Lecture 4: Reciprocity and the Impact of Beliefs

Transcription

1 Lecture 4: Reciprocity and the Impact of Beliefs Experimental evidence that more than pure distributional concern matters: e.g. Blount (995), Bolle, Kritikos (999), Charness, Rabin (00), Engelmann, Strobel (004), Gneezy et al (000).. Reciprocity and the Gift Exchange Experiment (Charness 004) Gift exchange game with exogenous matching of buyers and sellers (no market) three treatments T: price p set unilaterally by buyer T: price set by random device T3: price set by third party (experimenter) () September 4, 03 / 6

2 Prediction with only distributional concerns: p-e relation the same for all treatments. Experimental results: p-e signi cantly steeper in T than in T and T3 =) Not only distributional concerns, but also reciprocity seems to have an impact. Reciprocity: A person behaves reciprocally, if he is kind to someone who is kind to him, and he is unkind to someone who is unkind to him. =) If only reciprocity, p should have no impact on e in T and T3. In literature "fairness" and "reciprocity" often confused, or words used interchangeable. () September 4, 03 / 6

3 . A Theory of Sequential (Pure) Reciprocity (Dufwenberg and Kirchsteiger 004) based on Rabin (993), which applies only to static games...the model Extensive form game with nite number of players and complete information strategies chosen by players determine "material payo " (i.e. money) what is "kind behavior" in a game? player s behavior (strategy) is kind to, if it intends to give a large material payo, large compared with set of possible payo s =) Intentions crucial () September 4, 03 3 / 6

4 How to observe intentions? - Example F f 0 D d 0 By choosing F, does intend to give a lot? Should think that s choice of F intends to give a lot to? () September 4, 03 4 / 6

5 Answer depends on beliefs of about s strategy, and on s belief about s belief about s strategy =) utility depends not only on strategy combination, but also on beliefs =) approach of psychological game theory required (see Lecture 5 for general description of PGT) () September 4, 03 5 / 6

6 Notation: Strategy set of i, A i ; a i A i rst order beliefs b ij : belief of i about strategy of j - probability distribution over A j. second order beliefs c ijk : belief of i about belief of j about strategy of k. Assumption: second order beliefs are point beliefs - c ijk probability distribution over A k (and not probability distribution over probability distribution...) π i (a): material payo of i (money or other objective measure; not vnm-utility) () September 4, 03 6 / 6

7 kindness of i toward j: k ij (a i, (b ik ) k6=i ) = π j (a i, (b ik ) k6=i ) belief of i about kindness of j towards i: λ iji (b ij, (c ijk ) k6=j ) = π i (b ij, (c ijk ) k6=j ) utility function: 0.5[ max α i A i π j (α i, (b ik ) k6=i ) + min α i A i π j (α i, (b ik ) k6=i )] 0.5[ max β j A j π i (β j, (c ijk ) k6=j ) + min β j A j π i (β j, (c ijk ) k6=j )] U i (a i, b ik, (c ijk ) k6=j ) = π i (a i, (b ik ) k6=i ) + Y ij kij (a i, (b ik ) k6=i )λ iji (b ij, (c ijk ) k6=j ) j6=i with Y ij measuring how sensitive i is to reciprocity concerns vis-a-vis j. () September 4, 03 7 / 6

8 sequential reciprocity equilibrium (SRE) optimisation in all subgames =) sequential rationality updating of perception of opponents kindness =) updating of incentives R: set of all nodes r which are roots of subgames a i (r): a i, except for path to r A i (r, a i ): set of all strategies of i, for which holds: consistent with r at all infosets but r and those on the path to r behavior as described by a i () September 4, 03 8 / 6

9 De nition: a is a Sequential Reciprocity Equilibrium, if for all i and all r: ) ai (r) is argmax A i (r,ai ) U i (a i, b ij (r), c iji (r)) ) aj (r) = b ij (r) for all j 6= i 3) ak (r) = c ijk (r) for all j 6= i, k 6= j Theorem: for all nite extensive form games an SRE exists () September 4, 03 9 / 6

10 .. Examples a) Sequential prisoners dilemma - Simple version of gift exchange game C D c d c d 0 0 () September 4, 03 0 / 6

11 Result : If player defects (chooses D), player also defects in every SRE. Intuition: For any belief and second order belief of, D is worse for than C. Hence, has no incentive to forego monetary payo in order to be kind to. Result : If player cooperates, the following holds in all SRE: a) If Y >, player cooperates. b) If Y < 0.5, player defects. c) If 0.5 < Y <, player cooperates with a probability of p = Y Y. Intuition: for high Y, the nice choice of is reciprocated by nice choice of. For low Y, monetary incentives dominate. () September 4, 03 / 6

12 Result 3: If Y < 0.5, defection is 0 s unique equilibrium behavior. Intuition: Since will for sure be unkind, even a very reciprocal player has no incentive to be kind to. Result 4: If Y >, 0 s equilibrium behavior is characterized by one of the following three possibilities: a) Player cooperates (regardless of Y ). b) Y > and player defects. c) Y > and player cooperates with probability q = Y Y. Intuition: If behaves reciprocally, there are some equilibria. might cooperate for monetary and for reciprocity reasons. If Y is high, there also exists an equilibrium where they get stuck in mutual distrust - selful lling beliefs. () September 4, 03 / 6

13 b) Centipede game Very low reciprocity motivation: Standard solution High reciprocity motivation: players stay to the end Medium range: only mixed strategy equilibria =) in games with perfect information pure SRE need not exist even for generic material payo s SRE not unique () September 4, 03 3 / 6

14 c) So long Sucker L R l r 3 0 ε 3 3 l r ε Player gets always "punished", irrespectively of what he does. () September 4, 03 4 / 6

15 d) Wage undercutting (D&K 000) Stage : two identical workers make simultaneously wage o ers. Stage : rm decides which worker to employ (at most one) Stage 3: employed worker decides about his e ort Result: If workers are su ciently motivated by reciprocity a) Low wage worker shirks, high wage worker provides high e ort b) If available, rm chooses high wage worker. () September 4, 03 5 / 6

16 This approach models only reciprocity, and does not take distributional concerns into account. Purpose is to model a speci c behavioral phenomenon in order to get a better understanding of particular economic observations (see e.g. Dufwenberg, Kirchsteiger 000) Models of distributional concerns and reciprocity Falk, Fischbacher (998) Charness, Rabin (00) () September 4, 03 6 / 6