A Laboratory Investigation of Multiperson Rationality and Presentation Effects

This paper reports the results of laboratory experiments in which subjects were presented with different two-person decision problems in both their extensive and normal forms. All games generated the same equilibrium outcomes. Our results indicate that the presentation of the decision problem significantly affects the strategy chosen. Surprisingly, these presentation effects were most prominent in the simplest games where differences in presentation would seem most transparent. It appears that subjects are much more likely to use (and fear) incredible threats when the problem is presented as a one-stage rather than as a multistage game. Journal of Economic Literature Classification Numbers: 026,215.     

The approximate purification of mixed strategies with finite observation sets

This note provides a finite result on the approximate purification of strategies in a class of two-person games.     

Decomposable solutions for N — person bargaining games

Two solutions for n-person bargaining games are defined and axiomatized, which exhibit the so called “decomposability property”. One of the solutions extends the two-person Kalai-Smorodinsky solution to n-person games. The other is the Raiffa solution.     

Competitive environments and protective behavior

The class of two-person competition games is introduced and analyzed. For any game in this class the set of Nash equilibria is convex and all Nash equilibria lead to the same payoff vector. Competition games are compared to other competitive environments such as unilaterally competitive games and rivalry games. Moreover, protective behavior within competitive environments is analyzed. For matrix games it is known that protective strategies profiles exactly correspond to proper equilibria. It is shown that this result can be extended to the class of unilaterally competitive games.     

Investigating the failure to best respond in experimental games

Experimental Economics - We examine strategic sophistication using eight two-person 3?×?3 one-shot games. To facilitate strategic thinking, we design a ‘structured’...     

Equilibrium Selection inn-Person Coordination Games

This paper investigates several approaches to equilibrium selection and the relationships between them. The class of games we study aren-person generalized coordination games with multiple Pareto rankable strict Nash equilibria. The main result is that all selection criteria select the same outcome (namely the risk dominant equilibrium) in two-person games, and that most equivalences break for games with more than two players. All criteria select the Pareto efficient equilibrium in voting games, of which pure coordination games are special cases.Journal of Economic LiteratureClassification Numbers: C70, C72, D82.     

An experimental study of costly coordination

This paper reports data for coordination game experiments with random matching. The experimental design is based on changes in an effort-cost parameter, which do not alter the set of Nash equilibria nor do they alter the predictions of adjustment theories based on imitation or best response dynamics. As expected, however, increasing the effort cost lowers effort levels. Maximization of a stochastic potential function, a concept that generalizes risk dominance to continuous games, predicts this reduction in efforts. An error parameter estimated from initial two-person, minimum-effort games is used to predict behavior in other three-person coordination games.     

Fictitious play in ‘one-against-all’ multi-player games

Summary. A compound game is an (n + 1) player game based on n two-person subgames. In each of these subgames player 0 plays against one of the other players. Player 0 is regulated, so that he must choose the same strategy in all n subgames. We show that every fictitious play process approaches the set of equilibria in compound games for which all subgames are either zero-sum games, potential games, or games. Received: July 18, 1997; revised version: December 4, 1998     

Communication in games of incomplete information: Two players

We study the effect of communication in two-person games of incomplete information. We show that for any two-player game of incomplete information, any rational mediated communication mechanism outcome (satisfying a Nash domination condition) can be implemented as the perfect Bayesian equilibrium of a cheap-communication extension of the original game.     

A Property of Nash Equilibria in Repeated Games with Incomplete Information

We consider two-person non-zero-sum infinitely repeated games with lack of information on one side. The characterization of Nash equilibrium payoffs obtained by Hart allows for complex strategies, which are actually required by some equilibrium payoffs in some games. We show that appropriate one-shot public communication mechanisms make Nash equilibrium payoffs achievable by means of simple strategies. Furthermore, these mechanisms satisfy a notion of self-fulfillment.Journal of Economic LiteratureClassification Numbers: D82, C72.     

The averaging mechanism

We consider two-person bargaining games and we assume that the agents want to apply two possibly different bargaining solutions. Problems arise if the allocations chosen by these bargaining solutions do not coincide. We propose a mechanism which mimics a step-by-step bargaining procedure in order to resolve such a conflict and give a characterization of the mechanism.     

How strategy sensitive are contributions?

Summary. We test six hypotheses for contributions in dilemma games, a category that includes the prisoner's dilemma and public goods games. Our experiment focuses specifically on the strategic interdependence of contributing behavior, and manipulates the strategy space of a two-person dilemma game especially designed for the task. The hypothesis that contributors have non-linear preferences over own and the other player's payoffs accurately matches the strategic pattern of contributing that we observe across treatments. None of the reasons for contributing advanced by the other hypotheses, whether alone or in additive combination, does so.     

Equilibrium play and best response to (stated) beliefs in normal form games

We report experimental results on a series of ten one-shot two-person 3×3 normal form games with unique equilibrium in pure strategies played by non-economists. In contrast to previous experiments in which game theory predictions fail dramatically, a majority of actions taken coincided with the equilibrium prediction (70.2%) and were best-responses to subjects' stated beliefs (67.2%). In constant-sum games, 78% of actions taken were predicted by the equilibrium model, outperforming simple K-level reasoning models. We discuss how non-trivial game characteristics related to risk aversion, efficiency concerns and social preferences may affect the predictive value of different models in simple normal form games.     

Voluntarily Separable Repeated Prisoner's Dilemma

Ordinary repeated games do not apply to real societies where one can cheat and escape from partners. We formulate a model of endogenous relationships that a player can unilaterally end and start with a randomly assigned new partner with no information flow. Focusing on two-person, two-action Prisoner's Dilemma, we show that the endogenous duration of partnerships generates a significantly different evolutionary stability structure from ordinary random matching games. Monomorphic equilibria require initial trust building, while a polymorphic equilibrium includes earlier cooperators than any strategy in monomorphic equilibria and is thus more efficient. This is due to the non-linearity of average payoffs.     

Playing the wrong game: An experimental analysis of relational complexity and strategic misrepresentation

It has been suggested that players often produce simplified and/or misspecified mental models of strategic decisions [Kreps, D., 1990. Game Theory and Economic Modeling. Oxford Univ. Press, Oxford]. We submit that the relational structure of players' preferences in a game is a source of cognitive complexity, and may be an important driver of such simplifications. We provide a classification of order structures in two-person games based on the properties of monotonicity and projectivity, and present experiments in which subjects construct representations of games of different relational complexity and subsequently play the games according to these representations. Experimental results suggest that relational complexity matters. More complex games are harder to represent, and this difficulty seems correlated with short term memory capacity. In addition, most erroneous representations are simpler than the correct ones. Finally, subjects who misrepresent the games behave consistently with such representations, suggesting that in many strategic settings individuals may act optimally on the ground of simplified and mistaken premises.     

On the convergence of reinforcement learning

This paper examines the convergence of payoffs and strategies in Erev and Roth's model of reinforcement learning. When all players use this rule it eliminates iteratively dominated strategies and in two-person constant-sum games average payoffs converge to the value of the game. Strategies converge in constant-sum games with unique equilibria if they are pure or if they are mixed and the game is 2×2. The long-run behaviour of the learning rule is governed by equations related to Maynard Smith's version of the replicator dynamic. Properties of the learning rule against general opponents are also studied.     

On the evolutionary selection of sets of Nash equilibria

It is well established for evolutionary dynamics in asymmetric games that a pure strategy combination is asymptotically stable if and only if it is a strict Nash equilibrium. We use an extension of the notion of a strict Nash equilibrium to sets of strategy combinations called ‘strict equilibrium set’ and show the following. For a large class of evolutionary dynamics, including all monotone regular selection dynamics, every asymptotically stable set of rest points that contains a pure strategy combination in each of its connected components is a strict equilibrium set. A converse statement holds for two-person games, for convex sets and for the standard replicator dynamic.     

Informational properties of the Nash solutions of two stochastic nonzero-sum games

Within the framework of stochastic two-person nonzero-sum games, we deal with two commonly used models in engineering and economics—namely, the LQG (Linear-Quadratic-Gaussian) and the duopoly problems. We investigate how variations in information available to either player affect the equilibrium Nash strategies for these two models, whose existence and uniqueness have been proven in the paper. We show that for the LQG model better information for either player results in lower average Nash costs for both players; whereas for the duopoly model better information for one player helps him alone to achieve a higher average Nash profit, and it hurts the other player in the sense that his average Nash profit decreases. We further relate these properties of the Nash solutions for these two games to some of the distinct features of zero-sum games and team problems.     

A note on equivalence of consistency and bilateral consistency through converse consistency

In the framework of (set-valued or single-valued) solutions for coalitional games with transferable utility, the three notions of consistency, bilateral consistency, and converse consistency are frequently used to provide axiomatic characterizations of a particular solution (like the core, prekernel, prenucleolus, Shapley value). Our main equivalence theorem claims that a solution satisfies consistency (with respect to an arbitrary reduced game) if and only if the solution satisfies both bilateral consistency and converse consistency (with respect to the same reduced game). The equivalence theorem presumes transitivity of the reduced game technique as well as difference independence on payoff vectors for two-person reduced games.     

Stated Beliefs and Play in Normal-Form Games

Using data on one-shot games, we investigate whether players' actions can be viewed as responses to underlying expectations about their opponent's behaviour. In our laboratory experiments, subjects play a set of 14 two-person 3×3 games and state beliefs about which actions they expect their opponents to play. The data sets from the two tasks are largely inconsistent. Rather, we find evidence that the subjects perceive the games differently when they (i) choose actions and (ii) state beliefs—their stated beliefs reveal deeper strategic thinking than their actions. On average, they fail to best respond to their own stated beliefs in almost half of the games. The inconsistency is confirmed by estimates of a unified statistical model that jointly uses the actions and the belief statements. There, we can control for decision noise and formulate a statistical test that rejects consistency. Effects of the belief elicitation procedure on subsequent actions are mostly insignificant.