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.     

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.     

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 note on “Evolution of Preferences”

A surprising result in Dekel et al. (2007) states that strict Nash equilibria might cease to be evolutionary stable when agents are able to observe a signal that fully reveals the opponent’s preferences, even if the frequency of the signal is very low. I show that when the signal a player receives on her opponent’s preferences is almost uninformative, all strict Nash equilibria are evolutionary stable, no matter the frequency of the signal.     

Equilibria in a class of aggregative location games

Consider a multimarket oligopoly, where firms have a single license that allows them to supply exactly one market out of a given set of markets. How does the restriction to supply only one market influence the existence of equilibria in the game? To answer this question, we study a general class of aggregative location games where a strategy of a player is to choose simultaneously both a location out of a finite set and a non-negative quantity out of a compact interval. The utility of each player is assumed to depend solely on the chosen location, the chosen quantity, and the aggregated quantity of all other players on the chosen location. We show that each game in this class possesses a pure Nash equilibrium whenever the players’ utility functions satisfy the assumptions negative externality, decreasing marginal utility, continuity, and Location–Symmetry. We also provide examples exhibiting that, if one of the assumptions is violated, a pure Nash equilibrium may fail to exist.     

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.     

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.     

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.     

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.     

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.     

Sequential bargaining with common values

We study the alternating-offers bargaining problem of assigning an indivisible and commonly valued object to one of two players who jointly own this object. The players are asymmetrically informed about the object’s value and have veto power over any settlement. There is no depreciation during the bargaining process which involves signalling of private information. We characterise the perfect Bayesian equilibrium outcome of this game which is unique if offers are required to be strictly increasing. Equilibrium agreement is reached gradually and non-deterministically. The better informed player obtains a rent.     

Theoretical approaches to lowest unique bid auctions

In this paper, we analyze a game called the lowest unique bid auction (LUBA). It is known that there is no closed form solution for the symmetric mixed strategy equilibria in LUBA. We propose two alternative approaches. We use weak dominance to identify upper bounds for players’ bids. Under symmetry, we provide a method to compute a mixed strategy equilibrium by utilizing the recursive structure of the winning chances of the bids under one assumption.     

NCAA basketball: when does recruiting talent translate into wins for power conferences?

NCAA men’s basketball teams are spending an increasing amount of money on recruiting expenditures in an effort to win more games. We examine the recruiting quality of both freshmen and returning players to determine the impact of recruiting on a power conference team’s success. In particular, we identify the point at which players of different recruiting quality begin to contribute to a team’s success and estimate the impact of returning players by year and position. We find that elite freshmen guards are the only position that immediately impacts a team’s success, while forwards and centers need one or more seasons to develop into a player that generates wins for his team. We conclude by noting how these results may inform coaches regarding the makeup of talent and positions on their roster.     

重复占优策略博弈在团队培训中的应用及启发——基于一个体验式训练项目的思考

博弈论是研究决策主体行为发生直接相互作用时的决策以及这种决策的均衡问题的一门科学。占优策略博弈是指存在始终优于对方反应的博弈,占优策略博弈可能存在合作解和非合作解。在对一个基于重复非合作占优策略博弈而设计的培训项目进行分析的基础上,探讨有限重复占优策略博弈的非合作解和合作解的差别,并借鉴这种差别对于团队管理的启发,在管理实践中充分考虑主体的充分理性和最大化偏好,以团队利益为前提,尊重个体合理利益,设计科学的激励约束制度,促成一个良性的竞争合作团队氛围。     

Adaptive learning in an asymmetric auction: genetic algorithm approach

Agent-based simulations are performed to study adaptive learning in the context of asymmetric first-price auctions. Non-linearity of the Nash equilibrium strategies is used to investigate the effect of task complexity on adaptive learning by varying the degree of approximation the agents can handle. In addition, learning in different information environments is explored. Social learning allows agents to imitate each other’s bidding strategies based on their relative success. Under individual learning agents are limited to their own experience. We observe convergence to steady states near the predicted equilibrium in all cases. The ability to learn non-linear functions helps the agents with a non-linear equilibrium strategy but hurts the agents with an almost linear one. Better information about the opponent population has a relatively modest impact. A larger number of strategies to experiment with and an ability to systematically compare strategies by holding a number of factors constant have a comparatively stronger beneficial effect.     

R&D Competition when firms Choose Variance

I consider an infinite-period race where players choose between low- and high-variance motion technologies. I provide sufficient conditions under which, in equilibrium, the leader chooses a safe technology and the laggard a risky one, thus formalizing the sports intuition that the laggard has nothing to lose. Various examples and empirical implications are presented.     

Uncertain information structures and backward induction

In everyday economic interactions, it is not clear whether each agent’s sequential choices are visible to other participants or not: agents might be deluded about others’ ability to acquire, interpret or keep track of data. Following this idea, this paper introduces uncertainty about players’ ability to observe each others’ past choices in extensive-form games. In this context, we show that monitoring opponents’ choices does not affect the outcome of the interaction when every player expects their opponents indeed to be monitoring. Specifically, we prove that if players are rational and there is common strong belief in opponents being rational, having perfect information and believing in their own perfect information, then, the backward induction outcome is obtained regardless of which of her opponents’ choices each player observes. The paper examines the constraints on the rationalization process under which reasoning according to Battigalli’s (1996) best rationalization principle yields the same outcome irrespective of whether players observe their opponents’ choices or not. To this respect we find that the obtention of the backward induction outcome crucially depends on tight higher-order restrictions on beliefs about opponents’ perfect information. The analysis provides a new framework for the study of uncertainty about information structures and generalizes the work by Battigalli and Siniscalchi (2002) in this direction.     

Evolution of repeated prisoner's dilemma play under logit dynamics

In an evolutionary set-up, we append an ecology of iterated prisoner's dilemma (IPD) game strategies, consisting of unconditional cooperators (AllC), unconditional defectors (AllD) and reactive players (TFT) with two repeated strategies that have received less attention in the evolutionary IPD game literature: the error-proof, “generous” tit-for-tat (GTFT) which, with a certain probability, re-establishes cooperation after a (possibly by mistake) defection of the opponent and the penitent, “stimulus–response” (WSLS) strategy that resets cooperation after the opponent punished for defection. An abundance of rock–paper–scissors like patterns is discovered in the 3×3 ecologies comprising Pavlovian and “generous” players. Interestingly, the evolutionary success of Pavlov seems to depend on the absence of unconditional cooperators in the ecologies investigated.     

Bargaining through Approval

The paper considers two-person bargaining under Approval Voting. It first proves the existence of pure strategy equilibria. Then it shows that this bargaining method ensures that both players obtain at least their mean utility level in equilibrium. Finally it proves that, provided that the players are partially honest, the mechanism triggers sincerity and ensures that no alternative Pareto dominates the outcome of the game.