Cost-sharing in directed networks: Experimental study of equilibrium choice and system dynamics

This study reports the results of an experiment on directed networks with positive externalities induced by cost-sharing. Subjects participated in a network game in which they had to choose between private and public transportations. If a player chose public transportation, then she shared the travel cost equally with other players making the same choice, whereas if she chose private transportation, then her travel cost was fixed. Travel costs on the private route were manipulated across the two experimental conditions. In one condition, these costs were homogeneous among players; in the other condition, they were heterogeneous among players and only privately known. We found that half (none) of the player groups in the homogeneous (heterogeneous) condition converged toward the efficient equilibrium. Examination of the system dynamics shows that convergence toward efficiency was facilitated by: (1) the existence of an intermediate equilibrium choice; and (2) strategic teaching by which a farsighted player chooses strategies with poor short-term payoff in order to shift group decisions to the efficient equilibrium and thereby increase her own long-term benefit.     

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.     

Sovereign debt: Reputation, seizure and reputation

The interaction between a creditor and a sovereign debtor is described as a ‘one-shot’ game with discrete actions—total or no debt-repudiation and seizure of asset holding abroad. Possible Nash equilibria where each player chooses an action as to maximize his expected payoff given his beliefs about the other player’s action and the implications of those actions on the players’ trustworthy reputation are identified. However, if reputation losses rise convexly with the players’ relative hostility, partial repudiation and seizure can be the preferred strategies. The preferred repudiation and seizure rates are analyzed under asymmetric and symmetric information about the state of the world. (JEL classification F34)     

Deadlines in stochastic contests

We consider a two-player contest model in which breakthroughs arrive according to privately observed Poisson processes. Each player’s process continues as long as she exerts costly effort. The player who collects the most breakthroughs until a predetermined deadline wins a prize.We derive Nash equilibria of the game depending on the deadline. For short deadlines, there is a unique equilibrium in which players use identical cutoff strategies, i.e., they continue until they have a certain number of successes. If the deadline is long enough, the symmetric equilibrium distribution of an all-pay auction is an equilibrium distribution over successes in the contest. Expected efforts may be maximal for a short or intermediate deadline.     

Optimal information exchange in contests

We study optimal exchange of private information in a two-player all-pay auction contest with independent private binary values. A benevolent information center who is informed about the players’ values facilitates the exchange of information by disclosing a signal publicly. The informativeness of the signal determines the monotonicity of the unique symmetric equilibrium and the players’ expected payoff. We characterize the upper bound of players’ expected payoff and the corresponding optimal signals utilizing such a relation between the informativeness and the payoff. When the players are ex ante sufficiently heterogeneous, the optimal signals work through an information-rent channel by inducing allocative efficient contests. When the players are ex ante sufficiently homogeneous, the optimal signals work through an unlevel-playing-field channel by inducing asymmetric contests. In order to guarantee efficient allocation, a regulator can punish any exchange of information when the players are sufficiently homogeneous and impose no restrictions when they are sufficiently heterogeneous.     

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.     

Optimal risk taking in an uneven tournament game with risk averse players

We analyze the optimal choice of risk in a two-stage tournament game between two players that have different concave utility functions. At the first stage, both players simultaneously choose risk. At the second stage, both observe overall risk and simultaneously decide on effort or investment. The results show that those two effects which mainly determine risk taking – an effort effect and a likelihood effect – are strictly interrelated. This finding sharply contrasts with existing results on risk taking in tournament games with symmetric equilibrium efforts where such linkage can never arise. Conditions are derived under which this linkage leads to a reversed likelihood effect so that the favorite (underdog) can increase his winning probability by increasing (decreasing) risk which is impossible in a completely symmetric setting.     

On the Choice of Risk and Effort in Tournaments—Experimental Evidence

We investigate a simple two‐person tournament in a controlled laboratory experiment. Each player chooses between two distributions of random shocks. After observing the overall risk, both players decide simultaneously on their effort. Theory predicts both players should choose the distribution with the higher variance of random shock, as this minimizes equilibrium effort. We show that the effort exerted is sensitive towards risk. The agents exert less effort if the random shock is high. However, agents do not learn to commit themselves by choosing a high risk in our experiment.     

Perseverance and suspense in tug-of-war

We study a tug-of-war game between two players using the lottery contest success function (CSF) and a quadratic cost (of effort) function. We construct a pure strategy symmetric Markov perfect equilibrium of this game, show that it is unique, and provide closed-form solutions for equilibrium strategies and values. In stark contrast to a model of tug-of-war with an all-pay auction CSF, players exert positive efforts until the very last battle in this equilibrium. We deliver a set of empirically appealing results on effort dynamics.     

A model of influence with a continuum of actions

We generalize a two-action (yes–no) model of influence to a framework in which every player has a continuum of actions, among which he has to choose one. We assume the set of actions to be an interval. Each player has an inclination to choose one of the actions. Due to the influence among players, the final decision of a player, i.e., his choice of one action, may be different from his original inclination. In particular, a coalition of players with the same inclination may influence another player with different inclination, and as a result of this influence, the decision of the player is closer to the inclination of the influencing coalition than his inclination was. We introduce a measure of such a positive influence of a coalition on a player. Several unanimous influence functions in this generalized framework are considered. Also the set of fixed points under a given influence function is analyzed. Furthermore, we study linear influence functions and discuss their convergence. For a linear unanimous function, we find necessary and sufficient conditions for the existence of the positive influence of a coalition on a player, and we calculate the value of the influence index. We also introduce a measure of a negative influence of a coalition on a player.     

Learning and decision costs in one-person games

This paper reports the results of a two-part data analysis of learning in a repeated costly decision experiment. In the first part we test payoff dominance under the hypothesis of expected payoff maximization. We utilize a dynamic probability distribution over decisions for each player, characterizing what each player has learned about the payoff function by the moments from these distributions. In the second part, we test the hypothesis of expected payoff maximization for players in each treatment group. Expected payoff maximization is supported but payoff dominance is not. Copyright © 1999 John Wiley & Sons, Ltd.     

Good neighbors are hard to find: computational complexity of network formation

We investigate the computational complexity of several decision problems in a simple strategic game of network formation. We find that deciding if a player has a strategy that guarantees him a certain payoff against a given strategy profile of the other players is an NP-complete problem. Deciding if there exists a strategy profile that guarantees a certain aggregate payoff is also NP-complete. Deciding if there is a Nash equilibrium in pure strategies which guarantees a certain payoff to each player is NP-hard. The problem of deciding if a given strategy profile is a Nash equilibrium is investigated as well. We would like to thank both referees for valuable corrections and suggestions and one referee in particular for a very detailed and beneficial critique. We thank Colin de la Higuera for fruitful discussions. Support by the French Ministry for Youth, Education and Research, through project SCSHS-2004-04 is gratefully acknowledged.     

The cost of Corporate Social Responsibility: the case of the Community Reinvestment Act

A Data Envelopment Analysis (DEA) cost minimization model is employed to estimate the cost to thrift institutions of achieving a rating of ‘outstanding’ under the anti-redlining Community Reinvestment Act, which is viewed as an act of voluntary Corporate Social Responsibility (CSR). There is no difference in overall cost efficiency between ‘outstanding’ and minimally compliant ‘satisfactory’ thrifts. However, the sources of cost inefficiency do differ, and an ‘outstanding’ rating involves annual extra cost of $6.547 million or, 1.2% of total costs. This added cost is the shadow price of CSR since it is not an explicit output or input in the DEA cost model. Before and after-tax rates of return are the same for the ‘outstanding’ and ‘satisfactory’ thrifts, which implies a recoupment of the extra cost. The findings are consistent with CSR as a management choice based on balancing marginal cost and marginal revenue. An incidental finding is that larger thrifts are less efficient.

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.     

Buying cooperation in an asymmetric environmental differential game

We consider a two-player asymmetric differential game of pollution control. One player is non-vulnerable to pollution, or unwilling to consider damages when choosing her production policy in a non-cooperative game. We characterize the feedback-Nash equilibrium and the cooperative solution. We establish conditions under which the vulnerable player can buy the cooperation of the non-vulnerable player to control her emissions. We further use the Nash bargaining solution to allocate the total cooperative dividend between the two players and propose a time-consistent decomposition overtime of the total payoff.     

First price package auction with many traders

This paper studies a first price package auction in which multiple sellers participate in addition to multiple buyers. We generalize the notion of the profit-target strategy which is first introduced as a truthful strategy in a first price package auction with a single seller by Bernheim and Whinston (1986b). We then show that the set of equilibrium payoffs in profit-target strategies is equal to the bidder-optimal core, and is also equal to the set of coalition-proof Nash equilibria. Using this result, we find that any equilibrium payoff vector is weakly Pareto-dominated by the VCG payoff vector for buyers, and that the Walrasian competitive equilibrium payoff vector is weakly Pareto-dominated by some equilibrium payoff vector for buyers, even if goods are substitutes. This contrasts with the first price package auction with a single seller, in which it is shown that if goods are substitutes, then those three outcomes are payoff-equivalent.     

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.     

Deterministic multi-player Dynkin games

A multi-player Dynkin game is a sequential game in which at every stage one of the players is chosen, and that player can decide whether to continue the game or to stop it, in which case all players receive some terminal payoff.We study a variant of this model, where the order by which players are chosen is deterministic, and the probability that the game terminates once the chosen player decides to stop may be strictly less than 1.We prove that a subgame-perfect ε-equilibrium in Markovian strategies exists. If the game is not degenerate this ε-equilibrium is actually in pure strategies.     

Sad-Loser contests

We consider asymmetric winner-reimbursed contests. It turns out that such contests (Sad-Loser) have multiple internal pure-strategy equilibria (where at least two players are active). We describe all equilibria and discuss their properties. In particular, we find (1) that an active player is indifferent among all her non-negative choices and her expected payoff is zero in any internal equilibrium, (2) that a higher-value (stronger) player always spends less than a lower-value (weaker) player and therefore always has a lower chance to win a Sad-Loser contest in any internal equilibrium, and (3) a sufficient condition for a net total spending to be higher in a Sad-Loser contest than in the corresponding asymmetric contest.     

Pay,productivity and aging in Major League Baseball

Using panels of player pay and performance from Major League Baseball (MLB), we examine trends in player productivity and salaries as players age. Pooling players of all ability levels leads to a systematic bias in regression coefficients. After addressing this problem by dividing players into talent quintiles, we find that the best players peak about 2 years later than marginal players, and development and depreciation of performance appear to be more pronounced for players with the highest ability levels. Within-career variation, however, is less pronounced than between-player variation, and the performance level of players within a given quintile will typically remain lower than the talent level for rookies in the next higher quintile. We also find preliminary evidence that free agents are paid proportionately to their production at all ability levels, whereas young players’ salaries are suppressed by similar amounts.