Dal Bó, Pedro ; Fréchette, Guillaume R. (2011) “The Evolution of Cooperation in Infinitely Repeated Games: Experimental Evidence." The American Economic Review, Volume 101, Number 1, February 2011 , pp. 411-429(19). DOI: http://dx.doi.org/10.1257/aer.101.1.411;brown.edu 提供的 [PDF] ; Download Data Set (355.92 KB) | Online Appendix (284.08 KB);
==notes by yinung==
========================= 合作 背叛 合作 (R, R) (12, 50) 背叛 (50,12) (25, 25) ========================= R＝32,40,48 繼續玩之機率：1/2, 3/4
…each subject participated in between 23 and 77 infinitely repeated games
…Previous experimental evidence has shown that subjects often fail to coordinate on a specific equilibrium when they play a small number of infinitely repeated games
…the level of cooperation decreases with experience and converges to low levels
…the level of cooperation does not necessarily increase and may remain at low levels even after significant experience is obtained.
…this evidence suggests that while being an equilibrium action may be a necessary condition for cooperation to arise with experience, it is not sufficient.
若 cooperation 是 risk dominant, 則合作隨著經驗增加而上升。
If we consider together all sessions for which cooperation is risk dominant, we find that cooperation increases on average as subjects gain experience….Risk dominance has been used as a selection criterion in the study of coordination games.
…These results show how difficult it is for experienced subjects to sustain high levels of cooperation. They cast doubt on the common assumption that agents will make the most of the opportunity to cooperate whenever it is possible to do so in equilibrium.
…In fact the impact of repetition on rates of cooperation was rather modest, leading Roth to conclude that the results are equivocal (Roth 1995). (註：這些結果皆來自於隨機結束的賽局 randomly terminated game. All of these papers used games with a randomly determined length.）
A usual criticism of the theory of infinitely repeated games is that it does not provide sharp predictions since there may be a multiplicity of equilibria. To address this issue, we present experimental evidence on the evolution of cooperation in infinitely repeated prisoner’s dilemma games as subjects gain experience. We show that cooperation may prevail in infinitely repeated games, but the conditions under which this occurs are more stringent than the subgame perfect conditions usually considered or even a condition based on risk dominance.
- Bereby-Meyer,Yoella, and Alvin E. Roth. 2006. “The Speed of Learning in Noisy Games: Partial Re-inforcement and the Sustainability of Cooperation.” American Economic Review, 96(4): 1029–42.
Dal Bó, Pedro. 2005. “Cooperation under the Shadow of the Future: Experimental Evidence from Infinitely Repeated Games.”American Economic Review, 95(5): 1591–1604.
這篇比較無限和有限重覆 PD game, 發現無限重覆賽局，在同樣條件下會有較高的合作率。[this paper] compares infinitely repeated and finitely repeated prisoner’s dilemma games of the same expected length and finds that cooperation is larger in the former as theory predicts.
- Dal Bó, Pedro. 2007. “Tacit Collusion under Interest Rate Fluctuations.” RAND Journal of Economics, 38(2): 533–40.
Hans-Theo Normann, Brian Wallace (2012) “The impact of the termination rule on cooperation in a prisoner’s dilemma experiment." International Journal of Game Theory, August 2012, Volume 41, Issue 3, pp 707-718. link to Springer; working版本：ucl.ac.uk 提供的 [PDF]；hhu.de 提供的 [PDF]；
Note by yinung
這篇原是 DICE 的 working paper (see 本站另一篇 PO 文），現在刊出來了。
主要結論 （in abstract):
此文研究3種 PD game 實驗回合結束的方式 (告知結束回合、不告知結束回合、隨機結束)，比較合作率之不同。 1. 三種結束方式不影響合作率 2. 隨機結束方式不會提高合作率 (相對於告知結束回合)；不同繼續玩 (隨機結束) 之機率高低 (continuation probability) 亦不影響合作率 3. 結束方式會影響 over time 和 end-game 行為 4. 預期玩的回合愈長，合作率愈高
三種 termination rule:
- finite horizon: Flood (1952) and Rapoport and Chammah (1965) … it is well known that stable cooperation does occur also in finitely repeated games
- unknown horizon: Fouraker and Siegel 1963
- random-stopping rule: to terminate the experiment (Roth and Murnighan1978; Axelrod 1980)
End-game effects: Morehous (1966) … defection rates increase towards the end of the game when the horizon of the game is known to be finite. … used a probabilistic termination rule so that “end-game effects were successfully avoided” (Axelrod 1984, p. 42). Murnighan and Roth (1983, p. 284) argue that “consideration of end-game play is less critical” with the random termination rule. Holt (1985, p. 320) makes the same point. 也有人在分析結果時，去掉最後幾回合 贊同應用隨機結束 With finitely many periods, the theory is bland; by contrast, the random termination rule “permits the nature of the equilibrium outcomes to be controlled” (Roth and Murnighan 1978, p. 191) Selten and Stoecker (1983) further noted that subjects learn to anticipate the endgame effect in that this effect is shifted to earlier rounds when a supergame with a finite horizon is repeated several times (see also Andreoni and Miller 1993) 隨機結束機率高者，使合作率較高 （與本文結果不一致），但重覆的實驗不能確定此一結果 Roth and Murnighan (1978) found that a random stopping rule with higher continuation probability does lead to more cooperation in the prisoner’s dilemma. However, in the modified setup analyzed in Murnighan and Roth (1983), this could not be confirmed. Dal Bo (2005) 發現隨機結束機率(在 supergame 玩 10 次後) 有重要影響
至少玩 22 回合 (??? 不知何義，待了解… 因為結束回合數不一定一樣), 4 情境、每情境有15組人
- RandomHigh (5/6 繼續機率)
- RandomLow (1/6 繼續機率)
supergame 不重覆 （？？）Subjects were rematched after the first supergame. 另外有 Shorter Horizon 額外較短實驗
- Known5 （9組，2人一組）
- Known10 （11組）
- Random5+5 (11組；至少5回，5/6 繼續機率，平均期望值＝10回，恰與 Known10 對照)
合作率 要小心合作率的定義！（各文獻不一定相同) 其 Table 2 中的合作率是 cooperte choices (不是 cooperate outcome, 所以22回合中才最多有 44 個 cooperate choices) 此文皆是利用 cooperate choice 來分析，Harvey Wichman (1970, J of Personality and Social Psychology) 也是用此定義 有的文獻合作率的數字是 cooperate outcome (兩人皆是 cooperate choice 才算合作） (引文)…In order to take the possible dependence of observations between paired players into account, we count each participating pair as one observation. Matrix of the game 從合作到背叛，邊際利得只有 1000－800 ＝ 200, 反而對手損失 750 較多；若因此而雙方開始不合作，每期邊際損失 250，根本不值得背叛；但是一旦被背叛，當次邊際損失 700， 要3次背叛才得以報復。
================================ 背叛 合作 背叛 (350, 350) (1000, 50) 合作 (50, 1000) (800, 800) ================================
文中所述之 5 個 results
- Result 1. The termination rule does not significantly affect average cooperation.
2種隨機結束和告知回合、不告知 等 treatment 合作率無顯著不同 (Kruskal-Wallis test)
- Result 2. There is a negative and significant time trend in treatments Known and RandomLow.
Known 和 RandomLow 合作率有下降趨勢 (significant time trend)
- Result 3. A significant end-game effect occurs in all treatments except Unknown。
- Result 4 The termination rule does not significantly affect average cooperation rates in treatments Known5, Known10, Random5+5.
- Result 5 The length of the horizon of the game significantly increases cooperation rates.
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|By:||Hitoshi Matsushima (The University of Tokyo)
Tomomi Tanaka (Economic Development & Global Education, LLC)
Tomohisa Toyama (Kogakuin University)
We examine repeated prisonersâ€™ dilemma with imperfect private monitoring and random termination where the termination probability is low. We run laboratory experiments and show subjects retaliate more severely when monitoring is more accurate. This experimental result contradicts the prediction of standard game theory. Instead of assuming full rationality and pure self-interest, we introduce naivete and social preferences, i.e., reciprocal concerns, and develop a model that is consistent with, and uniquely predicts, the observed behavior in the experiments. Our behavioral model suggests there is a trade-off between naivete and reciprocity. When people are concerned about reciprocity, they tend to make fewer random choices.
Cooperation in prisoner’s dilemma games can usually be sustained only if the game has an infinite horizon. We analyze to what extent the theoretically crucial distinction of finite vs. infinite-horizon games is reflected in the outcomes of a prisoner’s dilemma experiment. We compare three different experimental termination rules in four treatments: a known finite end, an unknown end, and two variants with a random termination rule (with a high and with a low continuation probability, where cooperation can occur in a subgame-perfect equilibrium only with the high probability). We find that the termination rules do not significantly affect average cooperation rates. Specifically, employing a random termination rule does not cause significantly more cooperation compared to a known finite horizon, and the continuation probability does not significantly affect average cooperation rates either. However, the termination rules may influence cooperation over time and end-game behavior. Further, the (expected) length of the game significantly increases cooperation rates. The results suggest that subjects may need at least some learning opportunities (like repetitions of the supergame) before significant backward induction arguments in finitely repeated game have force. —
|Keywords:||Prisoner’s dilemma,Repeated games,Infinite-horizon games,Experimental economics|