Learning games

This paper presents a model of learning about a game. Players initially have little knowledge about the game. Through playing the same game repeatedly, each player not only learns which action to choose but also constructs a personal view of the game. The model is studied using a hybrid payoff matrix of the prisoner's dilemma and coordination games. Results of computer simulations show that (1) when all the players are slow at learning the game, they have only a partial understanding of the game, but might enjoy higher payoffs than in cases with full or no understanding of the game; (2) when one player is quick in learning the game, that player obtains a higher payoff than the others. However, all can receive lower payoffs than in the case in which all players are slow learners.     

A new relative skill measure for games with chance elements

An interesting aspect of games is the relative extent to which a player can positively influence his results by making appropriate strategic choices. This question is closely related to the issue of how to distinguish between games of skill and games of chance. The distinction between these two types of games is definitely interesting from a juridical point of view. In this paper we present a modification of an existing measure of the skill level of a game, which has served as a juridical tool for the classification of games. The main difference is that this new definition does not automatically classify incomplete information games without chance moves as games of skill. Copyright © 2004 John Wiley & Sons, Ltd.     

Strategic behavior in non-atomic games

In order to remedy the possible loss of strategic interaction in non-atomic games with a societal choice, this study proposes a refinement of Nash equilibrium, strategic equilibrium. Given a non-atomic game, its perturbed game is one in which every player believes that he alone has a small, but positive, impact on the societal choice; and a distribution is a strategic equilibrium if it is a limit point of a sequence of Nash equilibrium distributions of games in which each player’s belief about his impact on the societal choice goes to zero. After proving the existence of strategic equilibria, we show that all of them must be Nash. We also show that all regular equilibria of smooth non-atomic games are strategic. Moreover, it is displayed that in many economic applications, the set of strategic equilibria coincides with that of Nash equilibria of large finite games.     

Strategic complementarities and nested potential games

This paper reports that every finite game of strategic complementarities is a nested pseudo-potential game defined by Uno [Uno, H., 2007. Nested potential games. Economics Bulletin 3(17), 1–8] if the action set of each player is one-dimensional, except possibly for one player.     

Repeated two-person zero-sum games with unequal discounting and private monitoring

We consider discounted repeated two-person zero-sum games with private monitoring. We show that even when players have different and time-varying discount factors, each player’s payoff is equal to his stage-game minmax payoff in every sequential equilibrium. Furthermore, we show that: (a) in every history on the equilibrium path, the pair formed by each player’s conjecture about his opponent’s action must be a Nash equilibrium of the stage game, and (b) the distribution of action profiles in every period is a correlated equilibrium of the stage game. In the particular case of public strategies in public monitoring games, players must play a Nash equilibrium after any public history.     

Discussie bijdrage

An analysis is given of the character of games of chance and games of skill. It is shown that every game is influenced by stochastic effects. Whether a game should be classified as a game of pure skill or as a game of chance depends on the rules of the game, on the evaluation of the outcomes of the game and on the possibilities the player has by applying his skill to cope with the random effects of the particular game situations.     

There is no ‘I’ in team: Career concerns,risk‐taking incentives,and team outcomes

The National Basketball Association contracting rules provide plausibly exogenous variation in career concerns near contract end. We use this setting to study how individual career concerns affect risk‐taking behavior and can sabotage team performance. Using the frequency and duration of player injuries from 1991 to 2013 we measure individual risk‐taking behavior. We find that the average player's likelihood of missing a game due to injury falls by 0.06 percentage points (or over 100% relative to the mean injury rate) in the final 3 months of his contract, and when missing games due to injury is unavoidable, his recovery time drops by 22 days. However, “elite” players with virtually no career concerns actually miss more games due to injury. Finally, we find that elite players missing too many games and “average” players playing before healthy, combine to hurt team performance. For each additional player in the last 3 months on contract, the win probability for that team falls by over 2.6%.     

What restrictions do Bayesian games impose on the value of information?

In a Bayesian game players play an unknown game. Before the game starts some players may receive a signal regarding the specific game actually played. Typically, information structures that determine different signals, induce different equilibrium payoffs. In two-person zero-sum games the equilibrium payoff measures the value of the particular information structure which induces it. We pose a question as to what restrictions Bayesian games impose on the value of information. We provide answers for two kinds of information structures: symmetric, where both players are equally informed, and one-sided, where only one player is informed.     

Averaged predictions and the learning of equilibrium play

The main objects here are noncooperative games in which all externalities occur via a one-dimensional variable. So-called mean-value iterates are used to approach Nash equilibrium. The proposed schemes generalize many received methods, and can be interpreted as learning taking place during repeated play. An important feature is that no player need be fully informed about the game structure. Particular examples include Cournot oligopolies and some nonatomic market games.     

Non-cooperative games on hyperfinite Loeb spaces

We present a particular class of measure spaces, hyperfinite Loeb spaces, as a model of situations where individual players are strategically negligible, as in large non-anonymous games, or where information is diffused, as in games with imperfect information. We present results on the existence of Nash equilibria in both kinds of games. Our results cover the case when the action sets are taken to be the unit interval, results now known to be false when they are based on more familiar measure spaces such as the Lebesgue unit interval. We also emphasize three criteria for the modelling of such game-theoretic situations—asymptotic implementability, homogeneity and measurability—and argue for games on hyperfinite Loeb spaces on the basis of these criteria. In particular, we show through explicit examples that a sequence of finite games with an increasing number of players or sample points cannot always be represented by a limit game on a Lebesgue space, and even when it can be so represented, the limit of an existing approximate equilibrium may disappear in the limit game. Thus, games on hyperfinite Loeb spaces constitute the ‘right' model even if one is primarily interested in capturing the asymptotic nature of large but finite game-theoretic phenomena.     

Piecewise closed-loop equilibria in differential games with regime switching strategies

We propose a new methodology exploring piecewise closed-loop equilibrium strategies in differential games with regime switching actions. We develop a general game with two players. Players choose an action that influences the evolution of a state variable, and decide on the switching time from one regime to another. Compared to the optimal control problem with regime switching, necessary optimality conditions are modified for the first player to switch. When choosing her optimal switching strategy, this player considers the impact of her choice on the other player’s actions and consequently on her own payoffs. In order to determine the equilibrium timing of regime changes, we derive conditions that help eliminate candidate equilibrium strategies that do not survive deviations in switching strategies. We then apply this new methodology to an exhaustible resource extraction game.     

Coalitional games with veto players: Consistency, monotonicity and Nash outcomes

Following Dagan et al. [Dagan, N., Volij, O., Serrano, R. (1997). A non-cooperative view on consistent bankruptcy rules, Games Econ. Behav. 18, 55–72], we construct an extensive form game for veto-balanced TU games in which a veto player is the proposer and the other players are responders. The set of Nash outcomes of this extensive form game is described, and compared to solutions of TU games such as the nucleolus, kernel and egalitarian core. We find necessary and sufficient conditions under which the nucleolus of the game is a Nash outcome.     

Bounded memory and permanent reputations

A recent result in repeated incomplete information games is that after an arbitrarily long history, any equilibrium of the continuation game must be an equilibrium of the complete information game. This result implies that reputation is a short-run phenomenon. We study a particular class of reputation games and show that bounded memory may lead to permanent reputations. In fact, for a particular range of parameters, a bounded memory player may never be able to learn anything at all.     

The testable implications of zero-sum games

We study collective choices from the revealed preference theory viewpoint. For every product set of individual actions, joint choices are called Nash-rationalizable if there exists a preference relation for each player such that the selected joint actions are Nash equilibria of the corresponding game. We characterize Nash-rationalizable joint choice behavior by zero-sum games, or games of conflicting interests. If the joint choice behavior forms a product subset, the behavior is called interchangeable. We prove that interchangeability is the only additional empirical condition which distinguishes zero-sum games from general non-cooperative games.     

Robust games: theory and application to a Cournot duopoly model

In this paper, the robust game model proposed by Aghassi and Bertsimas (Math Program Ser B 107:231–273, 2006) for matrix games is extended to games with a broader class of payoff functions. This is a distribution-free model of incomplete information for finite games where players adopt a robust-optimization approach to contend with payoff uncertainty. They are called robust players and seek the maximum guaranteed payoff given the strategy of the others. Consistently with this decision criterion, a set of strategies is an equilibrium, robust-optimization equilibrium, if each player’s strategy is a best response to the other player’s strategies, under the worst-case scenarios. The aim of the paper is twofold. In the first part, we provide robust-optimization equilibrium’s existence result for a quite general class of games and we prove that it exists a suitable value \(\epsilon \) such that robust-optimization equilibria are a subset of \(\epsilon \)-Nash equilibria of the nominal version, i.e., without uncertainty, of the robust game. This provides a theoretical motivation for the robust approach, as it provides new insight and a rational agent motivation for \(\epsilon \)-Nash equilibrium. In the last part, we propose an application of the theory to a classical Cournot duopoly model which shows significant differences between the robust game and its nominal version.     

Performance evaluation of National Football League teams

Most recent empirical analyses of production in the sports economic literature have focused on Major League Baseball. This paper extends that literature by analysing football production in the National Football League (NFL). Using the Poisson regression model, we measure the performance of NFL teams and head coaches. The measure is based on a production process where player skills are converted into games won. The evidence reveals that quality coaching is an important component in the production process. It appears that efficient coaching can account for an additional three to four victories in a given season. Copyright © 2000 John Wiley & Sons, Ltd.     

Weakly balanced contributions and the weighted Shapley values

We provide a concise characterization of the class of positively weighted Shapley values by three properties, two standard properties, efficiency and marginality, and a relaxation of the balanced contributions property called the weak balanced contributions property. Balanced contributions: the amount one player gains or loses when another player leaves the game equals the amount the latter player gains or loses when the former player leaves the game. Weakly balanced contributions: the direction (sign) of the change of one player’s payoff when another player leaves the game equals the direction (sign) of the change of the latter player’s payoff when the former player leaves the game. Given this characterization, the symmetric Shapley value can be “extracted”from the class of positively weighted Shapley values by either replacing the weak balanced contributions property with the standard symmetry property or by strengthening the former into the balanced contributions property.     

Best response dynamics with level-n expectations in two-stage games

This paper analyzes behavior in repeatedly played two-stage games, where players choose actions in both stages according to best replies using level-n expectations about the opponent׳s actions in both stages. Level-n expectations are recursively defined in a way that a player holding level-n expectations correctly predicts the action of an opponent holding level-(n−1) expectations. A general conceptual framework to study such dynamics for two-stage games is developed and it is shown that, contrary to results for single-stage games, the fixed points of the dynamics depend on the level of the expectations. In particular, for level-0 expectation, fixed points correspond to a Nash equilibrium of a simultaneous move version of the game, whereas (under certain conditions) fixed points converge towards the subgame perfect equilibrium of the two-stage game if the level of expectations goes to infinity. The approach is illustrated using a two-stage duopoly game, where firms in the first stage invest in activities reducing their marginal costs and in the second stage engage in Cournot competition. Conditions for local stability of the fixed points are derived for different levels of expectations and it is shown that level-2 expectations are sufficient to move the fixed-point of the dynamics to a close neighborhood of the subgame perfect equilibrium. Furthermore, it is demonstrated that although firms benefit from unilateral increases in the level of expectations, an increase of n by all firms reduces all profits.     

Strong Nash equilibrium in games with common and complementary local utilities

A rather general class of strategic games is described where the coalitional improvements are acyclic and hence strong Nash equilibria exist: The players derive their utilities from the use of certain facilities; all players using a facility extract the same amount of local utility therefrom, which amount depends both on the set of users and on their actions, and is decreasing in the set of users; the ultimate utility of each player is the minimum of the local utilities at all relevant facilities. Two important subclasses are “games with structured utilities,” basic properties of which were discovered in 1970s and 1980s, and “bottleneck congestion games,” which attracted researchers’ attention quite recently. The former games are representative in the sense that every game from the whole class is isomorphic to one of them. The necessity of the minimum aggregation for the existence of strong Nash equilibria, actually, just Pareto optimal Nash equilibria, in all games of this type is established.