The impact of the termination rule on cooperation in a prisoner’s dilemma experiment

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 提供的 [PDF] 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

  • Know
  • Unknow
  • 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。
2種隨機結束和告知回合都有 End-game effect (在最後幾回合作率顯著逐漸下降的情況)。 折現率~= 4/13 時,{C,C} is a subgame perfect Nash equilibrium outcome of the infinitely repeated game if and only if the discount factor is larger than 4/13 ~ 0.31.
  • 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.


Kruskal-Wallis test
Testing for differences in cooperation with all treatments jointly does not suggest significant results


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 versus 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.


  1. Andreoni J, Miller JH (1993) Rational cooperation in the finitely repeated prisoners’ dilemma: experimental evidence. Econ J 103: 570–585 CrossRef
  2. Angelova V, Bruttel LV, Güth W, Kamecke U (2011) Can subgame perfect equilibrium threats foster cooperation? An experimental test of finite-horizon folk theorems. Econ Inq (forthcoming)
  3. Axelrod R (1984) The evolution of cooperation. Basic Books, New York
  4. Axelrod R (1980) More effective choice in the prisoner’s dilemma. J Confl Resolut 24: 379–403 CrossRef
  5. Benoit J-P, Krishna V (1985) Finitely repeated games. Econometrica 53(4): 905–922 CrossRef
  6. Benoit J-P, Krishna V (1987) Nash equilibria of finitely repeated games. Int J Game Theory 16(3): 197–204 CrossRef
  7. Bolton GE, Ockenfels A (2000) ERC: a theory of equity, reciprocity and competition. Am Econ Rev 90: 166–193 CrossRef
  8. Bruttel L, Kamecke U (2012) Infinity in the lab. How do people play repeated games? Theory Dec 72(2): 205–219
  9. Bruttel LV, Güth W, Kamecke U (2012) Finitely repeated prisoners’ dilemma experiments without a commonly known end. Int J Game Theory 41(1): 23–47 CrossRef
  10. Dal Bo P (2005) Cooperation under the shadow of the future: experimental evidence from infinitely repeated games. Am Econ Rev 95: 1591–1604 CrossRef
  11. Engle-Warnick J, Slonim RL (2004) The evolution of strategies in a repeated trust game. J Econ Behav Organ 55: 553–573 CrossRef
  12. Fehr E, Schmidt KM (1999) A theory of fairness, competition and cooperation. Q J Econ 114: 817–868 CrossRef
  13. Feinberg RM, Husted TA (1993) An experimental test of discount effects on collusive behavior in dupoly markets. J Ind Econ 41(2): 153–160 CrossRef
  14. Fischbacher U (2007) Z-Tree, Zurich toolbox for readymade economic experiments. Exp Econ 10(2): 171–178 CrossRef
  15. Flood MM (1952) Some experimental games. Research Memorandum RM-789. RAND Corporation, Santa Monica, CA
  16. Fouraker L, Siegel S (1963) Bargaining behavior. McGraw-Hill, New York
  17. Gonzales LG, Güth W, Levati V (2005) When does the game end? Public goods experiments with non-definite and non-commonly known time horizons. Econ Lett 88(2): 221–226 CrossRef
  18. Hollander M, Wolfe DA (1999) Nonparametric statistical methods. Wiley, New York
  19. Holt CH (1985) An experimental test of the consistent—conjectures hypothesis. Am Econ Rev 75: 314–325
  20. Kaplan T, Ruffle B (2006) Which way to cooperate? Working Paper, Ben-Gurion University
  21. Kreps DM, Milgrom P, Roberts J, Wilson R (1982) Rational cooperation in the finitely repeated prisoner’s dilemma. J Econ Theory 27(2): 245–252 CrossRef
  22. Luce RD, Raiffa H (1957) Games and decisions: introduction and critical survey. Wiley, New York
  23. Morehous LG (1966) One-play, two-play, five-play, and ten-play runs of prisoner’s dilemma. J Confl Resolut 10: 354–361 CrossRef
  24. Murnighan JK, Roth AE (1983) Expecting continued play in prisoner’s dilemma games: a test of three models. J Confl Resolut 27: 279–300 CrossRef
  25. Neyman A (1999) Cooperation in repeated games when the number of stages is not commonly known econometrica. Econometrica 67: 45–64 CrossRef
  26. Orzen H (2008) Counterintuitive number effects in experimental oligopolies. Exp Econ 11(4): 390–401 CrossRef
  27. Rapoport A, Cammah AM (1965) Prisoner’s dilemma. A study in conflict and cooperation. University of Michigan Press, Ann Arbor
  28. Roth AE (1995) Bargaining experiments. In: Kagel J, Roth AE (eds) Handbook of experimental economics. Princeton University Press, Princeton, pp 253–348
  29. Roth AE, Murnighan JK (1978) Equilibrium behavior and repeated play of the prisoners’ dilemma. J Math Psychol 17: 189–198 CrossRef
  30. Samuelson L (1987) A note on uncertainty and cooperation in a finitely repeated prisoner’s dilemma. International Journal of Game Theory 16(3): 187–195 CrossRef
  31. Selten R, Stoecker R (1986) End behavior in finite prisoner’s dilemma supergames. J Econ Behav Organ 7: 47–70 CrossRef
  32. Selten R, Mitzkewitz M, Uhlich GR (1997) Duopoly strategies programmed by experienced players. Econometrica 65: 517–556 CrossRef
  33. Suetens S, Potters J (2007) Bertrand Colludes more than Cournot. Exp Econ 10(1): 71–77 CrossRef
  34. Stahl DO (1991) The graph of prisoner’s dilemma supergame payoffs as a function of the discount factor. Games Econ Behav 3: 360–384 CrossRef

2 comments on “The impact of the termination rule on cooperation in a prisoner’s dilemma experiment

  1. 引用通告: PD game 相關文獻回顧 (以合作為觀點) | 行為 & 經濟實驗文獻注 (備忘) by 楊奕農

  2. 引用通告: The impact of the termination rule on cooperation in a prisoner’s dilemma experiment | 行為 & 經濟實驗文獻注 (備忘) by 楊奕農


在下方填入你的資料或按右方圖示以社群網站登入: 標誌

您的留言將使用 帳號。 登出 /  變更 )

Google photo

您的留言將使用 Google 帳號。 登出 /  變更 )

Twitter picture

您的留言將使用 Twitter 帳號。 登出 /  變更 )


您的留言將使用 Facebook 帳號。 登出 /  變更 )

連結到 %s