Learning in experimental games

Summary Experimental games typically involve subjects playing the same game a number of times. In the absence of perfect rationality by all players, the subjects may use the behavior of their opponents in early rounds to learn about the extent of irrationality in the population they face. This makes the problem of finding the Bayes-Nash equilibrium of the experimental game much more complicated than finding the game-theoretic solution to the ideal game without irrationality. We propose and implement a computationally intensive algorithm for finding the equilibria of complicated games with irrationality via the minimization of an appropriate multi-variate function. We propose two hypotheses about how agents learn when playing experimental games. The first posits that they tend to learn about each opponent as they play it repeatedly, but do not learn about the population parameters through their observations of random opponents (myopic learning). The second posits that both types of learning take place (sequential learning). We introduce a computationally intensive sequential procedure to decide on the informational value of conducting additional experiments. With the help of that procedure, we decided after 12 experiments that our original model of irrationality was unsatisfactory for the purpose of discriminating between our two hypotheses. We changed our models, allowing for two different types of irrationality, reanalyzed the old data, and conducted 7 more experiments. The new model successfully discriminated between our two hypotheses about learning. After only 7 more experiments, our approximately optimal stopping rule led us to stop sampling and accept the model where both types of learning occur.We acknowledge the financial support from NSF grant #SES9011828 to the California Institute of Technology. We also acknowledge the able research assistance of Mark Fey, Lynell Jackson and Jeffrey Prisbrey in setting up the experiments, recruiting subjects and running the experiments. We acknowledge the help of the Jet Propulsion Laboratory and its staff members for giving us access to their Cray XMP/18, and subsequently their Cray YMP2E/116.     

VIDEO GAMES AND CRIME

Psychological studies find that video game play is associated with markers for violent and antisocial attitudes. It is plausible that these markers indicate either whetted or sated preferences for antisocial behavior. I investigate whether a proxy for video gaming is associated with the prevalence of various crimes and find evidence that gaming is associated with significant declines in crime and death rates. These results are robust to various alternative specifications. Other youth‐related leisure activities—sports and movie viewing—generate smaller or no effects. These results cast doubt on the desirability of proposed restrictions on video game marketing. (JEL L86, D18, I18)     

Learning and equilibrium as useful approximations: Accuracy of prediction on randomly selected constant sum games

There is a good deal of miscommunication among experimenters and theorists about how to evaluate a theory that can be rejected by sufficient data, but may nevertheless be a useful approximation. A standard experimental design reports whether a general theory can be rejected on an informative test case. This paper, in contrast, reports an experiment designed to meaningfully pose the question: “how good an approximation does a theory provide on average.” It focuses on a class of randomly selected games, and estimates how many pairs of experimental subjects would have to be observed playing a previously unexamined game before the mean of the experimental observations would provide a better prediction than the theory about the behavior of a new pair of subjects playing this game. We call this quantity the model’s equivalent number of observations, and explore its properties. This research was supported by a grant from the U.S. National Science Foundation and the USA–Israel Binational Science Foundation. We are very grateful for helpful conversations with David Budescu, Jerry Busemeyer, Gary Chamberlain, Paul Feigin, Dave Krantz, Jack Porter, Tom Wallsten, Wolfgang Viechtbauer, and Richard Zeckhauser.     

Did you really take a hit? Understanding how video games playing affects individuals

This study addresses the important and recurring question of whether playing video games is detrimental to the socio-economic development of a person. It does this by using novel data from the Taking Part Survey in England to establish whether games playing is associated with particular socio-economic characteristics and/or other forms of cultural participation. The results do not indicate any obviously negative effects of video games playing: rather, those who play are typically better educated and no less wealthier, and games players are also more likely than non-games players to participate in other forms of culture, particularly active forms of participation. These findings are reinforced when comparing the characteristics of individuals who did and did not play video games when younger.     

Generalised weakened fictitious play

A general class of adaptive processes in games is developed, which significantly generalises weakened fictitious play [Van der Genugten, B., 2000. A weakened form of fictitious play in two-person zero-sum games. Int. Game Theory Rev. 2, 307–328] and includes several interesting fictitious-play-like processes as special cases. The general model is rigorously analysed using the best response differential inclusion, and shown to converge in games with the fictitious play property. Furthermore, a new actor–critic process is introduced, in which the only information given to a player is the reward received as a result of selecting an action—a player need not even know they are playing a game. It is shown that this results in a generalised weakened fictitious play process, and can therefore be considered as a first step towards explaining how players might learn to play Nash equilibrium strategies without having any knowledge of the game, or even that they are playing a game.     

Strategic complexity in repeated extensive games

This paper studies a machine (finite automaton) playing a two-player repeated game of a simple extensive-form game with perfect information. We introduce a new complexity measure called multiple complexity which incorporates a strategyʼs responsiveness to information in the stage game as well as the number of states of the machine. We completely characterize the Nash equilibrium of the machine game. In the sequential-move prisonerʼs dilemma, cooperation can be sustained as an equilibrium.     

ON THE NEUTRALITY OF COALITION FORMATION IN A PURE BARGAINING PROBLEM*

We consider the sequential bargaining game à la Stahl–Binmore–Rubinstein with random proposers, juxtaposing an ex ante coalition formation stage to their bargaining game. On the basis of the expected outcomes in the negotiation over how to split a dollar, players can form coalitions in a sequential manner, within each of which they can redistribute their payoffs. It turns out that the grand coalition does form, and that each player receives his discounted expected payoff, which is obtained by playing as a single player in the negotiation, although there could be many equilibria in the bargaining stage.     

Contributing for Myself,but Free riding for My Group?

Many real‐world decisions are made by individuals as representatives of groups. Most research, however, studies either individuals or groups as decision‐makers. This paper explores whether there is a general difference between a decision made as an individual and as a representative of a group in the context of a public good game. We conducted a series of experiments to test this question and to understand mechanisms contributing to potential differences. We found that representatives contributed less than individuals when they could not communicate with their constituency. However, when they could discuss their strategy before playing, they contributed at least as much as individual decision‐makers. Furthermore, when they could justify their decision after playing, they contributed even more than individual decision‐makers. We discuss potential reasons for this and directions for future research.     

The effects of old and new media on children’s weight

Childhood obesity rates have recently been rising in many countries. It has been suggested in the literature that changes in children’s media exposure may contribute to explaining this trend. I investigate whether or not this hypothesis is supported by data. I contribute to the literature by focusing not only on television but also on new media – computers and video games. The Child Development Supplement to the Panel Study of Income Dynamics is used for the analysis. To address the endogeneity of children’s media exposure, I use dynamic and panel data models. This is another improvement upon the existing literature. Additionally, an extensive list of control variables is included in the regressions. I find that video game playing or computer use has no effect on children’s body weight. On the other hand, television viewing may increase children’s body weight slightly.     

ASYMMETRY OF CUSTOMER LOSS AND RECOVERY UNDER ENDOGENOUS PARTNERSHIPS: THEORY AND EVIDENCE*

This article is inspired by real‐world phenomena that firms lose customers based on imprecise information and take a long time to recover. If consumers are playing an ordinary repeated game with fixed partners, there is no clear reason why recovery happens slowly. However, if consumers are playing an endogenously repeated game, a class of simple efficient equilibria exhibits the asymmetry of fast loss and slow recovery of customers after a bad signal. Exit is systematic, but formation of a new partnership is random. We also give empirical evidence of our equilibria at an individual‐firm level.     

Is limit pricing evolutionarily stable?

We have analyzed the question whether or not limit-pricing behavior can be viewed as a strategy, which survives the replicator dynamics in a game in which firms change strategies through imitation. Our results show that, regardless of the proportion of low cost incumbents in the population, potential entrants that play the strategy "enter" when they observe limit output are wiped out. The unique ESS consists of all low cost incumbents playing the "limit output" strategy, and all high cost incumbents playing the "monopoly output". This ESS includes potential entrants playing "stay out" after observing the limit output, and playing "enter" if monopoly output is observed in the first stage. Hence, limit pricing survives the replicator dynamics and appears in the stable state of the population.JEL Classification: B25, C73, L11We would like to thank an anonymous referee for valuable comments on an earlier version of this paper. Correspondence to: U. Soytas     

ARE TELEVISION AND VIDEO GAMES REALLY HARMFUL FOR KIDS?

Are watching television (TV) and playing video games really harmful for children's development? By using a unique longitudinal dataset with detailed information on children's development and health, we examined the causal effect of hours of TV watched or of video games played on school‐aged children's problem behavior, orientation to school, and obesity. The results suggested that the answer to the question is yes, but the magnitude of the effect is sufficiently small to be considered as negligible. The results were robust to within‐twin‐fixed effects. (JEL I10, I20)     

Market Games in Large Economies with a Finite Number of Types

We study market games derived from an exchange economy with a continuum of agents, each having one of finitely many possible types. The type of agent determines his initial endowment and utility function. It is shown that, unlike the well-known Shapley–Shubik theorem on market games (Shapley and Shubik in J Econ Theory 1:9–25, 1969), there might be a (fuzzy) game in which each of its sub-games has a non-empty core and, nevertheless, it is not a market game. It turns out that, in order to be a market game, a game needs also to be homogeneous. We also study investment games – which are fuzzy games obtained from an economy with a finite number of agents cooperating in one or more joint projects. It is argued that the usual definition of the core is inappropriate for such a model. We therefore introduce and analyze the new notion of comprehensive core. This solution concept seems to be more suitable for such a scenario. We finally refer to the notion of feasibility of an allocation in games with a large number of players. Some of the results in this paper appear in a previous draft distributed by the name “Cooperative investment games or Population games”. An anonymous referee of Economic Theory is acknowledged for his/her comments     

An Experimental Study of Communication and Coordination in Noncooperative Games

This paper reports the results of an experiment designed to test the usefulness of alternative solution concepts to explain players' behavior in noncooperative games with preplay communication. In the experiment subjects communicate byplain conversationprior to playing a simple game. In this setting, we find that the presumption ofindividualisticandindependentbehavior underlying the concept of Nash equilibrium is inappropriate. Instead, we observe behavior to becoordinatedandcorrelated. Statistical tests reject Nash equilibrium as an explanation of observed play. The coalition proof correlated equilibrium of the game, however, explains the data when the possibility of errors by players is introduced.Journal of Economic LiteratureClassification Numbers: C72, C92.     

Extensive and strategic forms: Games and models for games

This paper aims to make precise, in the context of epistemic models for games, some relations between the normal or strategic form representation of a game and the extensive or dynamic form representation. It is argued, first, that epistemic models defined for strategic form representations provide all the materials necessary for defining models for corresponding extensive form representations of the game, models that provide information about the way the game is played that is sufficient to evaluate the rationality of the choices that the players make, and are disposed to make, in the course of playing the dynamic game. Second, two definitions of rationality are compared — one for strategy choices in the normal form representation, and one for the individual choices that the player is disposed to make in the course of playing the dynamic game. It is shown that they are essentially equivalent in games with perfect recall. The main focus is on the intuitive foundational assumptions about rationality and dynamic choice that are needed to motivate the definitions. It is argued that to evaluate the rationality of a player's choices in a dynamic context, it is essential to distinguish passive knowledge (knowledge about nature and about the prior beliefs and strategy choices of other players that is based on observation and inference) fromactive knowledge (knowledge about one's own choices and future choices that is grounded in one's decisions).     

The role of strategic uncertainty in games: An experimental study of cheap talk and contracts in the Nash demand game

We utilise results from a human-subjects experiment to examine the connection between strategic uncertainty and outcomes in games. Our basic game is a Nash demand game where one player has an outside option available. A “chat” treatment allows bargainers to send cheap-talk messages prior to playing the basic game, and in a “contracts” treatment, they can additionally propose and accept binding contracts. We propose that strategic uncertainty comprises at least two facets: “coordination-type”, which is lower in the chat game than in the basic game, and “rationality-type”, which is lower in the contracts game than in the chat game. We find that both types of strategic uncertainty impact bargaining outcomes: moving from the basic game to the chat game, and thence to contracts, improves several aspects of outcomes, such as higher efficiency, less opting out and less under-demanding. Other results include a treatment effect on the types of agreements that are reached.     

A New Bargaining Set of an N-Person Game and Endogenous Coalition Formation

The two most fundamental questions in cooperative game theory are: When a game is played, what coalitions will be formed and what payoff vectors will be chosen? Few solution concepts or theories in the current literature provide satisfactory answers to both questions; answers are especially lacking for the first one. In this paper we introduce a new bargaining set, which is the first solution concept in cooperative game theory that provides answers to both fundamental questions endogenously. Journal of Economic Literature Classification Number: C71.     

Trust and trustworthiness in games: An experimental study of intergenerational advice

This paper investigates the development of conventions of trust in what we call intergenerational games, i.e., games played by a sequence of non-overplapping agents, who pass on advice on how to play the game across adjacent generations of players. Using the trust game of Berg et al. (1995) as our experimental decision problem, advice seems to decrease the amount of trustthat evolves when this game in played in an inter-generational manner in that it decreases the amount of money sent from Senders to Returners. Ironically, advice increases trustworthinessin that Returners tend to send more back. Further, subjects appear to follows conventions of reciprocity in that they tend to Send more if they think the Returners acted in a “kind” manner, where kind means the Sender sent more money than the receiver expected. Finally, while we find a causal relationship running from trustworthiness to trust, the opposite can not be established. We note that many of our results can only be achieved using the tools offered by inter-generational games. The inter-generational advice offered provides information not available when games are played in their static form. Combining that information with elicited beliefs of the Senders and Returners adds even more information that can be used to investigate the motives that subjects have for doing what they do. Electronic supplementary material Electronic supplementary material is available for this article at and accessible for authorised users. JEL Classification C91 · C72 Resources for this research were provided by National Science Foundation grants SBR-9709962 and SBR-9709079 and by both the Center for Experimental Social Science and the C.V. Starr Center for Applied Economics at New York University. We would like to thank Shachar Kariv for both his comments and research assistance. We also thank Mikhael Shor and Judy Goldberg for research assistance, and Yevgeniy Tovshteyn for computer programming.     

Reciprocity Game

This paper shows that reciprocity comes from the desire to cooperate in finitely repeated prisoner's dilemma game. Before playing the finitely repeated prisoner's dilemma game, players choose the reciprocity level and commit to it, and the reciprocity level is public information. There are T equilibria if the prisoner's dilemma game is repeated for T periods, and each equilibrium is associated with different levels of cooperation. Further, if players choose their reciprocity levels sequentially, then the most cooperative equilibrium will be the unique equilibrium. However, reciprocity does not matter for the one‐period game and the infinitely repeated game.     

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.