Infinite-horizon Game Research Articles

Game theory and the kula

The paper expounds a non-cooperative game that can be interpreted as a model of the system of kula that was described by Bronislaw Malinowski in his Argonauts of the Western Pacific. The game of kula is an infinite-horizon game with an arbitrary, but fixed, number n of players. It generates pure norms of direct reciprocity, pure norms of indirect reciprocity, and mixed norms whereby a player who deviates is punished both by the individual who has been harmed and by a third party.

Bargaining and efficiency in networks

We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain over a unit surplus. Players who reach agreement are removed from the network without replacement. The global logic of efficient matchings and the local nature of bargaining, in combination with the irreversible exit of player pairs following agreements, create severe hurdles to the attainment of efficiency in equilibrium. For many networks all Markov perfect equilibria of the bargaining game are inefficient, even as players become patient. We investigate how incentives need to be structured in order to achieve efficiency via subgame perfect, but non-Markovian, equilibria. The analysis extends to an alternative model in which individual players are selected according to some probability distribution, and a chosen player can select a neighbor with whom to bargain.

Markov equilibria in a model of bargaining in networks

We study the Markov perfect equilibria (MPEs) of an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain. Players who reach agreement are removed from the network without replacement. We establish the existence of MPEs and show that MPE payoffs are not necessarily unique. A method for constructing pure strategy MPEs for high discount factors is developed. For some networks, we find that all MPEs are asymptotically inefficient as players become patient.

Bargaining in Stationary Networks

We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain. Players who reach agreement are replaced by new players at the same positions in the network. We show that all equilibria are payoff equivalent. The payoffs and the set of agreement links converge as players become patient. Several new concepts—mutually estranged sets, partners, and shortage ratios—provide insights into the relative strengths of the positions in the network. We develop a procedure to determine the limit equilibrium payoffs for all players. Characterizations of equitable and nondiscriminatory networks are also obtained. (JEL C78, D85)

Inter-firm knowledge diffusion, market power, and welfare

We propose an infinite-horizon quantity-setting differential game with learning spillovers and organizational forgetting to analyze the optimal management decisions affecting the evolution of the stock of know-how, and, in turn, the dynamics of productive efficiency. Specifically, we study the long run impact of inter-firm knowledge diffusion on market power, i.e. the ability of a firm to raise the price above the marginal cost, and welfare. We consider two types of processes through which knowledge is acquired: (i) passive learning, or learning-by-doing, where managers do not actively invest in information and (ii) active learning, or learning-by-investing, where managers acquire new and additional information through specific investments in human capital. We show that: under (i), knowledge diffusion reduces market power; under (ii), knowledge diffusion reduces market power as long as learning spillovers are sufficiently important. From a welfare viewpoint, we also show that: under (i), knowledge diffusion is always welfare-enhancing; under (ii), weak spillovers are required in order for knowledge diffusion to be welfare-enhancing.

Equilibria in an infinite horizon game with an incumbent, entry and switching costs

We consider an infinite horizon game in which a consumer incurs a switching cost when changing supplier. There is an incumbent firm at the beginning of the game, and a new entrant in each period, all with a production cost of 0. We identify the set of possible profits of the incumbent when the common discount rate of all the agents is greater than 1/2: it is the interval [0, σ/(1 − δ)]. There exist equilibria in which the incumbent sells the good to the consumer in every period at any price in [0, σ]. The result also holds with several entrants and several consumers.

Why Does the WTO Have an Antidumping Agreement?

This paper develops a model that offers a plausible interpretation for the empirical observation of diffusion of antidumping (AD) laws amongst WTO members. Contrary to the regnant belief that this proliferation is driven mainly by retaliatory motives, our model shows that adoption and use of AD laws involve a cooperative act amongst WTO members in an infinite horizon game. The diffusion of AD laws need not result in a Prisonner.s dilemma. Instead we show that the AD agreement of the WTO can serve as a risk-sharing vehicle amongst WTO members in response to adverse shocks incurred by domestic producers.

Bargaining and Efficiency in Networks

We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain over a unit surplus. Players who reach agreement are removed from the network without replacement. The global logic of efficient matchings and the local nature of bargaining, in combination with the irreversible exit of player pairs following agreements, create severe hurdles to the attainment of efficiency in equilibrium. For many networks all Markov perfect equilibria of the bargaining game are inefficient, even as players become patient. We investigate how incentives need to be structured in order to achieve efficiency via subgame perfect, but non-Markovian, equilibria. The analysis extends to an alternative model in which individual players are selected according to some probability distribution, and a chosen player can select a neighbor with whom to bargain.

Existence of perfect equilibria in a class of multigenerational stochastic games of capital accumulation

In this paper we introduce a model of multigenerational stochastic games of capital accumulation where each generation consists of m different players. The main objective is to prove the existence of a perfect stationary equilibrium in an infinite horizon game. A suitable change in the terminology used in this paper provides (in the case of perfect altruism between generations) a new Nash equilibrium theorem for standard stochastic games with uncountable state space.

Firm Regulation and Profit Sharing: A Real Option Approach

Abstract To avoid the extremely high profit levels found in the recent experience of public utilities' regulation, some regulators have introduced a profit-sharing (PS) rule that revises prices to the benefit of consumers. However, in order to be successful, a PS rule should satisfy appropriate incentive conditions.In this paper, we study the incentive properties of a second best PS mechanism designed by the regulator to induce a regulated monopolist to divert its "excessive" profits to the customers. In a real option model where a regulated monopolist manages a long-term franchise contract and the regulator has the option to revoke the contract if there is serious welfare loss, we first endogenously derive the welfare maximising PS rule under the verifiability of profits. We then explore the dynamic efficiency of this PS rule under the non-verifiability of profits and study the firm's incentive to comply with it in an infinite-horizon game. Finally, we derive the price adjustment path which follows the adoption of a PS rule in a price cap regulation.We show that the riskiness of the distribution of the firm's future profits and the regulator's cost in revoking the franchise contract are key factors in determining the equilibrium properties of a dynamic PS rule.

Designing Efficient Mechanisms for Dynamic Bilateral Trading Games

This paper studies the problem of allocating a good between two players in each period of an infinite-horizon game. The players' valuations in each period are private information, and the valuations change over time. We analyze two special cases for the dynamics of valuations: correlated where players' valuations are exogenous but serially correlated; and learning by doing, where a player's past consumption improves his current distribution of valuations, but his valuations are otherwise uncorrelated. We analyze conditions under which there exists an efficient, Bayesian incentive-compatible (BIC), individually rational (IR), budget-balanced (BB) mechanism, when the mechanism designer has commitment power. We consider

The Impact of the Termination Rule on Cooperation in a Prisoner's Dilemma Experiment

Cooperation in prisoner's dilemma games can usually be sustained only if the game has an infinite horizon. We analyze to what extent the theoretically crucial distinction of finite vs. infinite-horizon games is reflected in the outcomes of a prisoner's dilemma experiment. We compare three different experimental termination rules in four treatments: a known finite end, an unknown end, and two variants with a random termination rule (with a high and with a low continuation probability, where cooperation can occur in a subgame-perfect equilibrium only with the high probability). We find that the termination rules do not significantly affect average cooperation rates. Specifically, employing a random termination rule does not cause significantly more cooperation compared to a known finite horizon, and the continuation probability does not significantly affect average cooperation rates either. However, the termination rules may influence cooperation over time and end-game behavior. Further, the (expected) length of the game significantly increases cooperation rates. The results suggest that subjects may need at least some learning opportunities (like repetitions of the supergame) before significant backward induction arguments in finitely repeated game have force.

On Dynamic Compromise

What prevents majorities from extracting surplus from minorities in legislatures? We study an infinite horizon game where a legislative body votes to determine distributive policy each period. Proposals accepted by a simple majority are implemented, otherwise the status quo allocation prevails. We construct a symmetric Markov perfect equilibrium that exhibits compromise in the following sense: if the initial status quo allocation is not too unequal, then the Markov process is absorbed into allocations in which more than a minimum winning majority receives a positive share of the social surplus with positive probability. The compromise is only sustainable if, starting from the unequal allocations, the Markov process is absorbed into allocations in which there is a complete absence of compromise. The compromise equilibrium exists when discounting is neither too small nor too large. We find that, contrary to intuition, the range of discount factors for which this equilibrium exists increases as the number of legislators increases. In this sense, compromise is easier in larger legislatures.

Decentralized learning from failure

We study decentralized learning in organizations. Decentralization is captured through Crawford and Haller's [Learning how to cooperate: optimal play in repeated coordination games, Econometrica 58 (1990) 571–595] attainability constraints on strategies. We analyze a repeated game with imperfectly observable actions. A fixed subset of action profiles are successes and all others are failures. The location of successes is unknown. The game is played until either there is a success or the time horizon is reached. We partially characterize optimal attainable strategies in the infinite horizon game by showing that after any fixed time, agents will occasionally randomize while at the same time mixing probabilities cannot be uniformly bounded away from zero.

Infinite horizon noncooperative differential games

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton–Jacobi equations. In the present paper, we consider a class of infinite horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value functions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.

Constructions of Nash Equilibria in Stochastic Games of Resource Extraction with Additive Transition Structure

A class of N-person stochastic games of resource extraction with discounted payoffs in discrete time is considered. It is assumed that transition probabilities have special additive structure. It is shown that the Nash equilibria and corresponding payoffs in finite horizon games converge as horizon goes to infinity. This implies existence of stationary Nash equilibria in the infinite horizon case. In addition the algorithm for finding Nash equilibria in infinite horizon games is discussed

Strategic Delegation in a Durable Goods Monopoly

This paper studies a simultaneous-move infinite-horizon delegation game in which the principal of a durable goods monopoly entrusts pricing decisions to a manager who enjoys monetary rewards but dislikes production effort. We show that cheap delegation enables the principal to attain the precommitment price plan in a time consistent Markov-perfect equilibrium. The paper analyzes the robustness of this result under alternative specifications of timing and objectives. We also provide a numerical characterization of the Markov-perfect equilibrium pricing and remuneration strategies for the case of linear-quadratic payoffs.

Construction of Nash equilibria in symmetric stochastic games of capital accumulation

A large class of nonzero-sum symmetric stochastic games of capital accumulation/resource extraction is considered. An iterative method leading to a Nash equilibrium in the infinite horizon game with the discounted evaluation is studied.

Forecast Horizon for a Class of Dynamic Games

In theory, a Markov perfect equilibrium of an infinite-horizon nonstationary dynamic game requires from the players the ability to forecast an infinite amount of data. In this paper, we prove that early strategic decisions are decoupled effectively from the tail game in nonstationary dynamic games with discounting and uniformly bounded rewards. This decoupling is formalized by the notion of a forecast horizon. In words, the first-period equilibrium strategies are invariant with respect to changes in the game parameters for periods beyond the forecast horizon. We illustrate our results in the context of dynamic games of exploitation of a common pool resource and make use of the rather natural monotonicity properties of finite-horizon equilibria.

Capital Controls: a Political Economy Approach

AbstractThe paper examines the economic consequences of political conflicts that arise when countries implement capital controls. In an overlapping‐generations model, agents vote on whether to open or close an economy to capital flows. The young (workers) receive income only from wages while the old (capitalists) receive income only from savings. The authors characterize the set of stationary equilibria for an infinite‐horizon game. Assuming dynamic efficiency, when the median representative is a worker (capitalist), capital‐importing countries will open (close) while capital‐exporting countries will close (open). These predicted patterns are consistent with data on liberalization policies over time and across various countries.