Inequality, cooperation, and growth: An experimental study

In a three player dynamic public goods experiment, social output today determines production possibilities tomorrow. In each period, players choose to sabotage, to cooperate, or to play best response. Sabotage harms social output and growth. Mutual cooperation maximises both. The property rights to social output are distributed unequally. Extent and skew of inequality are varied. We observe equilibrium play in most cases. There is also substantial cooperation, but little sabotage. Our exogenous variations of inequality are neutral to growth, neither negatively correlated to cooperation, nor positively correlated to sabotage. The neutrality result is shown to be sensitive to the dynamic nature of the employed game.     

Building rational cooperation on their own: Learning to start small

We report experimental results for a twice‐played prisoners' dilemma in which the players can choose the allocation of the stakes across the two periods. Our point of departure is the assumption that some (but not all) people are willing to cooperate, as long as their opponent is sufficiently likely to do so. The presence of such types can be exploited to enhance cooperation by structuring the twice‐played prisoners' dilemma to “start small,” so that the second‐stage stakes are larger (but not too much larger) than the first‐stage stakes. We compare conditions where the allocation of stakes is chosen exogenously to conditions where it is chosen by the players themselves. We show that players gravitate toward the payoff‐maximizing strategy of starting small in a twice‐played prisoners' dilemma. Intriguingly, the salutary payoff effects of doing so are larger than those that arise when the same allocation is exogenously chosen.     

Contributions to Cournot oligopoly theory

This work contributes to a number of questions concerning oligopoly models. In particular, uniqueness of the Cournot equilibrium point is demonstrated under the assumption that either the unit price function is differentiable and the derivative is strictly negative or the cost functions are strictly convex. Also, under the assumption of either strictly decreasing unit price function or strictly convex cost functions, it is shown that (a) the total production level at equilibrium increases with entry of additional players, (b) that cooperation between some of the players necessarily entails profit for the others, and (c) cooperative grouping causes decrease in production levels.     

The tragedy of the commons revisited: The importance of group decision-making

We use a laboratory experiment to compare the way groups and individuals behave in an inter-temporal common pool dilemma. The experimental design distinguishes between a non-strategic problem where players (individuals or groups of three) make decisions without interaction and a strategic part where players harvest from a common pool. This allows us to correct for differences between individuals and groups in the quality of decisions when testing for differences in competitiveness. Group decisions are either made by majority rule or unanimity. The results show that groups are less myopic than individuals (i.e., they make qualitatively better decisions) but that they are more competitive than individuals when placed in a strategic setting. The net result for groups deciding by majority rule is that they make less efficient decisions in the strategic game than individuals do. We are able to show that this is caused by the median voter departing from her original preference in early periods with a shrinking pool. When groups have to make unanimous decisions they start playing the strategic game more efficiently then individuals do, but they rapidly become more competitive with repetition of the game.     

A Common Pool Resource Game with Sequential Decisions and Experimental Evidence

We describe a common pool resource game in which players choose how much of the stock to extract in a sequential manner. There are two choices and one represents taking a larger proportion of the stock than the other. After a player makes a choice, the remaining stock grows at a constant rate. We consider a game with a finite number of alternating moves. It is shown that changes in the larger proportion of the stock that the players are allowed to take and the growth rate affect equilibrium, but have little effect on behavior in the laboratory. In addition to observing more cooperation than predicted, we observe that parameters that are strategically irrelevant affect behavior. The results of this research might help policy makers in developing adequate policies to prevent overexploitation of some natural renewable resources.     

Framing and Misperception in Public Good Experiments

Earlier studies have found that framing has a substantial impact on the degree of cooperation observed in public good experiments. We show that the way the public good game is framed affects misperceptions about the incentives of the game. Moreover, we show that such framing‐induced differences in misperceptions are linked to the framing effect on subjects' cooperation behavior. When we do not control for the different levels of misperceptions between frames, we observe a significant framing effect on subjects' cooperation preferences. However, this framing effect becomes insignificant once we remove subjects who misperceive.     

The absent-minded prisoner

If one of two rational players is absent-minded for at least three rounds, cooperation in a prisoners’ dilemma with a finite number of repetitions is possible. If both players are absent-minded, even two rounds of absent-mindedness can be enough for cooperation in these rounds and all rounds before. Sufficient conditions for the existence of a cooperative equilibrium are derived, a plausible interpretation of absent-mindedness in the case of many repetitions is given.     

Public good agreements under the weakest‐link technology

We analyze the formation of public good agreements under the weakest‐link technology. Whereas policy coordination is not necessary for symmetric players, it matters for asymmetric players; however, this fails in the absence of transfers. By contrast, with a transfer scheme, asymmetry may be an asset for cooperation. We characterize various types and degrees of asymmetry and relate them to the stability of self‐enforcing agreements. Asymmetric distributions of autarky public good provision levels (also representing asymmetric interests in cooperation) that are positively skewed tend to be conducive to the stability of agreements. We show that under such conditions, even a coalition including all players can be stable. However, asymmetries that foster stability (instability) tend to be associated with low (high) gains from cooperation.     

Group efforts when performance is determined by the “best shot”

We investigate the private provision of a public good whose level is determined by the maximum effort made by a group member. Costs of effort are either commonly known or privately known. For symmetric perfect-information games, any number of players may be active and we characterize the unique (mixed-strategy) equilibrium in which active contributors use the same strategy. Increasing the number of active players leads to stochastically lower individual efforts and level of the public good. When information is private, the symmetric equilibrium is in pure strategies. Increasing the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Comparative statics with respect to costs and levels of risk aversion are derived. Finally, whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency.     

Group membership,team preferences,and expectations

Group membership is a powerful determinant of social behaviour in a variety of experimental games. Its effect may be channelled primarily via the beliefs of group members, or directly change their social preferences. We report an experiment with a prisoner's dilemma with multiple actions, in which we manipulate players’ beliefs and show that group identity has a consistent positive effect on cooperation only when there is common knowledge of group affiliation. We also test the robustness of the minimal group effect using three different manipulations: one manipulation fails to induce group identity, and we observe an unsystematic effect of group membership when knowledge of affiliation is asymmetric.     

Lab and framed lab versus natural experiments: Evidence from a risky choice experiment

Some recent papers have studied data from TV game shows to examine the behaviour of individuals towards risk. It is generally agreed that data from these shows are useful in detecting individual risk aversion in the field, with both “real life” subjects and incentives. Field experiments also include some interesting reality features that could affect individuals’ behaviour and possibly lead to different findings. This paper aims at investigating lab versus field evidence in risk taking attitudes, especially controlling for framing effects. To assess whether the behaviour of subjects in the field is consistent with that of experimental subjects, we designed an experiment to mimic (with experimental rewards and subjects) the rules of a well-known Italian TV game show, Affari Tuoi, in two different settings: a traditional lab setting, where the game was played individually (109 subjects) (Treatment 1); and a framed lab, in which the experiment was replicated in the Italian public television (RAI) studio where the show was actually recorded, with a smaller sample of undergraduate students (33) and in the presence of an audience (Treatment 2). Our comparison between the two different settings aims at establishing whether the presence of an audience, or of a situation that reproduces the stress that contestants must experience in the TV studio, can affect experimental subjects’ choices. We did not find any significant evidence of framing effects: students behave in a similar way in the two lab settings, responding essentially to incentives. Comparing the risk attitudes shown by experimental subjects in the two lab treatments with those exhibited by the contestants in the field, we found that contestants in the TV show are generally more risk averse than students in the lab.     

Cooperation in Stochastic OLG games

This paper builds on Cremer's (Quart. J. Econ. 101 (1986) 33) seminal analysis which shows that (almost) complete cooperation can be achieved as an equilibrium in a game played by overlapping generations of players if the institution in which players cooperate is infinitely lived.We analyze a similar model in which the costs of cooperation are subject to random shocks. Even if these random shocks are very small, the range of parameters for which cooperation can be sustained is decreased considerably in comparison to the deterministic case. Furthermore, we show how the efficient outcome can be approximated if the level of cooperation can be varied continuously and the cooperation technology has decreasing or constant returns to scale, while this is not possible in the case of increasing returns to scale.     

Rational cooperation in the finitely repeated prisoners' dilemma

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 the finitely repeated game. We show here how incomplete information about one or both players' options, motivation or behavior can explain the observed cooperation. Specifically, we provide a bound on the number of rounds at which Fink may be played, when one player may possibly be committed to a “Tit-for-Tat” strategy.     

THE EXPECTED NUMBER OF COOPERATORS IN THE N‐PERSON PRISONERS’ DILEMMA WITH RANDOM ORDER MOVES*

The present paper examines the possibility of cooperation occurring in the N‐person prisoners’ dilemma, played with randomly ordered moves and imperfect information. To take into account imperfect cooperation, the degree of cooperation is evaluated in terms of the expected number of cooperators. It is shown that: (i) the expected number of cooperators is maximized at an equilibrium, a combination of strategies each of which is a type of trigger strategy; (ii) the maximum expected number of cooperators is attained in a state of imperfect cooperation for a range of payoff functions; and (iii) the maximum expected number of cooperators is non‐decreasing when the size of players’ group increases.     

An experiment on learning with limited information: nonconvergence, experimentation cascades, and the advantage of being slow

We present the results of an experiment on learning in a continuous-time low-information setting. For a dominance solvable version of a Cournot oligopoly with differentiated products, we find little evidence of convergence to the Nash equilibrium. In an asynchronous setting, characterized by players updating their strategies at different frequencies, play tends toward the Stackelberg outcome which favors the slower player. Convergence is significantly more robust for a “serial cost sharing” game, which satisfies a stronger solution concept of overwhelmed solvability. As the number of players grows, this improved convergence tends to diminish, seemingly driven by frequent and highly structured experimentation by players leading to a cascading effect in which experimentation by one player induces experimentation by others. These results have implications both for traditional oligopoly competition and for a wide variety of strategic situations arising on the Internet.     

Stability and Largeness of the Core

In general, there are examples of TU games where the core is stable but is not large. In this paper, we show that the extendability condition introduced by Kikuta and Shapley (1986, “Core Stability in n-Person Games,” Mimeo) is sufficient for the core to be stable as well as large, for TU games with five or fewer players. We provide a counter example when the number of players is six. We then introduce a stronger extendability condition and show that it is necessary and sufficient for the core to be large. Our proof makes use of a well-known result from the theory of convex sets. Journal of Economic Literature Classification Number: C71.     

Cooperation in Experimental Games of Strategic Complements and Substitutes

We conduct a laboratory experiment aimed at examining whether strategic substitutability and strategic complementarity have an impact on the tendency to cooperate in finitely repeated two-player games with a Pareto-inefficient Nash equilibrium. We find that there is significantly more cooperation when actions exhibit strategic complementarities than in the case of strategic substitutes. The difference is to some extent driven by a difference in the speed with which some pairs reach stable full cooperation, but mainly by differences in choices of pairs that do not succeed in reaching full cooperation.     

A resolution of N-person prisoners' dilemma

Summary. The purpose of this paper is to show that cooperation is possible in N-person prisoners' dilemma if players have incomplete information on the order of their moves. We consider a modified version of N-person prisoners' dilemma in which players sequentially move in the order determined by Nature, and during the play they get some information about others. We provide an information partition and a condition of payoffs for which cooperation is attained in an equilibrium. Further, for the case that full cooperation is not attained, we examine the largest partial cooperation which is achieved in an equilibrium. Received: March 5, 1995; revised version August 11, 1996     

Farsightedness in a Coalitional Great Fish War

We explore the implications of the farsightedness assumption on the conjectures of players in a coalitional Great Fish War model with symmetric players, derived from the seminal model of Levhari and Mirman (Bell J Econ 11:649–661, 1980). The farsightedness assumption for players in a coalitional game acknowledges the fact that a deviation from a single player will lead to the formation of a new coalition structure as the result of possibly successive moves of her rivals in order to improve their payoffs. It departs from mainstream game theory in that it relies on the so-called rational conjectures, as opposed to the traditional Nash conjectures formed by players on the behavior of their rivals. For values of the biological parameter and the discount factor more plausible than the ones used in the current literature, the farsightedness assumption predicts a wide scope for cooperation in non-trivial coalitions, sustained by credible threats of successive deviations that defeat the shortsighted payoff of any prospective deviator. Compliance or deterrence of deviations may also be addressed by acknowledging that information on the fish stock or on the catch policies actually implemented may be available only with a delay (dynamic farsightedness). In that case, the requirements are stronger and the sizes and number of possible farsighted stable coalitions are different. In the sequential move version, which could mimic some characteristics of fishery models, the results are not less appealing, even if the dominant player or dominant coalition with first move advantage assumption provides a case for cooperation with the traditional Nash conjectures.     

Punishment,counterpunishment and sanction enforcement in a social dilemma experiment

We present the results of an experiment that explores the sanctioning behavior of individuals who experience a social dilemma. In the game we study, players choose contribution levels to a public good and subsequently have multiple opportunities to reduce the earnings of the other members of the group. The treatments vary in terms of individuals’ opportunities to (a) avenge sanctions that have been directed toward themselves, and (b) punish others’ sanctioning behavior with respect to third parties. We find that individuals do avenge sanctions they have received, and this serves to decrease contribution levels. They also punish those who fail to sanction third parties, but the resulting increase in contributions is smaller than the decrease the avenging of sanctions induces. When there are five rounds of unrestricted sanctioning, contributions and welfare are significantly lower than when only one round of sanctioning opportunities exists, and welfare is lower than at a benchmark of zero cooperation. We thank James Andreoni, participants in seminars at Emory University, the University of Wisconsin-Madison, the University of New South Wales, the University of Sydney, Deakin University, the 2004 North American Regional Meetings of the ESA in Tucson, Arizona, USA, the 2004 IMEBE Meetings in Cordoba, Spain, and the 2005 SAET meetings in Vigo, Spain, for constructive and helpful comments. We thank Elven Priour for programming and organization of the sessions. Instructions for the experiment are available from the authors.