The control of game form recognition in experiments: understanding dominant strategy failures in a simple two person “guessing” game

This paper focuses on instructions and procedures as the reasons that subjects fail to behave according to the predictions of game theory in two-person “guessing game” (beauty contest game) experiments. In this game, two individuals simultaneously choose a number between 0 and 100. The winner is the person whose chosen number is the closest to 2/3 of the average of the two numbers. The weakly dominant strategy is zero. Because of the simplicity of the game, the widespread failure of subjects to choose the weakly dominant strategy has been interpreted as evidence of some fundamental inability to behave strategically. By contrast, we find that subjects’ behavior reflects a lack of understanding of the game form, which we define as the relationships between possible choices, outcomes and payoffs. To a surprising degree, subjects seem to have little understanding of the experimental environment in which they are participating. If subjects do not understand the game form, the experimental control needed for testing game theory is lost. The experiments reported here demonstrate that the failure to act strategically is related to how the game is presented. We test how well subjects are able to recognize the game under a variety of different presentations of the game. Some subjects fail to recognize the game form when it is presented abstractly. When the game is transformed into a simple isomorphic game and presented in a familiar context, subjects do choose weakly dominant strategies. While our results confirm the ability of subjects to make strategic decisions, they also emphasize the need to understand the limitations of experimental subjects’ ability to grasp the game as the experimenter intends. Given these limitations, we provide suggestions for better experimental control.     

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.     

Behavior in one-shot traveler’s dilemma games: model and experiments with advice

We report results of one-shot traveler’s dilemma game experiments to test the predictions of a model of introspection. The model describes a noisy out-of-equilibrium process by which players reach a decision of what to do in one-shot games. To test the robustness of the model and to compare it to other models of introspection without noise, we introduce non-binding advice. Advice has the effect of coordinating all players’ beliefs onto a common strategy. Experimentally, advice is implemented by asking subjects who participated in a repeated traveler’s dilemma game to recommend an action to subjects playing one-shot games with identical parameters. In contrast to observations, models based on best-response dynamics would predict lower claims than the advised. We show that our model’s predictions with and without advice are consistent with the data.      

Network architecture, salience and coordination

This paper reports the results of an experimental investigation of dynamic games in networks. In each period, the subjects simultaneously choose whether or not to make an irreversible contribution to the provision of an indivisible public good. Subjects observe the past actions of other subjects if and only if they are connected by the network. Networks may be incomplete so subjects are asymmetrically informed about the actions of other subjects in the same network, which is typically an obstacle to the attainment of an efficient outcome. For all networks, the game has a large set of (possibly inefficient) equilibrium outcomes. Nonetheless, the network architecture makes certain strategies salient and this in turn facilitates coordination on efficient outcomes. In particular, asymmetries in the network architecture encourage two salient behaviors, strategic delay and strategic commitment. By contrast, we find that symmetries in the network architecture can lead to mis-coordination and inefficient outcomes.     

Stationary solution to the overlapping generations model of fiat money: Experimental evidence

Overlapping generations model of fiat money yields an infinity of competitive equilibrium solutions, only one of which is stationary. Economies reported in this paper involved a sequence of overlapping generations of three or four individuals; each individual lived for two periods. In their young age individuals were endowed with chips that could be traded for fiat money wish the individuals of the old generation. In their old age, individuals could exchange their units o flat money for the consumption good. Results of the experiments exhibit some support for the stationary solution. The results are robust to two designs of exchange institutions (double oral auctior and supply schedule auction) and to two different endogenous ways of converting money into chips at the end of the game (average price prevailing during the last period the game is actually played and the average price forecast made during the last period the game is actually played).A preliminary version of this paper was presented at the meeting of the Economic Science Association and at the Federal Reserve Bank of Minneapolis. The authors are grateful for comments received from various participants at both presentations. Financial support was provided by the McKnight Foundation, the Honeywell Foundation, National Science Foundation (SES 89-12552), and Richard. M. and Margaret Cyert Family Funds.     

On the informational content of advice: a theoretical and experimental study

This paper examines the market for advice and the underlying perception that advice is useful and informative. We do this by first providing a theoretical examination of the informational content of advice and then by setting up a series of experimental markets where this advice is sold. In these markets we provide bidders with a demographic profile of the “experts” offering advice.The results of our experiment generate several interesting findings. The raw bid data suggest that subjects bid significantly more for data than they do for advice. Second, in the market for advice there appears to be no consensus as to who are the best advisors although on average economists demand the highest mean price and women suffer a discount. In addition, we find that whether a subject suffers from a representativeness bias in the way he or she processes data has an impact on how he or she bids for advice and on his or her willingness to follow it once offered. Finally, we find that on average people impute a low level of informativeness onto advice, consistent with their bidding behavior for data versus advice.This work was done under grant number SES-0425118 of the National Science Foundation. The authors would like to recognize the Center for Experimental Social Science at New York University for its additional support. We also acknowledge the help of Elizabeth Potamites for her research assistance.     

Learning and transfer in signaling games

We explore how learning to play strategically in one signaling game promotes strategic play in a related signaling game. Following convergence to a pooling equilibrium, payoffs are changed to only support separating equilibria. More strategic play is observed following the change in payoffs than for inexperienced subjects in control sessions, contrary to the prediction of a fictitious play learning model. Introducing a growing proportion of sophisticated learners, subjects who anticipate responders’ behavior following the change in payoffs, enables the model to capture the positive cross-game learning observed in the data. Research support form the National Science Foundation grant number SBR9809538 is gratefully acknowledged. We have received research support from Jo Ducey, Guillaume Frechette, Steve Lehrer, and Carol Kraker Stockman. We have benefitted from comments of Eric Bettinger, John Ham, Jim Rebeitzer, Bob Slonim and seminar participants at Case Western Reserve University, Ohio State University, the University of Mississippi, the University of Illinois, and Purdue University. The usual caveat applies.     

Learning dynamics for mechanism design: An experimental comparison of public goods mechanisms

In a repeated-interaction public goods economy, incomplete information and dynamic behavior may affect the realized outcomes of mechanisms known to be efficient in a complete information one-shot game. An experimental test of five public goods mechanisms indicates that subjects with private information appear to best respond to recent observations. This provides predictions about which mechanisms will generate convergence to their efficient equilibrium allocations. These predictions match the experimental result that globally stable efficient mechanisms realize the highest efficiency in practice. The simplicity of the suggested best response model makes it useful in predicting stability of mechanisms not yet tested.     

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.     

Teaching Nash Equilibrium and Dominance: A Classroom Experiment on the Beauty Contest

Abstract: The authors' aim in this article was to show how the use of classroom experiments may be a good pedagogical tool to teach the Nash equilibrium (NE) concept. The basic game is a version of the beauty contest game (BCG), a simple guessing game in which repetition lets students react to other players' choices and converge iteratively to the equilibrium solution. The authors perform this experiment with undergraduate students with no previous training in game theory. After four rounds, they observe a clear decreasing tendency in the average submitted number in all groups. Thus, the findings show that by playing a repeated BCG, students quickly learn how to reach the NE solution.     

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.     

Selection dynamics and adaptive behavior without much information

This paper uses the experimental method to investigate behavior in a coordination game when the information available to subjects is limited to their feasible choices and their experienced payoffs. In the experiment subjects converge to an absorbing state at rates that are orders of magnitude faster than reinforcement learning algorithms, but slower than under complete information. This state is very close to a mutual best response outcome. All cohorts converged to the market statistic predicted by the interior equilibrium regardless of the information conditions or the stability conditions. Eric Battalio programmed the graphical user interface. The National Science Foundation and Texas Advanced Research Program provided financial support. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or the Texas Advanced Research Program.     

Correlated equilibria and local interactions

Summary This paper shows that Nash equilibria of a local-interaction game are equivalent to correlated equilibria of the underlying game.This work was done while George Mailath and Larry Samuelson were visiting the University of Bonn, whose hospitality is gratefully acknowledged. We thank Ken Binmore for helpful discussions. This is a revision of Section 2 of Mailath, Samuelson, and Shaked [6]. Financial support from the National Science Foundation and the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn, is gratefully acknowledged.     

Who knows it is a game? On strategic awareness and cognitive ability

We examine strategic awareness in experimental games, that is, the question of whether subjects realize they are playing a game and thus have to form beliefs about others’ actions. We conduct a beauty contest game and elicit measures of cognitive ability and beliefs about others’ cognitive ability. We show that the effect of cognitive ability is highly non-linear. Subjects below a certain threshold choose numbers in the whole interval and their behavior does not correlate with beliefs about others’ ability. In contrast, subjects who exceed the threshold avoid choices above 50 and react very sensitively to beliefs about the cognitive ability of others.     

You are not in my boat: common fate and discrimination against outgroup members

Sharing a common fate with some people but not others may affect how economic agents behave in economic transactions, quite independently of strategic incentives. We present an experimental test of the effect of perceptions of common fate on the inducement of economic discrimination in bilateral settings. In settings where the bargaining power was all with one subject (the dictator game and a ‘unilateral power game’), about half of the subjects engaged in negative discrimination: insiders were not treated better relative to control sessions, but outsiders were treated worse. Discrimination may be induced by a more conflictual perception of the decision problem.     

The implications of a third-order translog demand system and some empirical results

In this paper a homogeneity and symmetry unconstrained variant of the “third-order translog”, representing a third-order approximation to an indirect utility function, is derived. Using Christensen, Jorgenson and Lau's (1975) data, and subsequent observations to 1986, we estimate this third-order translog, and show that it dominates the homogeneity and symmetry unconstrained translog. The restrictions implied by homogeneity and symmetry, conditional on our third-order translog are also rejected. These results are of interest since they demonstrate that standard restrictions implied by the theory of consumer behaviour can be rejected even in the context of a fairly general model which has been estimated using aggregate, time-series data. However, it has been found that the inclusion of dynamic effects sometimes leads to non-rejection of theoretical restrictions (Anderson and Blundell 1983). Unfortunately, the use of aggregate, time-series data to estimate a dynamic, third-order translog would be problematic, since exact aggregation restrictions are imposeda priori and observations are limited. This suggests that it would be more appropriate to estimate the third-order translog using cross-sectional micro-data sets with many observations. This type of data is now available, and can be used to estimate systems of consumer demands. Furthermore, these data sets contain extensive information on demographic characteristics which, having recently been found to be important determinants of demand (Barnes and Gillingham 1984) can be included in general demand systems of the third-order translog type.     

Match choice and Ghettoization in evolutionary games

I study how having a choice about who to play affects the conventions that arise in a population playing a 2×2 common interest game. Match choice allows agents playing “fragile but efficient” strategies to isolate themselves, raising their returns but making it harder for outsiders to duplicate their success. When agents are myopic, the second effect dominates: long run play can shift toward either the risk-dominant equilibrium (with common interests in matching) or toward the inefficient equilibrium (with opposing interests in matching). In contrast, when agents are patient, supra-Nash payoffs can be sustained.     

A study of zero-out auctions: testbed experiments of a process of allocating private rights to the use of public property

Summary This paper applies an experimental testbed methodology to the evaluation of a proposed mechanism for allocating the right to land at the New York airports. The mechanism is called a zero-out auction because it is supposed to allocate the rights efficiently like an auction while leaving all of the consumer's surplus with the buyers (as opposed to allocating some to the seller as would be the case with an ordinary auction). A new behavioral hypothesis is introduced to account for limited rationality of individuals and unusual behaviors of the process. The axiom, called theunbiased expectations hypothesis, does a good job of modeling individual behavior in the context of a game model.We acknowledge the financial support of the National Science Foundation and the Caltech Laboratory for Experimental Economics and Political Science. Comments by David Grether, John Ledyard, Michael Levine, Jennifer Reinganum, and Richard Sutch have been very helpful.     

Multilateral subsidy games

This paper examines the rationale for multilateral agreements to limit investment subsidies. The welfare ranking of symmetric multilateral subsidy games is shown to depend on whether or not investment levels are “friendly”, raising rival profits in total, and/or strategic complements, raising rival profits at the margin. In both Cournot and Bertrand competition, when spillovers are low and competition is intense (because goods are close substitutes), national-welfare-maximizing governments over-subsidize investment, and banning subsidies would improve welfare. When spillovers are high, national governments under-subsidize from a global welfare perspective, but the subsidy game is welfare superior to non-intervention. For helpful comments we are grateful to two referees, to Arijit Mukherjee, and to participants in seminars at Prague and UCD, at the EEA Conference in Lausanne and at the GEP Conference on “New Directions in International Trade Theory” at the University of Nottingham, June 2007. Dermot Leahy acknowledges the support of the Science Foundation Ireland Research Frontiers Programme (Grant MAT 017).     

Recommended play and correlated equilibria: an experimental study

This study reports a laboratory experiment wherein subjects play a hawk–dove game. We try to implement a correlated equilibrium with payoffs outside the convex hull of Nash equilibrium payoffs by privately recommending play. We find that subjects are reluctant to follow certain recommendations. We are able to implement this correlated equilibrium, however, when subjects play against robots that always follow recommendations, including in a control treatment in which human subjects receive the robot “earnings.” This indicates that the lack of mutual knowledge of conjectures, rather than social preferences, explains subjects’ failure to play the suggested correlated equilibrium when facing other human players. We are grateful for financial support provided by the Purdue University Faculty Scholar program and the Asociación Méxicana de Cultura, as well as for the valuable research assistance provided by Shakun Datta and Marikah Mancini. We received helpful comments from Shurojit Chatterji, David Cooper, Arthur Schram, Ricard Torres, an anonymous referee, and from conference and seminar participants at Royal Holloway, the University of Amsterdam, Purdue University, the Economic Science Association and the Society for the Advancement of Economic Theory.