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111 Justifiable Choice Yuval Heller Tel-Aviv University (Part of my Ph.D. thesis supervised by Eilon Solan) Bonn Summer School July 2009

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22 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

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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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44 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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55 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 x y B A x C(B)x C(A) y C(B) x, y A B y C(A) x x y A C(A)={x,y} C(B)={x,y} x C(A) y C(B) x C(B) \ B=AU{x}

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6 Insights from the Psychological literature 6 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)

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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 7

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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 8

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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 choice 9

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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: z C(x,z), z C(y,z), z C(x,y,z) Remark: z is dominated by alternatives in the convex hull of x & y (mixtures ) 10

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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: y conv(A) x C(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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12 Applying CARNI in Different Models of Choice 12

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Model 1 - Taste Justifications Von-Neumann-Morgensterns framework: X - Finite set of outcomes Alternatives: lotteries over X (A (X)) 3 Axioms imposed on choice: Continuity ( f or all g :{ f | f C({f,g}) is closed, { f | {f}=C({f,g}) is open ) Independence ( f C(A) g+(1- )f C( g+(1- )A) ) CARNI (instead of WARP in vN-Ms model) 13

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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 14

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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 u y in U such that x is better than y w.r.t. to u y Allows the choice of unjustified alternative, which is not best w.r.t. to any of the utilities 15

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Model 2 – Belief Justifications Anscombe-Aumanns 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 s S, f(s) C(A(s)) f C(A) WARP over unambiguous (constant) alternatives 16

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Anscombe-Aummans Framework A (X) S Set of acts (alternatives) f1f1 f2f2 f3f3 states of nature S (finite set) DM Lotteries over X X finite set of outcomes

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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 18

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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 p g in P such that f is better than g w.r.t. to p g Allows the choice of unjustified alternative, which is not best w.r.t. to any of the utilities 19

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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 20

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21 Concluding Remarks & Related Literature 21

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22 Models Primitive & Binariness Multiple-priors may be interpreted as models for choice when ambiguitys 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) 22

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23 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) 23

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24 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 24

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25 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 25

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26 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 knowledgeable 26

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27 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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28 Questions & Comments? Y. Heller (2009), Justifiable choice, mimeo.

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