Folk Theorem Research Articles

Two-Person Adversarial Games are Zero-Sum: An elaboration of a folk theorem

The observation that every two-person adversarial game is an affine transformation of a zero-sum game is traceable to Luce and Raiffa (1957) and made explicit in Aumann (1987). Recent work of Adler et al. (2009) and of Raimondo (2023) in increasing generality, proves what has so far remained a conjecture. We present two proofs of an even more general formulation: the first draws on multilinear utility theory developed by Fishburn and Roberts (1978); the second is a consequence of Adler et al.’s 2009 proof itself for a special case of a two-player game in which each player has a set of three actions.

Folk theorems in repeated games with switching costs

We study how switching costs affect the subgame perfect equilibria in repeated games. We show that (i) the Folk Theorem holds whenever the players are patient enough; (ii) the set of equilibrium payoffs is obtained by considering the payoffs of a simple one-shot auxiliary game; and (iii) the switching costs have a negative impact on a player in the infinitely undiscounted repeated game but can be beneficial for him in a finitely repeated game or in a discounted game.

Folk theorem under bankruptcy

In this paper, we examine Subgame Perfect Nash Equilibria (SPE) within an infinite horizon oligopolistic market, accounting for the possibility of firm bankruptcy. Central to our dynamic analysis is the concept of Bankruptcy-Free (BF) allocations. The consideration of bankruptcy introduces several distinctive features not encompassed by standard models of infinitely repeated games. Notably, when firms are sufficiently patient towards future outcomes, achieving certain collusive outcomes as equilibria may not be possible. We discern qualitatively different results depending on the number of firms in the market; in a duopoly, only specific BF allocations can be sustained as SPE, while in scenarios with more than two firms, allocations outside the BF set can also be sustained as SPE. In both instances, it is crucial that the payoffs are above the minimax BF payoff, a condition more restrictive than the standard minimax payoff advocated in Folk Theorems.

Entrepreneurial marketing strategies, resources and social disadvantage: exploring the role of resources and minority status among US agrofood enterprises

PurposeThis paper aims to investigate associations between firm resources and reliance on entrepreneurial marketing (EM) channels among agrofood ventures. It accounts for agropreneur gender and racial/ethnic status in the context of marketing channel portfolio composition. The authors examine the established assumption that resource limitations drive EM and whether socially disadvantaged status of agropreneurs is associated with marketing strategy beyond standard resourcing measures.Design/methodology/approachUsing 2015 Local Foods Marketing Practices Survey data, the authors apply linear regression to investigate differences in the use of EM channels, accounting for resources, social status and other factors.FindingsLimited-resource ventures rely more on consumer-oriented channels that require EM practices. Socially disadvantaged entrepreneurs favor these channels, even when accounting for resources. Notably, ventures headed by men of color rely more on the most customer-centric local foods marketing channel.Research limitations/implicationsFuture research should investigate how social and human capital influences the use of EM.Practical implicationsEntrepreneurial support policy and practice for agropreneurs should be cautious about the “double-burden” folk theorem of intersectional disadvantage and review how to best direct resources on EM to groups most likely to benefit.Originality/valueThis paper uses a unique, restricted, nation-wide, federal data set to examine relationships between resource endowments, social status and the composition of agrofood enterprises’ marketing channel portfolios. To the best of the authors’ knowledge, it is the first to include racial- and ethnic-minority status of agropreneurs and to account for intersectionality with gender.

Uniformly strict equilibrium for repeated games with private monitoring and communication

AbstractCooperation through repetition is an important theme in game theory. In this regard, various celebrated “folk theorems” have been proposed for repeated games in increasingly more complex environments. There has, however, been insufficient attention paid to the robustness of a large set of equilibria that is needed for such folk theorems. Starting with perfect public equilibrium as our starting point, we study uniformly strict equilibria in repeated games with private monitoring and direct communication (cheap talk). We characterize the limit equilibrium payoff set and identify the conditions for the folk theorem to hold with uniformly strict equilibrium.

The tripartite auction folk theorem

We formally study two bidder first-price, second-price, and all-pay auctions with known values, deriving the equilibrium payoffs and strategies and showing when all three yield the same equilibrium payoffs to the bidders. This latter result, the tripartite auction theorem, does not hold for all auctions, in particular it can fail for symmetric auctions with high stakes and in auctions with very low stakes.

Daily commuting

Workers generally commute on a daily basis, so we model commuting as a repeated game. The folk theorem implies that for sufficiently large discount factors, the repeated commuting game has as a Nash equilibrium any feasible strategy that is uniformly better than the minimax strategy payoff for a commuter in the one shot game, repeated over the infinite horizon. This includes the efficient equilibria. An example where the efficient payoffs strictly dominate the one shot Nash equilibrium payoffs is provided. Our conclusions pose a challenge to congestion pricing in that equilibrium selection could be at least as effective in improving welfare. We examine evidence from St. Louis to determine what equilibrium strategies are actually played in the repeated commuting game.

Games on graphs with coalitions

This paper models games where the strategies are nodes of a graph G (we denote them as G-games) and in presence of coalition structures. The cases of one-shot and repeated games are presented. In the latter situation, coalitions are assumed to move from a strategy to another one under the constraint that they are adjacent in the graph. We introduce novel concepts of pure and mixed equilibria which are comparable with classical Nash and Berge equilibria. A Folk Theorem for G-games of repeated type is presented.

The folk theorem for the prisoner's dilemma with endogenous private monitoring

We study the repeated prisoner's dilemma with private monitoring under the assumption that the monitoring structure is endogenously chosen by the players in each period. We allow the players to choose from all possible monitoring structures. If the players disagree on the monitoring structure they would like, the realized monitoring structure is determined by a function that aggregates their choices. When one player can dictate the monitoring structure, then the repetition of the stage Nash is the only sequential equilibrium outcome. In contrast, when no player can dictate the monitoring structure, we provide conditions on the aggregation function under which any strictly individually rational and feasible payoff vector can be supported in sequential equilibrium.

Economists' Folk Theorems and Problems with the US Energy Transition

Proposition: For the challenges facing the US energy transition to renewable electricity, and renewable energy generally, organized US wholesale electricity markets are part of the problem.

Commitment Games with Conditional Information Disclosure

The conditional commitment abilities of mutually transparent computer agents have been studied in previous work on commitment games and program equilibrium. This literature has shown how these abilities can help resolve Prisoner’s Dilemmas and other failures of cooperation in complete information settings. But inefficiencies due to private information have been neglected thus far in this literature, despite the fact that these problems are pervasive and might also be addressed by greater mutual transparency. In this work, we introduce a framework for commitment games with a new kind of conditional commitment device, which agents can use to conditionally disclose private information. We prove a folk theorem for this setting that provides sufficient conditions for ex post efficiency, and thus represents a model of ideal cooperation between agents without a third-party mediator. Further, extending previous work on program equilibrium, we develop an implementation of conditional information disclosure. We show that this implementation forms program ε-Bayesian Nash equilibria corresponding to the Bayesian Nash equilibria of these commitment games.

Monitoring versus Discounting in Repeated Games

We study how discounting and monitoring jointly determine whether cooperation is possible in repeated games with imperfect (public or private) monitoring. Our main result provides a simple bound on the strength of players' incentives as a function of discounting, monitoring precision, and on‐path payoff variance. We show that the bound is tight in the low‐discounting/low‐monitoring double limit, by establishing a public‐monitoring folk theorem where the discount factor and the monitoring structure can vary simultaneously.

Folk Theorem under Bankruptcy

Folk Theorem under Bankruptcy

Bad apples in symmetric repeated games

We study large‐population repeated games where players are symmetric but not anonymous, so player‐specific rewards and punishments are feasible. Players may be commitment types who always take the same action. Even though players are not anonymous, we show that an anti‐folk theorem holds when the commitment action is “population dominant,” meaning that it secures a payoff greater than the population average payoff. For example, voluntary public goods provision in large populations is impossible when commitment types never contribute, even if monetary rewards can be targeted to contributors; however, provision is possible if noncontributors can be subjected to involuntary fines. A folk theorem under incomplete information provides a partial converse to our result. Along the way, we develop some general results on symmetric games with incomplete information and/or repeated play.

New Section: Folk Theorems

New Section: Folk Theorems

Why Does the Folk Theorem Do Not Seem to Work When It Is Mostly Needed?

Why Does the Folk Theorem Do Not Seem to Work When It Is Mostly Needed?

Why Does the Folk Theorem Do Not Seem to Work When it is Mostly Needed?

Why Does the Folk Theorem Do Not Seem to Work When it is Mostly Needed?

The Open Society and Its Complexities

<i>The Open Society and Its Complexities</i>

Intertemporal Hedging and Trade in Repeated Games With Recursive Utility

Two key features distinguish the general class of recursive preferences from the standard model of dynamic choice: (i) agents may care about the intertemporal distribution of risk, and (ii) their rates of time preference, rather than being fixed, may vary with the level of consumption. We investigate what these features imply in the context of a repeated strategic interaction. First, we show that opportunities for intertemporal trade may expand the set of feasible payoffs relative to that in a static interaction. Two distinct sources for such trade are identified: endogenous heterogeneity in the players' rates of time preference and a hedging motive pertaining to the intertemporal distribution of risk. The set of equilibrium payoffs may on the other hand shrink drastically as many efficient outcomes become unsustainable no matter the level of patience. This “antifolk” result occurs when the players prefer stage outcomes to be positively correlated rather than independent across time. Intuitively, such preferences make it inefficient to offset short‐term losses with future gains, while this is needed to ensure that security levels are met on path. We also establish a folk theorem: if security levels are met on path, such play can be sustained in a subgame perfect equilibrium provided that the players are sufficiently patient.

Some examples for stable and historic behavior in replicator equations

The evolutionary dynamics of zero-sum and non zero-sum games under replicator equations could be drastically different from each other. In zero-sum games, heteroclinic cycles naturally occur whenever the species of the population supersede each other in cyclic fashion (like for the Rock–Paper–Scissors game). In this case, the highly erratic oscillations may cause the divergence of the time averages. In contrast, it is a common belief that all “reasonable” replicator equations of non-zero sum games satisfy “The Folk Theorem of Evolutionary Game Theory” which asserts that (i)a Nash equilibrium is a rest point; (ii)a stable rest point is a Nash equilibrium; (iii)a strictly Nash equilibrium is asymptotically stable; (iv)any interior convergent orbit evolves to a Nash equilibrium. In this paper, we propose two distinct classes of replicator equations generated by Schur-convex potential functions which exhibit two opposing phenomena: stable/predictable and historic/unpredictable behavior. In the latter case, the time averages of the orbit will slowly oscillate during the evolution of the system and do not converge to any limit. This will eventually cause the divergence of higher-order repeated time averages.