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Justifiable Choice

Yuval HellerTel-Aviv University

(Part of my Ph.D. thesis supervised by Eilon Solan)

http://www.tau.ac.il/~helleryu/

Bonn Summer School

July 2009

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Contents

Introduction Choice with incomplete preferences and justifications

Violating WARP and binariness

Convex axiom of revealed non-inferiority (CARNI)

Applications of the new axiom: Taste-justifications

Belief-justifications

Related literature & concluding remarks

Incomplete Preferences

Most existing models of rational choice assume

complete psychological preferences

Rationality does not imply completeness: DMs may be indecisive when comparing 2 alternatives

Complicated alternatives

Multiple objectives (multi-criteria decision making)

Group decision making (social choice)

Aumann (62), Bewely (86), Dubra et al. (04), Mandler (05)3

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Choice Correspondence - C

C specifies the choosable alternatives: C(A) A.

A - a closed and non-empty set of alternatives

Interpretation: When facing A, DM always chooses an act in C(A)

All acts in C(A) are sometimes chosen

The unique choice in C(A) is not modeled explicitly Interpretations: justifications, subjective randomization

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Weak Axiom of Revealed Preferences (WARP)

WARP is often violated when preferences are incomplete

Example: x, y are incomparable acts, x’ is a little bit better than x

xy BA xC(B)xC(A)

yC(B)

x, y AB

yC(A)

x’xyAC(A)={x,y}

C(B)={x’,y}

xC(A)

yC(B)xC(B)\

B=AU{x’}

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Insights from the Psychological literature

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Behavior depends on payoff-irrelevant information

DM has several ways to evaluate acts, each with a

different justification (rationale) Observable information determines which justification to use

The chosen act: the best according to this justification

Examples: Availability heuristics, Anchoring (Tversky &

Kahneman, 74), Framing effect (Tversky & Kahneman, 81),

Reason-based choice (Shafir, Simonson & Tversky, 93)

Taste Justifications

Influence tastes over consequences

Example (regret justification, Zeelenberg et al., 96) Choice between safe & risky lotteries of equal attractiveness

(when feedback is only on the chosen lottery)

Having feedback on the risky lottery caused people to

choose it more often

Similar phenomena in real-life: Dutch postcode lottery

Your lottery number = Your postcode / address

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Belief Justifications

Influence beliefs over state of nature

Example (mood justification, Wright and Bower, 92): Happy/sad moods were induced (by focusing on happy/sad

personal experiments)

Induced mood influenced evaluation of ambiguous events

Happy people are optimistic: higher probability for positive

ambiguous events

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Weak Axiom of Revealed Non-Inferiority (Eliaz & Ok, 05)

WARNI: alternative is chosen if it is not revealed

inferior to any chosen alternative (WARP WARNI)

x is revealed inferior to y if x is not chosen in any set that

includes y

WARNI binariness: Choice is binary if it maximizes a binary relation (x is

chosen in A iff it is chosen in any couple in A)

Justifications often induce non-binary choice9

Example for Violating Binariness (Taste Justifications)

Alice chooses a restaurant for lunch x - serves meat, y - serves chicken

z – randomly serves either meat, chicken or fish

Incomplete preferences: Indecisive between meat &

chicken (uses justifications), fish is a little bit worse

Plausible choice: zC(x,z), zC(y,z), zC(x,y,z)

Remark: z is dominated by alternatives in the convex

hull of x & y (mixtures )10

Convex axiom of revealed non-inferiority (CARNI)

CARNI: x is chosen in A if it is not inferior to any

alternative in the convex hull of C(A) x is revealed inferior to y, if: yconv(A) xC(A)

WARP + independence CARNI

Why comparing to conv(A) (= not choosing z): Choice between x & y according to a toss of a coin

Multiple choices of z are strictly worse then multiple

choices between y & x

No justification (linear ordering) supports z 11

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Applying CARNI in Different Models of Choice

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Model 1 - Taste Justifications

Von-Neumann-Morgenstern’s framework: X - Finite set of outcomes

Alternatives: lotteries over X (A(X))

3 Axioms imposed on choice: Continuity ( for all g :{ f | fC({f,g}) is closed ,

{ f | {f}=C({f,g}) is open )

Independence ( fC(A) g+(1-)f C(g+(1-)A) )

CARNI (instead of WARP in vN-M’s model)

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Theorem 1 - Taste Justifications(multiple utilities)

C satisfies continuity, independence & CARNI

C has a multi-utility representation: A unique (up to

positive-linear transformations) closed and convex set U

of vN-M utility function, such that a lottery is chosen

iff it is best w.r.t. to some utility in U

Interpretation: Justification triggers the DM to think

primarily about a particular “anchoring” utility in U

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Relation with Eliaz-Ok (04)

Eliaz & Ok assume WARNI instead of CARNI

Their representation: a lottery x is chosen iff for

each y in the set there is a utility uy in U such that x

is better than y w.r.t. to uy

Allows the choice of unjustified alternative, which

is not best w.r.t. to any of the utilities

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Model 2 – Belief Justifications

Anscombe-Aumann’s framework (1963) : S – finite set of states of nature, X - finite set of outcomes Alternatives (acts): functions that assign lottery for each state Notation: f(s) – the constant function that assigns in all states

the lottery that f assigns in s

3 new axioms: Non-triviality: there is A, s.t. C(A)A Monotonicity: For all sS, f(s)C(A(s)) fC(A) WARP over unambiguous (constant) alternatives

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Anscombe-Aumman’s Framework

A (X)S

Set of acts (alternatives)

f1

f2

f3

states of natureS (finite set)

DM

Lotteries over X

X

finite set of outcomes

0.7 + 0.3

0.4 + 0.6

0.5 + 0.5

0.5 + 0.5

0.1 + 0.9

0.8 + 0.2

Theorem 2 - Taste Justifications(multiple priors)

C satisfies continuity, independence, CARNI,

non-triviality, monotonicity and unambiguous WARP

C has a multi-prior representation: A unique

closed and convex set P of priors and a unique vN-M

utility u, such that an alternative is chosen iff it is best

w.r.t. to some prior in P

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Relation with Bewely (02) & Lehrer-Teper (09)

They axiomatize preference relation Implict assumption: choice is binary

Our axioms = their axioms + CARNI

Their representation: an act f is chosen iff for each g in the

set there is a prior pg in P such that f is better than g w.r.t.

to pg

Allows the choice of unjustified alternative, which is not

best w.r.t. to any of the utilities

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Models for Both kinds of Justifications (Tastes & Beliefs)

Ok, Ortoleva & Riella (08) present a few axiomatic

models for prefernces that generelize multiple utilities

and mutliple priors A model that is either multy-utility or multi-prior

3 models of different kinds of state-dependant utilities

One can add CARNI to all of these axiomatic models

get the analog justification representations

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Concluding Remarks & Related Literature

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Model’s Primitive & Binariness

Multiple-priors may be interpreted as models for

choice when ambiguity’s evaluation has different

justifications

Most existing models combine the different

justifications into binary preferences ()

We demonstrate why justifications should be combined

into a non-binary choice correspondence (C)

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Global Binariness

Our models have a “global-binariness” property: Preferences (=binary choices) over the couples in A do

not reveal the choice in A

The preferences over all the couples in the grand set (or

at-least in conv(A)) reveal the choice in A

A few examples for non-binary choice models: Social choice - Batra & Pattanaik (72), Deb (83)

Preferences of elements over sets - Nehring (97)

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Status-Quo Justification

Violating WARP in a dynamic environment may be

vulnerable to money pumps

This can be avoided by a status-quo justification: DM uses justifications that are consistent with past choices

Example: choosing the most recently chosen act in C(A)

A related formal construction in Bewley (2002)

Strong empirical psychological support

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Conjectural Equilibrium (Battigalli, 87)

Each player has partial information about the actions of the others. In equilibrium she plays a best response against one of the consistent action profiles Similar concepts in the learning literature: Fudenberg & Levine

(93), Kalai & Lehrer (93), Rubinstein & Wolinsky, (94)

Modeled by belief-justifications: Each player has a set of priors – P

A common set when information is symmetric

Justification triggers each player into a specific prior

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Attitude to Uncertainty (Belief-Justifications)

Example: |S|=2, X={x,y}, y=C(x,y), P includes a segment around 0.5

Let: g= (x,y), f=(0.5x+0.5y,0.5x+0.5y)

Minimax model (Gilboa-Schmeidler, 1989) predicts: f g Our model predicts that both acts are choosable

Heath & Tversky (1991) – people are: Uncertainty-averse – when DM feels ignorant or uninformed

Uncertainty-seeker – when DM feels knowledgeable26

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Summary

We present a new axiom, CARNI, which behaviorally

describes (non-binary) choice when there are

incomplete preferences and multiple justifications

A convex variation of Ok-Eliaz (04) WARNI axiom

We apply the new axiom in different choice models:

Taste justifications (multiple utilities)

Belief justifications (multiple priors)

Generalizations (a la Ok, Ortoleva & Riella, 08)

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Questions & Comments?

Y. Heller (2009), Justifiable choice, mimeo.

http://www.tau.ac.il/~helleryu/weaker.pdf