Strategic complexity in repeated extensive games

This paper studies a machine (finite automaton) playing a two-player repeated game of a simple extensive-form game with perfect information. We introduce a new complexity measure called multiple complexity which incorporates a strategyʼs responsiveness to information in the stage game as well as the number of states of the machine. We completely characterize the Nash equilibrium of the machine game. In the sequential-move prisonerʼs dilemma, cooperation can be sustained as an equilibrium.     

Cooperation in international environmental negotiations due to a preference for equity

This paper demonstrates that cooperation in international environmental negotiations can be explained by preferences for equity. Within a N-country prisoner’s dilemma in which agents can either cooperate or defect, in addition to the standard non-cooperative equilibrium, cooperation of a large fraction or even of all countries can establish a Nash equilibrium. In an emission game, however, where countries can choose their abatement level continuously, equity preferences cannot improve upon the standard inefficient Nash equilibrium. Finally, in a two stage game on coalition formation, the presence of equity-interested countries increases the coalition size and leads to efficiency gains. Here, even a stable agreement with full cooperation can be reached.     

Reciprocity Game

This paper shows that reciprocity comes from the desire to cooperate in finitely repeated prisoner's dilemma game. Before playing the finitely repeated prisoner's dilemma game, players choose the reciprocity level and commit to it, and the reciprocity level is public information. There are T equilibria if the prisoner's dilemma game is repeated for T periods, and each equilibrium is associated with different levels of cooperation. Further, if players choose their reciprocity levels sequentially, then the most cooperative equilibrium will be the unique equilibrium. However, reciprocity does not matter for the one‐period game and the infinitely repeated game.     

Dynamic group formation in the repeated prisoner?s dilemma

We consider dynamic group formation in repeated n-person prisoner?s dilemma. Agreements in coalitional bargaining are self-binding in that they are supported as subgame perfect equilibria of repeated games. Individuals are allowed to renegotiate the cooperating group agreement through a process of voluntary participation. We prove that a cooperating group forms as an absorbing state of a Markov perfect equilibrium after a finite number of renegotiations if and only if the group is Pareto efficient, provided that individuals are patient. The cooperating group can only expand.     

防范国企经营者合谋：不对称信息的“囚徒困境”博弈

如何防范国企经营者和主管部门官员之间合谋侵占国有企业的生产剩余,一直是政府部门和学术界关注的焦点问题。本文根据Laffont和Martimort等在委托-代理框架下提出的分权制思想,构建了防范国企经营者合谋的"囚徒困境"博弈模型。但由于基于"囚徒困境"博弈所构建的双重审核机制可能产生过高的审查成本,且其实施受到主管官员有限责任的限制,因此通过引入不对称信息进一步改进了"囚徒困境"博弈,使不合谋的均衡结果能在更宽松的条件下实现。     

The supercore for normal-form games

This paper analyzes the supercore of a system derived from a normal-form game. For the case of a finite game with pure strategies, we define a sequence of games and show that the supercore coincides with the set of Nash equilibria of the last game in that sequence. This result is illustrated with the characterization of the supercore for the n-person prisoner's dilemma. With regard to the mixed extension of a normal-form game, we show that the set of Nash equilibrium profiles coincides with the supercore for games with a finite number of Nash equilibria.     

Characterizing belief-free review-strategy equilibrium payoffs under conditional independence

This paper proposes and studies a tractable subset of Nash equilibria, belief-free review-strategy equilibria, in repeated games with private monitoring. The payoff set of this class of equilibria is characterized in the limit as the discount factor converges to one for games where players observe statistically independent signals. As an application, we develop a simple sufficient condition for the existence of asymptotically efficient equilibria, and establish a folk theorem for N-player prisoner?s dilemma. All these results are robust to a perturbation of the signal distribution, and hence remain true even under almost-independent monitoring.     

On the Evolution of Pareto‐Optimal Behavior in Repeated Coordination Problems

I characterize the asymptotic behavior of a society facing a repeated‐common‐interest game. In this society, new individuals enter to replace their “parents” at random times. Each entrant has possibly different beliefs about others' behavior than his or her predecessor. A self‐confirming equilibrium (SCE) belief process describes an evolution of beliefs in this society consistent with a self‐confirming equilibrium of the repeated game. The main result shows that for any common‐interest game, the Pareto‐dominant equilibrium is a globally absorbing state of the behavioral dynamics when the SCE beliefs of new entrants satisfy certain independence and full‐support properties. This result does not involve either of the usual assumptions of myopia or large inertia common in evolutionary models, nor is this result possible if only Nash rather than self‐confirming equilibria are considered.     

Maximum‐revenue versus Optimum‐welfare Export Taxes: a Delegation Game*

In the Eaton and Grossman (1986 ) Bertrand duopoly model of strategic export taxes, both countries may be better off if they both delegate to policymakers who maximize tax revenue rather than welfare. However, both countries delegating to policymakers who maximize tax revenue is not a Nash equilibrium unless the degree of product substitutability is sufficiently high. For a wide range of values for the degree of product substitutability, the game is a prisoner's dilemma where both countries are better off delegating to policymakers who maximize tax revenue but both will delegate to policymakers who maximize welfare in the Nash equilibrium.     

Rational conjectures and evolutionary beliefs in public goods games

This paper applies conjectural variations (CVs) to a model of public good provision and shows that CVs are superior to Nash beliefs. In addition to imposing consistency, as Bresnahan, I show that consistent conjectures (CCs) are obtained from individual payoff maximization. CCs emerge as the unique subgame perfect Nash equilibrium (NE) in a two-stage game in which beliefs are chosen in Stage 1 and quantities in Stage 2. There is an individual payoff advantage to non-Nash behavior, generating a Prisoner's Dilemma in conjectures in addition to the usual free-rider problem associated with public goods. The correct and payoff maximizing conjecture is the unique equilibrium in an evolutionary framework against a player with Nash conjectures. The consistent conjecture equilibrium is the unique evolutionary equilibrium when both players conjectures evolve. Hence, the NE prediction is too optimistic when players have rational conjectures.     

Pricing decisions in a dual channels system with different power structures

This paper investigates the effect of product substitutability and relative channel status on pricing decisions under different power structures of a dual exclusive channel system where each manufacturer distributes its goods through a single exclusive retailer but two goods are substitute. A linear demand based on the utility function of a representative consumer is assumed, and three game scenarios(Manufacturer Stackelberg, Retailer Stackelbeg and Vertical Nash) are examined under symmetric and asymmetric related channel status. It is shown that no power structure is always the best for the entire supply chain though all members on supply chain have incentive to lead the Stackelberg game. Meanwhile, the vertical Nash game is an equilibrium for the members, however, a Prisoner's dilemma necessarily incurs for the entire supply chain because the Retailer Stackelberg or the Manufacturer Stackelberg can gain the better performance than that in vertical Nash for the entire supply chain when the product substitutability is moderate or higher and the asymmetric relative channel status is moderate, while consumers always get the most welfare from the vertical Nash game.     

Evolutionary Stability in the Investment-Opportunism-Retaliation Game

Can economically efficient outcomes be obtained and sustained in the absence of externally enforced property rights? We study the evolutionary properties of a game that exhibits two well-defined Nash equilibria: one generates an inefficient outcome while the other set generates an efficient outcome supported by the potential for retaliation. Although standard forward-looking refinements eliminate the efficient equilibrium, neither equilibrium type satisfies strict evolutionary stability criteria. However, both types of equilibrium define strategies that are neutrally stable, which makes them vulnerable to drift in dynamic environments. We conduct computer simulation experiments in which players learn adaptively via a tournament selection mechanism called sophisticated experimentation. Our simulations demonstrate that while the system spends a disproportionately high proportion of time in the inefficient equilibrium set, the efficient equilibrium is pervasive as the system drifts back and forth between the equilibrium sets, never settling on one or the other.     

On Learning to Cooperate

This paper shows that experimentation can have the effect of inducing outcomes that are “far” from Nash in a learning model similar to that studied by D. Fudenberg and D. K. Levine (1993, Econometrica61, 547-573). The game under consideration is a version of a finitely repeated prisoner's dilemma, and the average length of players' lifetimes is arbitrarily large, but fixed. If the number of stages in the prisoner's dilemma is sufficiently large, then experimentation induces cooperation within the society on an arbitrarily large scale, even though the beliefs of almost all players are eventually within an arbitrary η>0 of being correct at all of the information sets in the game. Journal of Economic Literature Classification Numbers: C72.     

Repeated games with present-biased preferences

We study infinitely repeated games with perfect monitoring, where players have β-δ preferences. We compute the continuation payoff set using recursive techniques and then characterize equilibrium payoffs. We then explore the cost of the present-time bias, producing comparative statics. Unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set is not monotonic in β or δ. Finally, we show how the equilibrium payoff set is contained in that of a repeated game with smaller discount factor.     

Interlinkage and Generous Tit‐for‐Tat Strategy

We investigate an infinitely‐repeated prisoners' dilemma with imperfect monitoring and consider the possibility that the interlinkage of the players' distinct activities enhances implicit collusion. We show a necessary and sufficient condition for the existence of a generous tit‐for‐tat Nash equilibrium. Such an equilibrium, if it exists, is unique. This equilibrium achieves approximate efficiency when monitoring is almost perfect, where the discount factors are fixed.     

The projection dynamic and the geometry of population games

The projection dynamic is an evolutionary dynamic for population games. It is derived from a model of individual choice in which agents abandon their current strategies at rates inversely proportional to the strategies' current levels of use. The dynamic admits a simple geometric definition, its rest points coincide with the Nash equilibria of the underlying game, and it converges globally to Nash equilibrium in potential games and in stable games.     

Global dynamics in repeated games with additively separable payoffs

This paper studies the global dynamics of a class of infinitely repeated two-player games in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that each player's action is a stationary function of the opponent's last action. We completely characterize IREs and their dynamics in terms of certain indifference curves. Our results are used to show that in a prisoners' dilemma game with mixed strategies, gradual cooperation occurs when the players are sufficiently patient, and that in a certain duopoly game, kinked demand curves emerge naturally.     

p-Dominance and Equilibrium Selection under Perfect Foresight Dynamics

This paper studies equilibrium selection based on a class of perfect foresight dynamics and relates it to the notion of p-dominance. A continuum of rational players is repeatedly and randomly matched to play a symmetric n×n game. There are frictions: opportunities to revise actions follow independent Poisson processes. The dynamics has stationary states, each of which corresponds to a Nash equilibrium of the static game. A strict Nash equilibrium is linearly stable under the perfect foresight dynamics if, independent of the current action distribution, there exists a consistent belief that any player necessarily plays the Nash equilibrium action at every revision opportunity. It is shown that a strict Nash equilibrium is linearly stable under the perfect foresight dynamics with a small degree of friction if and only if it is the p-dominant equilibrium with p<1/2. It is also shown that if a strict Nash equilibrium is the p-dominant equilibrium with p<1/2, then it is uniquely absorbing (and globally accessible) for a small friction (but not vice versa). Set-valued stability concepts are introduced and their existence is shown. Journal of Economic Literature Classification Numbers: C72, C73.     

Competitive behavior in market games: Evidence and theory

We explore whether competitive outcomes arise in an experimental implementation of a market game, introduced by Shubik (1973) [21]. Market games obtain Pareto inferior (strict) Nash equilibria, in which some or possibly all markets are closed. We find that subjects do not coordinate on autarkic Nash equilibria, but favor more efficient Nash equilibria in which all markets are open. As the number of subjects participating in the market game increases, the Nash equilibrium they achieve approximates the associated competitive equilibrium of the underlying economy. Motivated by these findings, we provide a theoretical argument for why evolutionary forces can lead to competitive outcomes in market games.     

Controls of the complex dynamics of a multi-market Cournot model

In this paper, a multi-market Cournot game is proposed based on a specific inverse demand function. The game is studied statically and dynamically. Puu's incomplete information approach, as a realistic method, is used to contract the corresponding dynamical model under this function. Therefore, some stability analysis is carried out on the model to detect the stability and instability conditions of the system's Nash equilibrium. Based on that analysis some dynamic phenomena such as bifurcation and chaos are found. Under certain assumption, chaos control is performed in order to control the monopolistic model. Furthermore, a dynamic multi-market Cournot model is introduced.