The Folk Theorem for Repeated Games with Time-Dependent Discounting

This paper defines a general framework to study infinitely repeated games with time-dependent discounting in which we distinguish and discuss both time-consistent and -inconsistent preferences. To study the long-term properties of repeated games, we introduce an asymptotic condition to characterize the fact that players become more and more patient; that is, the discount factors at all stages uniformly converge to one. Two types of folk theorems are proven without the public randomization assumption: the asymptotic one, that is, the equilibrium payoff set converges to the feasible and individual rational set as players become patient, and the uniform one, that is, any payoff in the feasible and individual rational set is sustained by a single strategy profile that is an approximate subgame perfect Nash equilibrium in all games with sufficiently patient discount factors. We use two methods for the study of asymptotic folk theorem: the self-generating approach and the constructive proof. We present the constructive proof in the perfect-monitoring case and show that it can be extended to time-inconsistent preferences. The self-generating approach applies to the public-monitoring case but may not extend to time-inconsistent preferences because of a nonmonotonicity result.

Reactive Strategies: An Inch of Memory, a Mile of Equilibria

We explore how an incremental change in complexity of strategies (“an inch of memory”) in repeated interactions influences the sets of Nash Equilibrium (NE) strategy and payoff profiles. For this, we introduce the two most basic setups of repeated games, where players are allowed to use only reactive strategies for which a probability of players’ actions depends only on the opponent’s preceding move. The first game is trivial and inherits equilibria of the stage game since players have only unconditional (memory-less) Reactive Strategies (RSs); in the second one, players also have conditional stochastic RSs. This extension of the strategy sets can be understood as a result of evolution or learning that increases the complexity of strategies. For the game with conditional RSs, we characterize all possible NE profiles in stochastic RSs and find all possible symmetric games admitting these equilibria. By setting the unconditional benchmark as the least symmetric equilibrium payoff profile in memory-less RSs, we demonstrate that for most classes of symmetric stage games, infinitely many equilibria in conditional stochastic RSs (“a mile of equilibria”) Pareto dominate the benchmark. Since there is no folk theorem for RSs, Pareto improvement over the benchmark is the best one can gain with an inch of memory.

A Few Bad Apples Spoil the Barrel: An Anti-Folk Theorem for Anonymous Repeated Games with Incomplete Information

We study anonymous repeated games where players may be “commitment types” who always take the same action. We establish a stark anti-folk theorem: if the distribution of the number of commitment types satisfies a smoothness condition and the game has a “pairwise dominant” action, this action is almost always taken. This implies that cooperation is impossible in the repeated prisoner's dilemma with anonymous random matching. We also bound equilibrium payoffs for general games. Our bound implies that industry profits converge to zero in linear-demand Cournot oligopoly as the number of firms increases. (JEL C72, C73, D83)

A complete folk theorem for finitely repeated games

This paper analyzes the set of pure strategy subgame perfect Nash equilibria of any finitely repeated game with complete information and perfect monitoring. The main result is a complete characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of the finitely repeated game. This model includes the special case of observable mixed strategies.

A Course in Microeconomic Theory

This book is a text in microeconomics that is both challenging and “user-friendly.” The work is designed for the first-year graduate microeconomic theory course and is accessible to advanced undergraduates as well. Placing unusual emphasis on modern noncooperative game theory, it provides the student and instructor with a unified treatment of modern microeconomic theory — one that stresses the behavior of the individual actor (consumer or firm) in various institutional settings. The author has taken special pains to explore the fundamental assumptions of the theories and techniques studied, pointing out both strengths and weaknesses. The book begins with an exposition of the standard models of choice and the market, with extra attention paid to choice under uncertainty and dynamic choice. General and partial equilibrium approaches are blended, so that the student sees these approaches as points along a continuum. The work then turns to more modern developments. Readers are introduced to noncooperative game theory and shown how to model games and determine solution concepts. Models with incomplete information, the folk theorem and reputation, and bilateral bargaining are covered in depth, followed by exploration of information economics. A closing discussion concerns firms as organizations and gives readers a taste of transaction-cost economics.

Decentralised Central Management: New Perspectives on the Socialist Planning Debate

The importance of the centralisation/decentralisation debate has been highlighted during the Covid-19 pandemic. This controversy is not new. The paper examines the two possibilities. – planning from the centre or, from the periphery and finally proposes a third way which combines the best of both. The debate is approached from various angles – a cybernetic perspective, socio-political perspective and an economic perspective. The core result is the creation of Decentralised Central Control (DCC) which allows optimal control at nodes/regions and data transmission and decision implementation is optimised. The DCC is predicated on the assumption that technology advance can create a system of central planning which is essentially decentralised and is not inhibited by the fatal flaws of traditional Central Planning such as leads and lags in decision making and information exchange on the back of a static and unchanging technology. The fractal system envisaged in this paper is almost a clone of the rational expectations configuration of the perfect market where there is instantaneous market clearing and near-perfect foresight for all agents on the basis of common knowledge. In this paper, we have demonstrated the notion of the problem of externalities and the divergence between private and social cost in common scarce resources and the solution offered by Eleanor Ostrom which is in fact a qualitative version of the Folk Theorem of Game Theory where perfect solutions arise once all parties realise a commonality purpose given a low discount rate and trigger price strategies.

Common learning and cooperation in repeated games

We study repeated games in which players learn the unknown state of the world by observing a sequence of noisy private signals. We find that for generic signal distributions, the folk theorem obtains using ex post equilibria. In our equilibria, players commonly learn the state, that is, the state becomes asymptotic common knowledge.

The Folk Theorem in Repeated Games With Anonymous Random Matching

We prove the folk theorem for discounted repeated games with anonymous random matching. We allow non‐uniform matching, include asymmetric payoffs, and place no restrictions on the stage game other than full dimensionality. No record‐keeping or communication devices—including cheap talk communication and public randomization—are necessary.

Repeated Game Analysis of a CSMA/CA Network under a Backoff Attack

We study a CSMA/CA (Carrier Sense Medium Access with Collision Avoidance) wireless network where some stations deviate from the defined contention mechanism. By using Bianchi’s model, we study how this deviation impacts the network throughput and show that the fairness of the network is seriously affected, as the stations that deviate achieve a larger share of the resources than the rest of stations. Previously, we modeled this situation using a static game and now, we use repeated games, which, by means of the Folk theorem, allow all players to have better outcomes. We provide analytical solutions to this game for the two player case using subgame perfect and correlated equilibria concepts. We also propose a distributed algorithm based on communicating candidate equilibrium points for learning the equilibria of this game for an arbitrary number of players. We validate approach using numerical simulations, which allows comparing the solutions we propose and discussing the advantages of using each of the methods we propose.