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

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111 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

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

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

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

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}

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)

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

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

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

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

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

12 Applying CARNI in Different Models of Choice 12

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

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

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

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

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 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 18

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

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

21 Concluding Remarks & Related Literature 21

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

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)

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

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