Equilibrium selection in similar repeated games: experimental evidence on the role of precedents

We report on an experiment examining behavior and equilibrium selection in two similar, infinitely repeated games, Stag Hunt and Prisoner’s Dilemma under anonymous random matching. We are interested in the role that historical precedents may play for equilibrium selection between these two repeated games. We find that a precedent for efficient play in the repeated Stag Hunt game does not carry over to the repeated Prisoner’s Dilemma game despite the possibility that efficient play can be sustained as an equilibrium of the indefinitely repeated game. Similarly, a precedent for inefficient play in the repeated Prisoner’s Dilemma game does not extend to the repeated Stag Hunt game. We conclude that equilibrium selection between similar repeated games may have less to do with historical precedents and might instead depend more on strategic considerations associated with the different payoffs of these similar repeated games.     

Three very simple games and what it takes to solve them

We study experimentally the nature of dominance violations in three minimalist dominance-solvable guessing games. Only about a third of our subjects report reasoning consistent with dominance; they all make dominant choices and almost all expect others to do so. Nearly two-third of our subjects report reasoning inconsistent with dominance, yet a quarter of them actually make dominant choices and half of those expect others to do so. Reasoning errors are more likely for subjects with lower working memory, intrinsic motivation and premeditation attitude. Dominance-incompatible reasoning arises mainly from subjects misrepresenting the strategic nature (payoff structure) of the guessing games.     

Payoff dominance and risk dominance in the observable delay game: a note

We examine whether the payoff dominant sequential-move (Stackelberg) outcome is realized when timing is endogenized. We adopt the observable delay game formulated by Hamilton and Slutsky [Games Econ Behav 2(1):29–46, 1990]. We find that if one sequential-move outcome is payoff dominant, either (i) the outcome both players prefer is the unique equilibrium; or (ii) two sequential-move outcomes are equilibria and the one both players prefer is risk dominant. In other words, no conflict between payoff dominance and risk dominance in the observable delay game exists, in contrast to other games such as (non pure) coordination games. We also find that even if one of two sequential-move outcomes is the unique equilibrium outcome in the observable delay game, it does not imply that the equilibrium outcome is payoff dominant to the other sequential-move outcome.      

Satisficing and optimality in 2×2 common interest games

Summary. The paper studies the evolution of cooperation when satisficing players repeatedly play a symmetric two-by-two game of common interest. We show that if initial aspiration levels are sufficiently close to the efficient payoff and aspiration adjusts at a sufficiently slow speed then the unique long run state will be the efficient outcome. In the special case of coordination games, the more tension there is between payoff dominance and risk dominance, the longer it takes for the system to lock into the payoff dominant outcome. Received: June 23, 1997; revised version: November 19, 1997     

When less information is good enough: experiments with global stag hunt games

There is mixed evidence on whether subjects coordinate on the efficient equilibrium in experimental stag hunt games under complete information. A design that generates an anomalously high level of coordination, Rankin et al. (Games Econo Behav 32(2):315–337, 2000), varies payoffs each period in repeated play rather than holding them constant. These payoff “perturbations” are eerily similar to those used to motivate the theory of global games, except the theory operates under incomplete information. Interestingly, that equilibrium selection concept is known to coincide with risk dominance, rather than payoff dominance. Thus, in theory, a small change in experimental design should produce a different equilibrium outcome. We examine this prediction in two treatments. In one, we use public signals to match Rankin et al. (2000)’s design; in the other, we use private signals to match the canonical example of global games theory. We find little difference between treatments, in both cases, subject play approaches payoff dominance. Our literature review reveals this result may have more to do with the idiosyncrasies of our complete information framework than the superiority of payoff dominance as an equilibrium selection principle.     

Playing safe in coordination games:: the roles of risk dominance, payoff dominance, and history of play

This paper presents results from a series of experiments designed to test the impact on subject behavior of changes in the risk dominance and payoff dominance characteristics of two player coordination games. The main finding is that changes in risk dominance significantly affect play of the subjects, whereas changes in the level of payoff dominance do not. Observed history of play also has an important influence on subject behavior, both when subjects are randomly rematched after each game and when they remain matched with the same individual for a sequence of games.     

Group play in games and the role of consent in network formation

We study games played between groups of players, where a given group decides which strategy it will play through a vote by its members. When groups consist of two voting players, our games can also be interpreted as network-formation games. In experiments on Stag Hunt games, we find a stark contrast between how groups and individuals play, with payoffs playing a primary role in equilibrium selection when individuals play, but the structure of the voting rule playing the primary role when groups play. We develop a new solution concept, robust-belief equilibrium, which explains the data that we observe. We provide results showing that this solution concept has application beyond the particular games in our experiments.     

Dominance solvable approval voting games

This work provides necessary and sufficient conditions for the dominance solvability of approval voting games. Our conditions are very simple since they depend just on the number of possible winners when voters play weakly undominated strategies. If there are at most two possible winners, then the game is dominance‐solvable and the outcome coincides with the Condorcet winner. If every candidate is a possible winner, the game is not dominance‐solvable. If none of the previous conditions holds, then the game need not be dominance‐solvable, and the outcome need not coincide with the Condorcet winner.     

Coordination and transfer

We study the ability of subjects to transfer principles between related coordination games. Subjects play a class of order statistic coordination games closely related to the well-known minimum (or weak-link) and median games (Van Huyck et al. in Am Econ Rev 80:234–248, 1990, Q J Econ 106(3):885–910, 1991). When subjects play a random sequence of games with differing order statistics, play is less sensitive to the order statistic than when a fixed order statistic is used throughout. This is consistent with the prediction of a simple learning model with transfer. If subjects play a series of similar stag hunt games, play converges to the payoff dominant equilibrium when a convention emerges, replicating the main result of Rankin et al. (Games Econ Behav 32:315–337, 2000). When these subjects subsequently play a random sequence of order statistic games, play is shifted towards the payoff dominant equilibrium relative to subjects without previous experience. The data is consistent with subjects absorbing a general principle, play of the payoff dominant equilibrium, and applying it in a new related setting.     

Incentive monotonicity and equilibrium selection in 2×2 matrix games

For the class of 2×2 matrix games with two strict Nash equilibria the paper introduces an equilibrium refinement called incentive monotonicity. It selects the risk-dominant equilibrium if interests are conflicting, while it remains silent in games with common interests. These results suggest that the equilibrium-selection problem might be more difficult in games with common interests, which is certainly the case if risk dominance and payoff dominance go in opposite directions.     

Experiment-inspired comments on John Roemer’s theory of cooperation

Abstract

We report on a nonsocial experiment where we find that all participants choose the dominant strategy in the experimental payoff, and compare it with the payoff-isomorphic, but social, Prisoner’s Dilemma treatment presented in a recent paper by Bosch-Domènech and Silvestre where 28% choose cooperation instead of the dominant strategy. The contrast reinforces Roemer’s emphasis on human cooperation. Next, we argue that Roemer’s Simple Kantian Equilibrium works well as a theory of cooperation under the assumption of monotonicity (positive or negative externalities), but not when efficient cooperation requires the division of labor by coordinating dissimilar tasks.     

On the Coalition-Proofness of the Pareto Frontier of the Set of Nash Equilibria

This article shows that the Pareto efficient frontier of the Nash equilibrium set of games with strategic substitutes is coalition-proof under the following conditions: (1) the game has three players, or, alternatively, a player's payoff depends on her own strategy and on the sum (but not on the composition) of other players' strategies; (2) an increase in a player's strategy either raises all other players' payoffs monotonically or reduces them monotonically; and (3) each player's payoff is strictly concave in her own strategy. Under these conditions, the Pareto dominance refinement is equivalent to the coalition-proof Nash equilibrium refinement.Journal of Economic LiteratureClassification Number: C72.     

Strategic Similarity and Emergent Conventions: Evidence from Similar Stag Hunt Games

This paper reports evidence on the origin of convention in laboratory cohorts confronting similar but not identical strategic situations repeatedly. The experiment preserves the action space of the game, while randomly perturbing the payoffs and scrambling the action labels in an effort to blunt the salience of retrospective selection principles. Hence, the similarity between stage games is reduced to certain strategic details, like efficiency, security, and risk dominance. Nevertheless, we do observe conventions emerging in half of the laboratory cohorts. When a convention emerges subjects's behavior conforms to the selection principles of efficiency rather than security or risk dominance. Journal of Economic Literature Classification Numbers: C72, C78, C92, D83.     

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.     

Infinite horizon common interest games with perfect information

We consider infinite horizon common interest games with perfect information. A game is a K-coordination game if each player can decrease other players' payoffs by at most K times his own cost of punishment. The number K represents the degree of commonality of payoffs among the players. The smaller K is, the more interest the players share. A K-coordination game tapers off if the greatest payoff variation conditional on the first t periods of an efficient history converges to 0 at a rate faster than K^−t as t→∞. We show that every subgame perfect equilibrium outcome is efficient in any tapering-off game with perfect information. Applications include asynchronously repeated games, repeated games of extensive form games, asymptotically finite horizon games, and asymptotically pure coordination games.     

Evolutionarily stable conjectures and other regarding preferences in duopoly games

We study the evolutionary selection of conjectures in duopoly games when players have other regarding preferences, i.e. preferences over payoff distributions. In both the Cournot and Bertrand duopoly games, the consistent conjectures are independent of other regarding preferences. Both duopoly games have evolutionarily stable conjectures that depend on other regarding preferences but that do not coincide with the consistent conjectures. For increasingly spiteful preferences, the evolutionarily stable conjectures implicate low quantities in the Cournot game and high prices in the Bertrand game, whereas the inverse relationships hold for the consistent conjectures. We discuss our findings in the context of ultimate and proximate causation.     

Characterization of the extreme core allocations of the assignment game

A characterization of the extreme core allocations of the assignment game is given in terms of the reduced marginal worth vectors. For each ordering in the player set, a payoff vector is defined where each player receives his or her marginal contribution to a certain reduced game played by his or her predecessors. This set of reduced marginal worth vectors, which for convex games coincide with the usual marginal worth vectors, is proved to be the set of extreme points of the core of the assignment game. Therefore, although assignment games are hardly ever convex, the same characterization of extreme core allocations is valid for convex games.     

Continuous-Time Strategy Selection in Linear Population Games

In an experimental evolutionary game framework we investigate whether subjects end up in a socially efficient state. We examine two games, a game where the socially efficient state is also an equilibrium and a game which has no equilibrium in pure strategies at all. Furthermore, we distinguish between a situation in which the subjects are completely informed about the payoff function and a situation in which they are incompletely informed. We observe that subjects spend the greater part of the time at or near the efficient state. If the efficient state is an equilibrium, they spend more time there than otherwise. Furthermore, incomplete information increases the time spent at the efficient state.     

Conventions: An evolutionary approach

Conventions are social institutions that solve recurrent coordination problems. In normative game theory, coordination games are considered problematic because of the multiplicity of equilibria. From a neoinstitutionalist perspective, however, this multiplicity should be an important part of the explanation of real-world institutions. The paper discusses the evolutionary (or “positive”) game-theoretical approach to the emergence of conventions. I note a precise sense in which conventions may be said to minimize transaction costs, but that they need not be efficient. Example applications to language, money, and the theory of the firm are discussed.     

Aggregate comparative statics

In aggregative games, each playerʼs payoff depends on her own actions and an aggregate of the actions of all the players. Many common games in industrial organization, political economy, public economics, and macroeconomics can be cast as aggregative games. This paper provides a general and tractable framework for comparative static results in aggregative games. We focus on two classes of games: (1) aggregative games with strategic substitutes and (2) nice aggregative games, where payoff functions are continuous and concave in own strategies. We provide simple sufficient conditions under which positive shocks to individual players increase their own actions and have monotone effects on the aggregate. The results are illustrated with applications to public good provision, contests, Cournot competition and technology choices in oligopoly.