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: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

可引述

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

本文之實驗

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.

本文所用無母數統計方法

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

Abstract

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.

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廣告

End Behavior in Sequences of Finite Prisoner’s Dilemma Supergames: A Learning Theory Approach

Reinhard and Selten (1986) “End Behavior in Sequences of Finite Prisoner’s Dilemma Supergames: A Learning Theory Approach." Journal of Economic Behavior & Organization

Volume 7, Issue 1, March 1986, Pages 47–70. http://dx.doi.org/10.1016/0167-2681(86)90021-1. systemsci.org 提供的 [PDF];

Abstract

A learning theory is proposed which models the influence of experience on end behavior in finite Prisoner’s Dilemma supergames. The theory is compared with experimental results. In the experiment 35 subjects participated in 25 Prisoner’s Dilemma supergames of ten periods each against anonymous opponents, changing from supergame to supergame. The typical behavior of experienced subjects involves cooperation until shortly before the end of the supergame. The theory explains shifts in the intended deviation period. On the basis of parameter estimates for each subject derived from the first 20 supergames, successful predictions could be obtained for the last five supergames.

 

PD game 相關文獻回顧 (以合作為觀點)

Rational cooperation in the finitely repeated prisoners’ dilemma

DM Kreps, P Milgrom… – Journal of Economic Theory, 1982 – Elsevier
A common observation in experiments involving finite repetition of the prisoners’ dilemma is that players do not always play the single-period dominant strategies (“finking”), but instead achieve some measure of cooperation. Yet finking at each stage is the only Nash equilibrium in
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Rational cooperation in the finitely repeated prisoner’s dilemma: Experimental evidence

J Andreoni… – The Economic Journal, 1993 – JSTOR
In the finitely repeated prisoner’s dilemma, it is well known that defection in every game is the unique dominant-strategy Nash equilibrium. This follows from the familiar backward-induction arguments. Kreps et al. (i 982), however, show that if there is incomplete information
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Notes by Yi-Nung

Kreps et al. (1982) 提出了一個引發許多後續研究的有趣模型,他們證明在有限重覆的 PD 賽局中,如果賽局雙方對其對手之行為模式有某種的不完全資訊的話,則在此有限重覆的賽局之早期中,雙方皆採合作策略是符合理性的。…(待續)

Cooperation without reputation: Experimental evidence from prisoner’s dilemma games

R Cooper, DV DeJong, R Forsythe… – Games and Economic …, 1996 – econ.ucsd.edu 0013 Cooperation without Reputation: Experimental Evidence from Prisoner’s Dilemma Games*
knowledge of rationality does not hold in this theoretical structure.
被引用 218 次
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Identifying cooperative behavior: some experimental results in a prisoner’s dilemma game

J Brosig – Journal of Economic Behavior & Organization, 2002 – Elsevier
that individuals with a cooperative disposition experience strong emotions that prevent them from cheating, then it should also be observed that, after communicating, these individuals will not exploit the expected cooperation of their partner in prisoner’s dilemma games.
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The evolution of cooperation in infinitely repeated games: Experimental evidence

PD Bo… – The American Economic Review, 2011 – ingentaconnect.com
criticism of the theory of infinitely repeated games is that it does not pro- vide solution has been provided to the problem of equilibrium selection: when both cooperation and defection Previous experimental evidence has shown that subjects often fail to coordinate on a spe- cific
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Tacit Cooperation, Strategic Uncertainty, and Coordination Failure: Evidence from Repeated Dominance Solvable Games

JB Van Huyck, JM Wildenthal… – Games and Economic …, 2002 – Elsevier
R. Cooper, DV DeJong, R. Forsythe and TW Ross, Cooperation without Reputation: Experimental Evidence from Prisoners P. Milgrom, J. Roberts and R. Wilson, Rational Cooperation in the and JK Murnighan, Equilibrium Behavior and Repeated Play of the Prisoner’s Dilemma.
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Endogenous transfers in the Prisoner’s Dilemma game: An experimental test of cooperation and coordination

G Charness, GR Fréchette… – Games and Economic Behavior, 2007 – Elsevier
We wished to not only test the general effectiveness of endogenous payments for cooperation We noted earlier that mutual cooperation is the unique action pair consistent with SPE As controls, we conducted one session for each game without the possibility of transfer payments
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The impact of the termination rule on cooperation in a prisoner’s dilemma experiment

HT Normann, B Wallace – University of London Royal Holloway …, 2006 – papers.ssrn.com
unknown horizon would be. Empirically, however, it is well known that stable cooperation does occur also in finitely repeated games. – A related concern is to avoid endgame effects. Morehous (1966) observed that defection
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Cooperation in infinitely repeated games: Extending theory and experimental evidence

M Blonski, P Ockenfels… – JW Goethe Universität Frankfurt, …, 2007 – luiss.it
A proper understanding of the determinants of cooperation is crucial for the social sciences.  In this paper we propose a novel theoretical tool that enhances this understanding. Accordingly we present experimental evidence designed to differentiate clearly among the prevailing
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Notes by Yi-Nung

這篇談及了讓人們之合作較容易形成的四個條件 (Mailath G. and L. Samuelson, 2006):
(i) 合作利得相對大於不合作
(ii) 對手採合作下, 不合作利得相對小
(iii) 因不合作而預期被報復 (或懲罰) 的恐懼或嚴重性高
(iv) 較有耐心或較重視未來價值者

AE Roth… – Journal of Mathematical Psychology, 1978 – Elsevier
Equilibrium behavior and repeated play of the prisoner’s dilemma. Two equilibrium indices are derived and are compared with two cooperation indices proposed by Behavior, communication, and assumptions about other people’s behavior in a commons dilemma situation.
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Notes by Yi-Nung

這篇是 PD game ending-rule 設成 q = (1-p) 的原始文獻 (其中也有實驗)

其中其實已經提到 ‘Tit-for-Tat’ 策略 (不只是從 Kreps et al. 1982, JET 來的):

… a player  to  play  his  cooperative  choice  in  the  first  period,  and  in  every  other period  to  play  the  same  choice  that  his  opponent  made  in  the  previous  period.

The [experimental] findings  also  indicated  that  the  last  play  of  an  n-period  game  may  result  in  a  reduction in  cooperative  choices.

應該把 ending-rule 設成機率的原因:

It  is  often  contended  in  the  literature  that  if  subjects  are  not  informed  of  the  number  periods  to  be played,  the  resulting  game yields  the  same  equilibria  as  the  infinite  game, since  no period  is  known  to  be  the  last.  However,  this  is  a considerable  oversimplification.  Since  it  is  apparent  that  the  game  must  eventually  terminate,  subjects  must  form  subctive  probabilities  greater  than  zero  that  a given  period  might  be  the  last.  Although  such obabilities  have  neither  been  observed  nor  controlled  by  experimenters,  we  shall  see that  they  play  a critical  role  in  determining  the  nature  of  equilibrium  outcomes.  (原因何在?)

P. 195

Morehous’s  (1966)  results  for  a game  with  a fixed duration  of  1, 2,  5,  or  10 periods  are  an  important  comparison.  He  reports  no  significant difference  in  the  amount  of  cooperation  evidenced  in  these  games,  although  the  results were  in  the  same  direction  as  our  results.

有關 face-to-face

E Williams – Psychological Bulletin, 1977 – psycnet.apa.org
1971) again used the Prisoner’s Dilemma, though only one of the pair was a subject, the other being a confederate. At certain points in the game, the confederate gave standardized friendly or unfriendly mes- sages in written form, by audio only, by audio-video, or face to face.
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The sound of silence in prisoner’s dilemma and dictator games

I Bohnet… – Journal of Economic Behavior & Organization, 1999 – Elsevier
exchanged in facetoface interactions which induces individuals to change their behavior, but rather the participants’ identification. While mutual identification allows for reciprocity, one-way identification excludes future social sanctions. In the prisoner’s dilemma (PD) game
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Collective action and the evolution of social norms

E Ostrom – The Journal of Economic Perspectives, 2000 – JSTOR
To the surprise of the experimenters, a higher level of cooperation occurred in the control groups that played the regular prisoner’s dilemma in both phases, especially for those who communicated on a facetoface basis. The
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Some consequences of e-mail vs. facetoface communication in experiment

N Frohlich… – Journal of Economic Behavior & …, 1998 – Elsevier
be that when there is substantial substance to communicate and debate because subjects face a complex decision, facetoface interaction helps establish what both individual and group interests dictate. The act of talking with one another may reveal the underlying dilemma.
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Do economists make bad citizens?

RH Frank, TD Gilovich… – The Journal of Economic Perspectives, 1996 – JSTOR
In naturally occurring social dilemmas, facetoface promises are sometimes an option, sometimes not. But when combined with our survey of charitable giving and the results of our prisoner’s dilemma experiments-as well as parallel findings reported by Marwell and Ames (1981
被引用 181 次相關文章全部共 13 個版本unimc.it 提供的 [PDF]

M Shubik – Journal of Conflict Resolution, 1970 – JSTOR
interaction. Third parties and writ- ten documents, as well as the avoidance of individual facetoface confrontation, are often used to remove sociopsychological factors. Let us return to our specific analysis of the Prisoner’s Dilemma. When
被引用 86 次相關文章全部共 6 個版本auth.gr 提供的 [PDF]

The value of a smile: Game theory with a human face

yinung: 看一些 smile 的照片後, 再進行實驗

JPW Scharlemann, CC Eckel, A Kacelnik… – Journal of Economic …, 2001 – Elsevier
Choosing the cooperative strategy is risky, because cheaters can always take advantage of the cooperator. In a prisoner’s dilemma game, defecting leaves the cheater better off and makes the cooperator worse off. In an exchange relationship the same is true.
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和電腦互動的文獻

A prisoner’s dilemma experiment on cooperation with people and human-like computers.

S Kiesler, L Sproull… – Journal of Personality and Social …, 1996 – psycnet.apa.org
Andreoni and Miller (1993) obtained moderately high coop- eration rates by having participants play a prisoner’s dilemma game with an acquaintance of the 3 As would be expected in the absence of visibility and facetoface communication, cooperation rates are lower in these
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與 end-game effect 有關

Normann and Wallace (2012, IJGT) The impact of the termination rule on cooperation in a prisoner’s dilemma experiment

此文提到其它 end-game 的文獻:(最早?) Morehous (1966), Axelrod (1984), Murnighan and Roth (1983), Holt (1985)

Andreoni and John H. Miller (1993) Rational Cooperation in the Finitely Repeated Prisoner’s Dilemma: Experimental Evidence

已注意到 end-game 但避談 (用刪去最後樣本來處理)

Wichman, Harvey. 1970. ‘‘Effects of Isolation and Communication on Cooperation in a Two-Person Game.’’ Journal of Personality and Social Psychology 16:114–20.

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

Date: 2011
By: Normann, Hans-Theo
Wallace, Brian
URL: http://d.repec.org/n?u=RePEc:zbw:dicedp:19&r=net
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
JEL: C72