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Justifiable Choice

Justifiable Choice. Yuval Heller Tel-Aviv University (Part of my Ph.D. thesis supervised by Eilon Solan) http://www.tau.ac.il/~helleryu/ Bonn Summer School July 2009. 1. 1. Contents. Introduction Choice with incomplete preferences and justifications Violating WARP and binariness

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Justifiable Choice

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  1. 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 1 1

  2. 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 2

  3. 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)

  4. Choice Correspondence - C • C specifies the choosable alternatives: f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 4

  5. 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AB x A B xC(A) xC(B) y yC(B) yC(A) C(A)={x,y} x xC(A) x’ \ A y xC(B) C(B)={x’,y} yC(B) B=AU{x’} 5

  6. Insights from the Psychological literature • Behavior depends on payoff-irrelevant information • DM has several ways to evaluate acts, each with a differentjustification (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) 6 6

  7. 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

  8. 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

  9. 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

  10. 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 )

  11. 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

  12. Applying CARNI in Different Models of Choice 12

  13. Model 1 - Taste Justifications • Von-Neumann-Morgenstern’s framework: • X - Finite set of outcomes • Alternatives: lotteries over X (AD(X)) • 3 Axioms imposed on choice: • Continuity ( for all g :{f | fC({f,g}) is closed , {f | {f}=C({f,g}) is open ) • Independence ( fC(A)ag+(1-a)f C(ag+(1-a)A) ) • CARNI (instead of WARP in vN-M’s model)

  14. 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 iffit 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

  15. Relation with Eliaz-Ok (04) • Eliaz & Ok assume WARNI instead of CARNI • Their representation: a lottery x is chosen ifffor 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

  16. 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 sS, f(s)C(A(s)) fC(A) • WARP over unambiguous (constant) alternatives

  17. DM S (finite set) states of nature Anscombe-Aumman’s Framework 0.7 + 0.3 f1 0.4 + 0.6 f2 0.5 + 0.5 0.5 + 0.5 f3 0.1 + 0.9 X AD(X)S 0.8 + 0.2 finite set of outcomes Set of acts (alternatives) Lotteries over X

  18. Theorem 2 - Taste Justifications(multiple priors) • C satisfies continuity, independence, CARNI,non-triviality, monotonicity and unambiguous WARP  C has a multi-prior representation: A uniqueclosed and convex set P of priors and a unique vN-M utility u, such that an alternative is chosen iffit is best w.r.t. to some prior in P

  19. 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 ifffor 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

  20. 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

  21. Concluding Remarks & Related Literature 21

  22. 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) 22

  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

  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

  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

  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

  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) 27

  28. Questions & Comments? • Y. Heller (2009), Justifiable choice, mimeo. http://www.tau.ac.il/~helleryu/weaker.pdf 28 28

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