadditive. Does any capacity function generate the ambiguityaversion present in the Ellsberg Paradox? Which capacities wouldcreate the paradox?

14. Multiple Priors

14. Multiple Priors

A natural story behind the Ellsberg paradox can be described asfollows. Given that the individual has not information about howmany black-yellow balls are, the individual forms a set of possiblepriors. Being uncertainty averse, considers the minimal expectedpayoff across these payoffs.

v(f ) = minµin∆

∫u(f (s))dµ(s)

This model, MMEU, is analyzed in Gilboa and Schmeidler (1989)and Casadesus-Masanell, Klibanoff and Ozdenoren (2000) for theAnscombe-Aumann and Savage’s frameworks.

14. Multiple PriorsOther multiple prior models:

I Variational preferences (Maccheroni, Marinacci and Rustichini,2006): v(f ) = minµin∆(

∫u(f (s))dµ(s) + c(p)) where c

describes ambiguity aversion. The MMEU case corresponds toc(p) = 0 for all p, while SEU corresponds to c(p) = 0 for aparticular p = q and c(p) =∞ otherwise.

I Multiplier Preferences (Hansen and Sargent, 2001; Strzalecki,2010): v(f ) = minµin∆(

∫u(f (s))dµ(s) + θR(p||q) where θ is

a trust parameter and R(p||q) describes the (entropy)distance from p to the true subjective probability q.

I Knightian Uncertainty (Bewley, 2002). Incomplete preferences(Pareto domination over priors).

I Sophisticated MMEU (Marinacci, 2002). Shows how theintersection between MMEU and sophistication looks like.

Exercise 22. Describe a set of plausible priors for the Ellsbergexperiment, explaining the intuition and the consequences, thatallows to resolve the paradox when considering the MMEU modelor its variations.

III Menu Preferences15. Indirect Utility Theory16. Preference for Flexibility17. Temptation and Self-Control

15. Indirect Utility Theory

15. Indirect Utility Theory

In the classical approach, a menu is valued as much as the bestalternative on it. The indirect utility function describes this idea.The value of a menu A is equal to v(A) = maxx∈A u(x).

Some axiomatic exercises in Nehring and Puppe (1996) andBallester, DeMiguel and Nieto (2003).

Exercise 23. Let X be a finite set of alternatives and % apreference on X (all non-empty subsets of X , or menus). We saythat the preference over menus is perfectly rational if for allA,B ∈ X , A % B =⇒ A ∼ A ∪ B % B. Prove that % is perfectlyrational if and only if there exists u : X → R such thatA % B ⇔ maxx∈A u(x) ≥ maxx∈B u(x).

16. Preference for Flexibility


Consider an individual who must decide in the morning a menu(restaurant) from which she will have to make a choice (lunch).With regards to this interpretation, Kreps (1979) discusses how theindividual may like the flexibility of having more options to choosefrom later in the day. The reason might be that she is not sureabout which her actual tastes will be at that moment. A keyproperty analyzed to represent this idea is the following:

Kreps (1979) and Dekel, Lipman and Rustichini (2001) deeplyanalyze this idea describing a model where states of nature arecompletely subjective (the possible different preferences of theindividual), providing finite or continuous versions of the followingmodel:

A % B ⇔∑s∈S

π(s) maxx∈A

U(x , s) ≥∑s∈S

π(s) maxx∈B

U(x , s)


The key axiom to characterize this preference is:

Monotonicity. A ⊆ B ⇒ B % A.

An alternative explanation on monotonic preferences on menus isfreedom of choice. See for instance Pattanaik and Xu (1990),Dutta and Sen (1996), or Alcalde-Unzu and Ballester (2004).

17. Temptation and Self-Control


Gul and Pesendorfer (2001)

Consider an individual who must decide what to eat for lunch. Shemay choose a vegetarian dish or a hamburger. In the morning,when she feels no hunger, she prefers the healthy, vegetarian dishbut at lunchtime, she experi- ences a craving for a hamburger

Consider an individual who must decide what to study for the day.She may choose an ambitious (and costly) plan of work or aminimal effort one. In the morning, when she feels no pressure, sheprefers the ambitious plan, but as soon as she starts working sheexperiences a craving for the minimal effort one.


To lessen the impact of her lunchtime craving, the decision-makermay seek to limit the options at lunchtime. For example, she maychoose a vegetarian restaurant. When this is not possible and theindividual is confronted with a menu that includes both thevegetarian meal and the hamburger, she may exerciseself-control, that is, resist the craving for the hamburger andchoose the vegetarian meal.

To lessen the impact of her daily craving, the decision-maker mayseek to limit the options at daily work. For example, she may studyin a place where the benefits of the minimal effort plan are notperceived. When this is not possible and the individual isconfronted with a menu that includes both plans of work, she mayexercise self-control, that is, resist the craving for the minimaleffort plan.


Recognizing Temptation: When there is no choice, the agent is nottempted and prefers the long-run menu x to the tempting y.However, when confronted with the menus x and x , y, theagent is not indifferent, and prefers to commit, avoidingtemptation. That is, x x , y.Recognizing Self-Control. When there is no choice, the agent isnot tempted and prefers the long-run menu x to the temptingy. When confronted with the menus y and x , y, the agentprefers the latter because she will self-control (maybe with somecost that does not exceed the benefit of x over y) and choose thebetter option x . That is, x , y y.Set-Betweeness: If A % B then A % A ∪ B % B.


Temptation Preferences with Self-Control:

U(A) = maxx∈A

u(x)−[maxy∈A

v(y)−v(x)] = maxx∈A

[u(x)+v(x)]−maxy∈A

v(y)

The individual considers the cost of temptation. Cost can bemeasured by the difference in short-run value to the optimalalternative.

Limiting case when cost is sufficiently high, OverwhelmingTemptation:

U(A) = maxx∈A:v(x)≥v(y) for all y∈A

u(x)

This model corresponds to Strotz’s idea (Strotz, 1955). An agentcommits/chooses in the first period knowing that a differentself/agent will choose in the second period.


Dekel and Lipman (2011) show a very interesting result on Gul andPesendorfer’s model of temptation. It can be represented as arandom Strotz model.

U(A) =∑i

maxx∈A:vi (x)≥vi (y) for all y∈A

u(x)

Hence x x , y y can be interpreted as: (1) The agentprefers to commit because there are chances of falling intemptation but (2) the agent prefers the good alternative availablebecause there are chances of self-controlling.Dekel, Lipman and Rustichini (2009) provide a complete overviewof Flexibility and Temptation.


Exercise 24. Consider a dieting agent who wishes to commitherself to eating only broccoli. There are two kinds of snacksavailable: chocolate cake and high-fat potato chips. Let b denotethe broccoli, c the chocolate cake and p the potato chips. Howcould you interpret the following ranking over menus, b b, cand b, p b, c , p? Prove whether this is possible under Guland Peserdorfer’s model of temptation.

Exercise 25. Consider again the dieting agent facing multipletemptations, but now suppose the two snacks available are high-fatchocolate ice cream i and low-fat chocolate frozen yogurt y . Howcould you interpret the following ranking over menus,b, y y, b b, i and b, i , y b, i? Prove whetherthis is possible under Gul and Peserdorfer’s model of temptation.

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(Advanced) Decision Theory - UAB Barcelonapareto.uab.es/mballester/Teaching_files/DecisionTheory7.pdfDecision theory appears to be at a crossroad, in more sense than one. Over half