I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA Beliefs in Repeated Games Masaki Aoyagi Guillaume R. Fréchette Sevgi Yuksel July 2020 1 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA M OTIVATION Social dilemmas capture many situations of interest: Cournot competition, public good provision, natural resource extraction, etc . Key tension: individually rational vs socially optimal. Long extensive theoretical litterature. Recent experimental literature: how cooperation varies with ◮ payoffs, ◮ monitoring, ◮ termination ( H and δ ). 2 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA M OTIVATION Two premises underlie the equilibrium predictions: ◮ beliefs are correct, and ◮ actions are best responses to those beliefs. In repeated interactions, this involves ◮ many contingencies, complex strategies, ◮ and sometimes multiple equilibria. → Making it challenging to form correct beliefs and to best respond. Goal. Study beliefs to better understand cooperation in the repeated Prisoners’ Dilemma (PD), comparing finite and indefinite games. 3 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA M OTIVATION Q UESTIONS 1. Are round beliefs accurate? 2. Is the relation between choices and beliefs different in the finite and indefinte games? 3. Can strategy choice be rationalized by beliefs? 4 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA M OTIVATION : I MPLEMENTATION T HE STUDY OF REPEATED GAMES AND OF BELIEFS PRESENTS CHALLENGES Simple repeated games: ◮ deterministic, ◮ with perfect monitoring. Simple belief elicitation: ◮ elicit beliefs over other’s choice in current round, ◮ recover beliefs over strategies econometrically. 5 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA D ESIGN 2 treatments (between subjects): Finite : Stage game repeated for 8 rounds. Indefinite : Stage game repeated with probability 7/8. - Block Random Design for the first 8 rounds. Random matching between supergames. Sessions end with the first supergame after 1 hour of play. Belief elicitation: ◮ Beliefs about other’s action in the current round. ◮ Introduced in the 5th supergame. ◮ BSR: Hossain-Okui (2013) modified by Wilson-Vespa (2018). Risk preferences at the end: ◮ Bomb task: Crosetto and Filippin (2013). 6 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA D ESIGN Table: S TAGE G AME (in ECU) C D C 51, 51 22, 63 D 63, 22 39, 39 7 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA Supergames Action Actions and Beliefs Treatment Session Only Early Late Finite 1 1, 2, 3, 4 5, 6, 7 8, 9 10, 11, 12 2 1, 2, 3, 4 5, 6, 7 8, 9 10, 11, 12 3 1, 2, 3, 4 5, 6, 7 8, 9, 10 11, 12, 13 4 1, 2, 3, 4 5, 6, 7 8 9, 10, 11 5 1, 2, 3, 4 5, 6, 7 8, 9, 10 11, 12, 13 6 1, 2, 3, 4 5, 6, 7 8, 9, 10 11, 12, 13 7 1, 2, 3, 4 5, 6, 7 8, 9 10, 11, 12 8 1, 2, 3, 4 5, 6, 7 8, 9 10, 11, 12 Indefinite 1 1, 2, 3, 4 5, 6, 7 8, 9, 10 2 1, 2, 3, 4 5, 6 7, 8, 9 3 1, 2, 3, 4 5 6, 7 4 1, 2, 3, 4 5 6, 7, 8 5 1, 2, 3, 4 5, 6, 7 8, 9 10, 11, 12 6 1, 2, 3, 4 5 6 7 1, 2, 3, 4 5, 6, 7 8, 9, 10 8 1, 2, 3, 4 5, 6 7, 8, 9 14 to 20 subjects per session: 302 subjects in total. Payment: $ 8 + choices from two supergames (pre/post) + beliefs in one. Earnings from $22.00 to $63.75 (with an average of $35.30). 8 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA A CTIONS 9 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA C OOPERATION O VER S UPERGAMES Finite Indefinite 1 1 .8 .8 Cooperation Rate Cooperation Rate .6 .6 Round 1 Other Coop. at t-1 Other Defect at t-1 Round 8 .4 .4 Early Late Early Late .2 .2 0 0 1 2 3 4 5 6 7 L-2 L-1 Last 1 2 3 4 5 6 7 L-2 L-1 Last Supergames Supergames 10 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA Result 1 ( Replication / Design Neutrality ) Reproduce typical qualitative data patterns . Confirms that: (i) Cooperation is history-dependent in both games. (ii) Cooperation evolves differently in Finite and Indefinite games. K EY DIFFERENCE : cooperation collapses at the end only in Finite games. 11 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA B ELIEFS 12 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA D ISTRIBUTION OF B ELIEFS BY T REATMENT AND R OUND Round 1 Round 2 Round 3 Round 4 1 1 1 1 Finite .8 Indefinite .8 .8 .8 .6 .6 .6 .6 Cdf Cdf Cdf Cdf .4 .4 .4 .4 .2 .2 .2 .2 0 0 0 0 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 Belief Belief Belief Belief Round 5 Round 6 Round 7 Round 8 1 1 1 1 .8 .8 .8 .8 .6 .6 .6 .6 Cdf Cdf Cdf Cdf .4 .4 .4 .4 .2 .2 .2 .2 0 0 0 0 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 Belief Belief Belief Belief Late supergames. Vertical lines indicate respective means. 13 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA Result 2 ( Question 2 ) Beliefs are different in Finite and Indefinite games. K EY DIFFERENCE : Beliefs collapse at the end of Finite games. 14 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA A CTIONS AND B ELIEFS 15 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA A CTIONS AND B ELIEFS Finite Indefinite 1 1 .8 .8 Cooperation Rate Cooperation Rate .6 .6 .4 .4 .2 .2 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Round Round Cooperation Rate Belief 95% Conf. Int. Coop. 95% Conf. Int. Belief Late supergames. 15 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA B ELIEFS E RRORS IN E ARLY VS . L ATE S UPERGAMES Finite Indefinite .15 .15 .1 .1 .05 .05 Belief error Belief error 0 0 -.05 -.05 -.1 -.1 Late supergames -.15 -.15 Early supergames 1 6 7 8 1 6 7 8 Round Round Belief error denotes average difference between beliefs and actions. 16 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA B ELIEFS E RRORS : R OUNDS 1 AND 7 Finite Indefinite 1 1 .8 .8 Cooperation Rate Cooperation Rate .6 .6 .4 .4 Early Late Early Late .2 .2 Coop-Round 1 Belief-Round 1 Coop-Round 7 Belief-Round 7 0 0 1 2 3 4 5 6 7 L-2 L-1 Last 1 2 3 4 5 6 7 L-2 L-1 Last Supergames Supergames 17 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA Result 3 ( Question 1 ) Overall, average beliefs are accurate . However, some systematic deviations are present and persistent. K EY DIFFERENCE : end-of-game optimism in Finite games and early pessimism in Indefinite games. 18 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA C ONDITIONAL B ELIEFS : F INITE Coop-Coop in Round 1 Coop-Defect in Round 1 70% 14% Cooperation Rate 0 .2 .4 .6 .8 1 Cooperation Rate 0 .2 .4 .6 .8 1 Actions Beliefs 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Round Round Defect-Coop in Round 1 Defect-Defect in Round 1 14% 2% Cooperation Rate Cooperation Rate 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Round Round Own choice listed first. Late supergames. Percentage of cases under the title. 19 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA C ONDITIONAL B ELIEFS : I NDEFINITE Coop-Coop in Round 1 Coop-Defect in Round 1 71% 12% Cooperation Rate 0 .2 .4 .6 .8 1 Cooperation Rate 0 .2 .4 .6 .8 1 Actions Beliefs 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Round Round Defect-Coop in Round 1 Defect-Defect in Round 1 12% 5% Cooperation Rate Cooperation Rate 0 .2 .4 .6 .8 1 0 .2 .4 .6 .8 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Round Round Own choice listed first. Late supergames. Percentage of cases under the title. 20 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA A CCURACY 20 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA A CCURACY Finite Indefinite Correct Within Correct Within Tercile 10% 5% Tercile 10% 5% Round 1 73 14 7 67 10 5 Round 2 CC 91 60 9 91 66 58 Round 1 CD 67 16 9 29 10 2 Actions DC 66 7 7 56 17 12 DD 67 8 8 79 0 0 Average 83 45 9 80 52 45 Round 1 actions are listed own action first, other action second: i.e. ( a i , a j ) . Average is weighted by the number of observations. 20 / 38
I NTRODUCTION D ESIGN A CTIONS AND B ELIEFS S TRATEGIES C ONCLUSION E XTRA C OOPERATIVE P ATH Finite Indefinite 1 1 .8 .8 Cooperation Rate Cooperation Rate .6 .6 .4 .4 .2 .2 Actions: Cooperative Path Beliefs: Cooperative Path 0 0 2 3 4 5 6 7 8 2 3 4 5 6 7 8 Round Round Cooperative path: joint cooperation up to that round. Late supergames. 21 / 38
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