Evidence on the equivalence of the strategic and extensive form representation of games

Our experiment tests whether strategically equivalent representations of games produce equivalent behavior when actually played. We examine representative members of the class of generic 2×2 extensive form games of perfect information and the equivalent strategic form games. Systematic differences exist between subjects’ choices in the strategic and extensive form representations. These differences cannot be attributed to differences in subjects’ ability to do backwards induction, in the salience of interpersonal preferences, or in optimization premiums between the two game forms. Instead, subjects in the extensive form are consistently more likely to allow the other player to make a meaningful choice.     

Internet cautions: Experimental games with internet partners

We report the results of experiments conducted over the internet between two different laboratories. Each subject at one site is matched with a subject at another site in a trust game experiment. We investigate whether subjects believe they are really matched with another person, and suggest a methodology for ensuring that subjects’ beliefs are accurate. Results show that skepticism can lead to misleading results. If subjects do not believe they are matched with a real person, they trust too much: i.e., they trust the experimenter rather than their partner. JEL Classification C9     

Eliciting beliefs: Proper scoring rules,incentives, stakes and hedging

Proper Scoring Rules (PSRs) are popular incentivized mechanisms to elicit an agent's beliefs. This paper combines theory and experiment to characterize how PSRs bias reported beliefs when (i) the PSR payments are increased, (ii) the agent has a financial stake in the event she is predicting, and (iii) the agent can hedge her prediction by taking an additional action. In contrast with previous literature, the PSR biases are characterized for all PSRs and all risk averse agents. Our results reveal complex distortions of reported beliefs, thereby raising concerns about the ability of PSRs to recover truthful beliefs in general decision-making environments.     

On strategic stability in discontinuous games

We identify a class of discontinuous normal-form games whose members possess strategically stable sets, defined according to an infinite-game extension of Kohlberg and Mertens’s (1986) equilibrium concept, and show that, generically, a set is stable if and only if it contains a single Nash equilibrium.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

Iterated strict dominance in general games

We offer a definition of iterated elimination of strictly dominated strategies (IESDS^*) for games with (in)finite players, (non)compact strategy sets, and (dis)continuous payoff functions. IESDS^* is always a well-defined order independent procedure that can be used to solve Nash equilibrium in dominance-solvable games. We characterize IESDS^* by means of a “stability” criterion, and offer a sufficient and necessary epistemic condition for IESDS^*. We show by an example that IESDS^* may generate spurious Nash equilibria in the class of Reny's better-reply secure games. We provide sufficient/necessary conditions under which IESDS^* preserves the set of Nash equilibria.     

Mindreading and endogenous beliefs in games

We argue that a Bayesian explanation of strategic choices in games requires introducing a psychological theory of belief formation. We highlight that beliefs in epistemic game theory are derived from the actual choice of the players, and cannot therefore explain why Bayesian rational players should play the strategy they actually chose. We introduce the players’ capacity of mindreading in a game theoretical framework with the simulation theory, and characterise the beliefs that Bayes rational players could endogenously form in games. We show in particular that those beliefs need not be ratifiable, and therefore that rational players can form action-dependent beliefs.     

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.      

Learning to communicate in cheap-talk games

We study learning in communication games. Our main finding is that a simple forward-looking learning rule leads to communication in a large class of games. This class is characterized by a partial-common-interest condition. In contrast, we show that a variety of purely backward looking dynamics may fail to guarantee communication. Memory is a partial substitute for looking forward: With long memory, backward-looking learning leads to communication in a class of games with perfect incentive alignment.     

Of mice and men: Within gender variation in strategic behavior

We study behavioral differences across and within genders in a family of ultimatum and dictator games. We find these differences are due not only to altruistic preferences but also beliefs about the strategic behavior of others. The behavior of men in strategic situations is not significantly more aggressive than women on average. But this average masks wide variation in intra-gender behavior. In particular, a sizable minority of males are “mice,” behaving timidly in strategic environments. Our experimental design shows that the standard ultimatum game can mask significant inter- and intra-gender differences in strategic behavior. These behavioral patterns in strategic environments are shown to be correlated with preferences for altruism in non-strategic settings. Such gender differences could well manifest themselves in real-world large-stakes transactions, such as salary negotiations.     

Interconnected games and international environmental problems

The purpose of this paper is to introduce the concept of interconnected games and to show its relevance for modeling international environmental problems. It is argued that an interconnected game approach to international environmental problems may enhance cooperation and provide an alternative to the use of financial side payments to induce countries to cooperate. Two types of interconnected games are distinguished in this paper, i.e. direct sum games and tensor games. In the former all the constituting isolated games are games in strategic form and in the latter they are repeated games. In both cases the interconnected game can be interpreted as a multiple objective game, but only the setting where a trade-off is made for the vector-payoffs is considered. In addition to the formal definition of these types of interconnected games, some elementary results concerning Nash equilibria of such games are derived.Folmer and v. Mouche: Landbouwuniversiteit Wageningen, Postbus 8130, 6700 EW Wageningen, The Netherlands; Ragland: University of Colorado at Boulder, Boulder, Colorado 80309-0256, U.S.A. An earlier version of this paper was presented at the EAERE conference in Stockholm, June 1991. The authors appreciate comments made by conference participants and journal referees.     

First- and second-order subjective expectations in strategic decision-making: Experimental evidence

We study first- and second-order subjective expectations (beliefs) in strategic decision-making. We elicit probabilistically both first- and second-order beliefs and apply the method to a Hide-and-Seek experiment. We study the relationship between choice and beliefs in terms of whether observed choice coincides with the optimal action given elicited beliefs. We study the relationship between first- and second-order beliefs under a coherence criterion. Weak coherence requires that if an event is assigned, according to first-order beliefs, a probability higher/lower/equal to the one assigned to another event, then the same holds according to second-order beliefs. Strong coherence requires the probability assigned according to first- and second-order beliefs to coincide. Evidence of heterogeneity across participants is reported. Verbal comments collected at the end of the experiment shed light on how subjects think and decide in a complex environment that is strategic, dynamic and populated by potentially heterogeneous individuals.     

Learning in games with strategic complementarities revisited

Fictitious play is a classical learning process for games, and games with strategic complementarities are an important class including many economic applications. Knowledge about convergence properties of fictitious play in this class of games is scarce, however. Beyond games with a unique equilibrium, global convergence has only been claimed for games with diminishing returns [V. Krishna, Learning in games with strategic complementarities, HBS Working Paper 92-073, Harvard University, 1992]. This result remained unpublished, and it relies on a specific tie-breaking rule. Here we prove an extension of it by showing that the ordinal version of strategic complementarities suffices. The proof does not rely on tie-breaking rules and provides some intuition for the result.     

Mixed equilibria in games of strategic complementarities

Summary. The literature on games of strategic complementarities (GSC) has focused on pure strategies. I introduce mixed strategies and show that, when strategy spaces are one-dimensional, the complementarities framework extends to mixed strategies ordered by first-order stochastic dominance. In particular, the mixed extension of a GSC is a GSC, the full set of equilibria is a complete lattice and the extremal equilibria (smallest and largest) are in pure strategies. The framework does not extend when strategy spaces are multi-dimensional. I also update learning results for GSC using stochastic fictitious play. Received: October 16, 2000; revised version: March 7, 2002 RID="*" ID="*" I am very grateful to Robert Anderson, David Blackwell, Aaron Edlin, Peter De Marzo, Ted O'Donoghue, Matthew Rabin, Ilya Segal, Chris Shannon, Clara Wang and Federico Weinschelbaum for comments and advise.     

Beliefs and rationalizability in games with complementarities

We propose two characteristics of beliefs and study their role in shaping the set of rationalizable strategy profiles in games with incomplete information. The first characteristic, type-sensitivity, is related to how informative a player thinks his type is. The second characteristic, optimism, is related to how “favorable” a player expects the outcome of the game to be. The paper has two main results: the first result provides an upper bound on the size of the set of rationalizable strategy profiles; the second gives a lower bound on the change of location of this set. These bounds are explicit expressions that involve type-sensitivity, optimism, and payoff characteristics. Our results generalize and clarify the well-known uniqueness result of global games (Carlsson and van Damme, 1993). They also imply new uniqueness results and allow us to study rationalizability in new environments. We provide applications to supermodular mechanism design (Mathevet, 2010b) and information processing errors.     

Uncertainty and insurance in strategic market games

Summary. For perfectly competitive economies under uncertainty, there is a well-known equivalence between a formulation with contingent goods and one with state-specific securities followed by spot markets for goods. In this paper, I examine whether this equivalence carries over to a particular form of imperfect competition. Specifically, I look at three Shapley-Shubik strategic market games: one with contingent commodities, one with Arrow securities traded under imperfect competition and one with Arrow securities traded under perfect competition. First I compare the feasibility constraints of these three games. Then I compare their equilibrium sets. As in Peck and Shell (1989), the only common equilibria between the first and the second game are those which involve no transfer of income across states. However, if the securities markets are competitive, then the set of equilibria of the contingent commodities game and the securities game coincide. Received: June 16, 1997; revised version: April 30, 1998     

Equilibrium selection in global games with strategic complementarities

We study games with strategic complementarities, arbitrary numbers of players and actions, and slightly noisy payoff signals. We prove limit uniqueness: as the signal noise vanishes, the game has a unique strategy profile that survives iterative dominance. This generalizes a result of Carlsson and van Damme (Econometrica 61 (1993) 989-1018) for two-player, two-action games. The surviving profile, however, may depend on fine details of the structure of the noise. We provide sufficient conditions on payoffs for there to be noise-independent selection.     

Monotone equilibria in Bayesian games of strategic complementarities

For Bayesian games of strategic complementarities, we provide a constructive proof of the existence of a greatest and a least Bayesian Nash equilibrium, each one in strategies that are monotone in type. Our main assumptions, besides strategic complementarities, are that each player's payoff displays increasing differences in own action and the profile of types and that each player's interim beliefs are increasing in type with respect to first-order stochastic dominance (e.g., types are affiliated). The result holds for general action and type spaces (single-, multi-, or infinite-dimensional; continuous or discrete) and no prior is assumed. We also provide the following comparative statics result: the greatest and least equilibria are higher if there is a first-order stochastic dominant shift in the interim beliefs. We apply this result to strategic information revelation in games of voluntary disclosure.     

On the effects of group identity in strategic environments

We examine differences in behavior between subjects interacting with a member of either the same or different identity group in both a centipede game and a series of stag hunt games. We find evidence that subjects interacting with outgroup members are more likely to behave as though best-responding to uniform randomization of the partner. We conclude that group identity not only affects player׳s social preferences, as identified in earlier research, but also affects the decision making process, independent of changes in the utility function.