Theory and experiment: What are the questions?

Smith, Vernon L. “Theory and experiment: What are the questions?." Journal of Economic Behavior & Organization 73.1 (2010): 3-15. [PDF];[my notes]


Smith 提到了 OPM (other person’s money) 問題,可以用以下的方式解決

We could give the constant positive sum ultimatum game economic content as follows: Each player provides $M of his own money. Some procedure is used for pairing the subjects, and determining who is to be Player 1 and who Player 2; this procedure in some variations might incorporate an earned and/or investment feature. It is understood that their pairing has economic significance in the sense that there are synergistic gains from the interaction equal to some fixed sum y > 2M. The experimenter provides only the surplus above 2M which represents the gains from specialization and exchange, as this is the one reliable source of a “free lunch” that converts economic systems into non-zero sum games. Hence, the total to be shared under the property right rules of the game is 2M + y, making it feasible for each to receive a share of the jointly created net gain above their pooled initial contribution, 2M.


New directions for modelling strategic behavior: Game-theoretic models of communication, coordination, and cooperation in economic relationships

Crawford, Vincent P. “New directions for modelling strategic behavior: Game-theoretic models of communication, coordination, and cooperation in economic relationships." Journal of Economic Perspectives 30.4 (2016): 131-50.


==original Abstract==

In this paper, I discuss the state of progress in applications of game theory in economics and try to identify possible future developments that are likely to yield further progress. To keep the topic manageable, I focus on a canonical economic problem that is inherently game-theoretic, that of fostering efficient coordination and cooperation in relationships, with particular attention to the role of communication. I begin with an overview of noncooperative game theory’s principal model of behavior, Nash equilibrium. I next discuss the alternative “thinking" and “learning" rationales for how real-world actors might reach equilibrium decisions. I then review how Nash equilibrium has been used to model coordination, communication, and cooperation in relationships, and discuss possible developments

Does Experience Affect Fairness and Reciprocity in Lab Experiments?

Date: 2016-07
By: Tiziana Medda (University of Cagliari)
Vittorio Pelligra (University of Cagliari)
Tommaso Reggiani (LUMSA University)
One of the most common criticisms about the external validity of lab experiments in economics concerns the representativeness of participants usually considered in these studies. The ever-increasing number of experiments and the prevalent location of research centers in university campuses produced a peculiar category of subjects: Students with high level of laboratory experience built through repeated participations in experimental sessions. We investigate whether the experience accumulated in this way biases subjects’ behaviour in a set of simple games widely used to study social preferences (Dictator Game, Ultimatum Game, Trust Game, and Prisoner’s Dilemma Game). Our main finding shows that subjects with a high level of experience in lab experiments do not behave in a significantly different way from novices.
Keywords: Experimental Methodology, External Validity, Experience, Lab Experiment
JEL: D03 D83 C91 C92

Individual Learning and Cooperation in Noisy Repeated Games

Date: 2013-07-06
By: Yuichi Yamamoto (Department of Economics, University of Pennsylvania)
We investigate whether two players in a long-run relationship can maintain cooperation when the details of the underlying game are unknown. Specifically, we consider a new class of repeated games with private monitoring, where an unobservable state of the world influences the payoff functions and/or the monitoring structure. Each player privately learns the state over time but cannot observe what the opponent learned. We show that there are robust equilibria in which players eventually obtain payoffs as if the true state were common knowledge and players played a “belief-free” equilibrium. We also provide explicit equilibrium constructions in various economic examples
Keywords: repeated game, private monitoring, incomplete information, belief-free equilibrium, ex-post equilibrium, individual learning
JEL: C72

Fairness norms can explain the emergence of specific cooperation norms in the Battle of the Prisoners Dilemma

Date: 2013-04-24
By: Fabian Winter (Max Planck Institute of Economics, Jena)
Cooperation norms often emerge in situations, where the long term collective benefits help to overcome short run individual interests, for instance in repeated Prisoner’s Dilemma (PD) situations. Often, however, there are different paths to cooperation, benefiting different kinds of actors to different degrees. This leads to payoff asymmetries even in the state of cooperation, and consequently can give rise to normative conflicts about which norms should be in place. This norm-coordination problem will be modeled as a Battle of the Sexes game (BoS) with different degrees of asymmetry in payoffs. We combine the PD and the BoS to the 3×3 Battle of the Prisoners Dilemma (BOPD) with several asymmetric cooperative and one non-cooperative equilibria. Bame theoretical and “behavioral" predictions are derived about the kind of norms that are likely to emerge under different shadows of the future and degrees of asymmetry and tested in a lab-experiment. Our experimental data show that game theory fairly well predicts the basic main effects of our experimental manipulations, but “behavioral" predictions perform better in describing the equilibrium selection process of emerging norms.
Keywords: Social norms, normative conflict, Prisoner’s Dilemma, coordination, experiment
JEL: Z13

Cooperation under the Shadow of the Future: Experimental Evidence from Infinitely Repeated Games

Bó, Pedro Dal (2005) “Cooperation under the Shadow of the Future: Experimental Evidence from Infinitely Repeated Games." American Economic Review, Volume 95, Number 5, December 2005 , pp. 1591-1604(14). ; 提供的 [PDF]  ; DOI: ;另見本站另一篇同作者在 2005AER 的文章


==notes by yinung==

本篇以有限重覆賽局 PD game 為對照,比較無限重覆賽局之影響。

主要針對 equilibrium actions and equilibrium outcome ,而非 equilibrium strategies (可能要參考 Dale O.  Stahl II (1991))

I focus on equilibrium actions and outcomes instead of equilibrium strategies,
I find that the percentage of outcomes in which both subjects cooperate is almost 19 percent when it is an equilibrium, whileit is less than 3 percent when it is not.


PD1                           PD2
=============================     =============================
          合作        背叛                  合作     背叛
 合作     (65, 65)   (10, 100)     合作     (75, 75)   (10, 100)
 背叛     (100,10)   (35, 35)      背叛     (100,10)   (45, 45)
=============================     =============================

有限賽局的設計(Finitely Repeated Games):

共有 1, 2,4 回合三種

子賽局均衡之推導,參見 Dale O.  Stahl II (1991)

the set of subgame perfect equilibria can be calculated using the results in Dale O.  Stahl II (1991).


…find strong evidence that the higher the probability of continuation, the higher the levels of cooperation.


針對 one-shot PD game; 合作率 9%; 無限重覆 (繼續機率=3/4) 合作率 38%。

…in the one-shot prisoner’s dilemma games studied here, the cooperation rate is 9 percent, for a probability of continuation of 3/4, it is 38 percent.

在有限重覆賽局中,也有終局效果 end-game effect,在最後一回合,合作率較低。

I find that the level of cooperation in the final round of the finitely repeated games is similar to the level of cooperation in one-shot games.


While there is an extensive literature on the theory of infinitely repeated games, empirical evidence on how “the shadow of the future" affects behavior is scarce and inconclusive. I simulate infinitely repeated prisoner’s dilemma games in the lab with a random continuation rule. The experimental design represents an improvement over the existing literature by including sessions with finite repeated games as controls and a large number of players per session (which allows for learning without contagion effects). I find that the shadow of the future matters not only by significantly reducing opportunistic behavior, but also because its impact closely follows theoretical predictions.


  • 子賽局均衡之推導和 rate of cooperation outcome 的定義 (??)
    Stahl, Dale O., II. “The Graph of Prisoner’s Dilemma Supergame Payoffs as a Function of the Discount Factor." Games and Economic Behavior, 1991, 3 (3), pp. 368 – 84.

The Evolution of Cooperation in Infinitely Repeated Games: Experimental Evidence

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:; 提供的 [PDF] ; Download Data Set (355.92 KB) | Online Appendix (284.08 KB);

==notes by yinung==

此文用的 PD game,進行無限重覆賽局的實驗。see also 另一篇也在 AER 2005 年刊出的和有限重覆 PD game 之比較(或參目本站的另一篇 note)。

          合作     背叛
 合作    (R, R)   (12, 50)
 背叛    (50,12)  (25, 25)
繼續玩之機率:1/2, 3/4


The 18 experimental sessions were conducted between July 2005 and March 2006. A total of 266 New York University undergraduates participated in the experiment, with an average of 14.78 subjects per session, a maximum of 20 and a minimum of 12. The subjects earned an average of $25.95, with a maximum of $42.93 and a minimum of $16.29. In the treatments with δ =
1/2 and δ = 3/4 the average number of rounds per match was 1.96 and 4.42 respectively, and the maximum was nine and 23 respectively.


…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.

Rational cooperation in the finitely repeated prisoner’s dilemma: Experimental evidence

Andreoni, J, Miller JH (1993) Rational cooperation in the finitely repeated prisoners’ dilemma: experimental evidence. Econ J 103: 570–585.  CrossRef

==from Google==
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 …
被引用 386 次相關文章全部共 14 個版本; 提供的 [PDF]

==notes by yinung==

此文沒有提及全部平均合作率, 只有 Figure 3 講到最後 2個 10 回合 PD game 結果

partners => 86% (1st round), above 50% (4-6 round), 0% (最後)

4 種情境 (皆玩 20 個 10回合之 PD game

  • Partners (14人, 兩人一組)

每回合對手皆相同 (電腦隨機決定配對之對手, 此後…)

  • Strangers (14人, 兩人一組)


  • Computer50

電腦報復 (tit-for-tat) 機率 50%

  • Computer0

電腦報復機率 1/1000

Matrix of the game

          合作     背叛
 合作     (7, 7)   (0, 12)
 背叛     (12,0)   (4, 4)

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.


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Behavioral Approach to Repeated Games with Private Monitoring

Date: 2013-03
By: Hitoshi Matsushima (The University of Tokyo)
Tomomi Tanaka (Economic Development & Global Education, LLC)
Tomohisa Toyama (Kogakuin University)
We examine repeated prisoners’ 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.