Strict Equilibria Research Articles

Ordinal imitative dynamics

This paper introduces an evolutionary dynamics based on imitate the better realization (IBR) rule. Under this rule, an agent in a population game imitates the strategy of a randomly chosen opponent whenever the opponent’s realized payoff is higher than his or her own. Such behavior generates a mean dynamics which depends on the order of payoffs, but not their magnitudes, and is polynomial in strategy utilization frequencies. We demonstrate that while the dynamics does not possess Nash stationarity or payoff monotonicity, under it pure strategies iteratively strictly dominated by pure strategies are eliminated and strict equilibria are locally stable. We investigate the relationship between the dynamics based on the IBR rule and the replicator dynamics. In trivial cases, the two dynamics are topologically equivalent. In Rock-Paper-Scissors games we conjecture that both dynamics exhibit the same types of behavior, but the partitions of the game set do not coincide. In other cases, the IBR dynamics exhibits behaviors that are impossible under the replicator dynamics.

Tournament-stable equilibria

Building on Arad and Rubinstein (2013), we introduce tournaments as simultaneous n-player games based on an m-player game g. A player meets each group of m−1 opponents m! times to play g in alternating roles. The winner of the tournament is the player who attains the highest accumulated score. We explore the relationship between the equilibria of the tournament and the equilibria of the game g and confirm that tournaments provide a refinement criterion. We compare it with standard refinements in the literature and show that it is satisfied by strict equilibria. We use our tournament model to study a selection of relevant economic applications, including risk-taking behavior.

Demand Adjustment in Coalitional Games

This paper associates a strategic n-person game with a given transferable utility game and studies its Nash equilibria. Strict equilibria in this model characterize those divisions of social surplus that can become conventions in the sense of Young (1993). It is shown that even in relatively simple games various inefficiencies can arise.

Double Round-Robin Tournaments

A tournament is a simultaneous n-player game that is built on a two-player game g. We generalize Arad and Rubinstein’s model assuming that every player meets each of his opponents twice to play a (possibly) asymmetric game g in alternating roles (using sports terminology, once at home and once away). The winner of the tournament is the player who attains the highest total score, which is given by the sum of the payoffs that he gets in all the matches he plays. We explore the relationship between the equilibria of the tournament and the equilibria of the game g. We prove that limit points of equilibria of tournaments as the number of players goes to infinity are equilibria of g. Such a refinement criterion is satisfied by strict equilibria. Being able to analyze arbitrary two-player games allows us to study meaningful economic applications that are not symmetric, such as the ultimatum game.

Imitation dynamics with payoff shocks

We investigate the impact of payoff shocks on the evolution of large populations of myopic players that employ simple strategy revision protocols such as the “imitation of success”. In the noiseless case, this process is governed by the standard (deterministic) replicator dynamics; in the presence of noise however, the induced stochastic dynamics are different from previous versions of the stochastic replicator dynamics (such as the aggregate-shocks model of Fudenberg and Harris in J Econ Theory 57(2):420–441, 1992). In this context, we show that strict equilibria are always stochastically asymptotically stable, irrespective of the magnitude of the shocks; on the other hand, in the high-noise regime, non-equilibrium states may also become stochastically asymptotically stable and dominated strategies may survive in perpetuity (they become extinct if the noise is low). Such behavior is eliminated if players are less myopic and revise their strategies based on their cumulative payoffs. In this case, we obtain a second order stochastic dynamical system where non-equilibrium states are no longer attracting and dominated strategies become extinct (a.s.), no matter the noise level.

Social activity and network formation

This paper develops a simple model in which a social hierarchy emerges endogenously when agents form a network for complementary interaction (“activity”). Specifically, we assume that agents are ex ante identical and their best response activity, as well as their value function, increases (strictly) concavely in the total activity of their neighbors in the network. There exists a unique and stable positive activity equilibrium on exogenous networks under mild conditions. When we endogenize network formation, equilibria become strongly structured: more active players have more neighbors, i.e., a higher degree, but tend to sponsor fewer links. Additionally, in strict equilibria, agents separate themselves into groups characterized by the symmetric activity of their members. The characteristic activity decreases in group size and the network is a complete multipartite graph.

Mean field games via controlled martingale problems: Existence of Markovian equilibria

Mean field games are studied in the framework of controlled martingale problems, and general existence theorems are proven in which the equilibrium control is Markovian. The framework is flexible enough to include degenerate volatility, which may depend on both the control and the mean field. The objectives need not be strictly convex, and the mean field interactions considered are nonlocal and Wasserstein-continuous. When the volatility is nondegenerate, continuity assumptions may be weakened considerably. The proofs first use relaxed controls to establish existence. Then, using a convexity assumption and measurable selection arguments, strict (non-relaxed) Markovian equilibria are constructed from relaxed equilibria.

Inertial Game Dynamics and Applications to Constrained Optimization

Aiming to provide a new class of game dynamics with good long-term rationality properties, we derive a second-order inertial system that builds on the widely studied "heavy ball with friction" optimization method. By exploiting a well-known link between the replicator dynamics and the Shahshahani geometry on the space of mixed strategies, the dynamics are stated in a Riemannian geometric framework where trajectories are accelerated by the players' unilateral payoff gradients and they slow down near Nash equilibria. Surprisingly (and in stark contrast to another second-order variant of the replicator dynamics), the inertial replicator dynamics are not well-posed; on the other hand, it is possible to obtain a well-posed system by endowing the mixed strategy space with a different Hessian-Riemannian (HR) metric structure, and we characterize those HR geometries that do so. In the single-agent version of the dynamics (corresponding to constrained optimization over simplex-like objects), we show that regular maximum points of smooth functions attract all nearby solution orbits with low initial speed. More generally, we establish an inertial variant of the so-called "folk theorem" of evolutionary game theory and we show that strict equilibria are attracting in asymmetric (multi-population) games - provided of course that the dynamics are well-posed. A similar asymptotic stability result is obtained for evolutionarily stable strategies in symmetric (single- population) games.

Robustness of public equilibria in repeated games with private monitoring

A repeated game with private monitoring is “close” to a repeated game with public monitoring (or perfect monitoring) when (i) the expected payoff structures are close and (ii) the informational structures are close in the sense that private signals in the private monitoring game can be aggregated by some public coordination device to generate a public signal whose distribution is close to the distribution of the public signal in the public monitoring game. We provide a sufficient condition for the set of uniformly strict perfect public equilibria for a public monitoring game to be robust in nearby private monitoring games in the sense that they remain equilibria with respect to the public signal that is generated by such public coordination devices with truthful reporting. Our sufficient condition requires that every player is informationally small in a well-defined sense.

Strict equilibria interchangeability in multi-player zero-sum games

The interchangeability property of Nash equilibria in two-player zerosum games is well-known. This paper studies possible generalizations of this property to multi-party zero-sum games. A form of interchangeability property for strict Nash equilibria in such games is established. It is also shown, by proving a completeness theorem, that strict Nash equilibria do not satisfy any other non-trivial properties.

Dynamics of human decisions

We study a dichotomous decision model, where individuals can make the decision yes or no and can influence the decisions of others.We characterize all decisions that form Nash equilibria.Taking into account the way individuals influence the decisions of others, we construct the decision tilings where the axes reflect the personal preferences of the individuals for making the decision yes or no. These tilings characterize geometrically all the pure and mixed Nash equilibria.We show, in these tilings, that Nash equilibria form degenerated hysteresis with respect to the dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hysteresis can help to explain the sudden appearance of social, political and economic crises.We observe the existence of limit cycles for the dynamics associated to situations where the individuals keep changing their decisions along time, but exhibiting a periodic repetition in their decisions.We introduce the notion of altruist and individualist leaders and study the way that the leader can affect the individuals to make the decision that the leader pretends.

Individually selecting among conventions - an evolutionary and experimental analysis

Conventions can be narrowly interpreted as coordinated ways of equilibrium play, telling all players which of possibly several equilibria to play or more broadly how to choose in a game without imposing the equilibrium property. Since coordination often takes place before learning about the game, one has to coordinate on a prescribing principle. For the subclass of 2×2-bimatrix games with two strict equilibria, we analyze the evolutionary stability of various such principles. In our experiment, we allow participants to coordinate on principles before playing various games. Based on between-subjects treatments, participants do so being completely (they know neither their role nor the game parameters), partially (they know either their role or the game parameters) ignorant, or with no veil of ignorance (they know their role and the game parameters).

Folk Theorems with Bounded Recall under (Almost) Perfect Monitoring, Second Version

A strategy profile in a repeated game has bounded recall L if play under the profile after two distinct histories that agree in the last L periods is equal. Mailath and Morris (2002, 2006) proved that any strict equilibrium in bounded-recall strategies of a game with full support public monitoring is robust to all perturbations of the monitoring structure towards private monitoring (the case of almost-public monitoring), while strict equilibria in unbounded-recall strategies are typically not robust. We prove the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. As a consequence, the perfect monitoring folk theorem is shown to be behaviorally robust under almost-perfect almost-public monitoring. That is, the same specification of behavior continues to be an equilibrium when the monitoring is perturbed from perfect to highly-correlated private.

Folk Theorems with Bounded Recall Under (Almost) Perfect Monitoring

A strategy profile in a repeated game has L bounded recall if play under the profile after two distinct histories that agree in the last L periods is equal. Mailath and Morris (2002, 2006) proved that any strict equilibrium in bounded-recall strategies of a game with full support public monitoring is robust to all perturbations of the monitoring structure towards private monitoring (the case of monitoring), while strict equilibria in unbounded-recall strategies are typically not robust. We prove that the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict. The general result uses calendar time in an integral way in the construction of the strategy profile. If the players' action spaces are sufficiently rich, then the strategy profile can be chosen to be independent of calendar time. Either result can then be used to prove a folk theorem for repeated games with almost-perfect almost-public monitoring.

Evolution of Preferences

We endogenize preferences using the “indirect evolutionary approach”. Individuals are randomly matched to play a two-person game. Individual (subjective) preferences determine their behaviour and may differ from the actual (objective) pay-offs that determine fitness. Matched individuals may observe the opponents' preferences perfectly, not at all, or with some in-between probability. When preferences are observable, a stable outcome must be efficient. When they are not observable, a stable outcome must be a Nash equilibrium and all strict equilibria are stable. We show that, for pure-strategy outcomes, these conclusions are robust to allowing almost perfect, and almost no, observability, with the notable exception that inefficient strict equilibria may fail to be stable with any arbitrarily small degree of observability (despite being stable with no observability).

Evolution of Preferences1

We endogenize preferences using the “indirect evolutionary approach”. Individuals are randomly matched to play a two-person game. Individual (subjective) preferences determine their behaviour and may differ from the actual (objective) pay-offs that determine fitness. Matched individuals may observe the opponents’ preferences perfectly, not at all, or with some in-between probability. When preferences are observable, a stable outcome must be efficient. When they are not observable, a stable outcome must be a Nash equilibrium and all strict equilibria are stable. We show that, for pure-strategy outcomes, these conclusions are robust to allowing almost perfect, and almost no, observability, with the notable exception that inefficient strict equilibria may fail to be stable with any arbitrarily small degree of observability (despite being stable with no observability).

On the Nash bargaining solution with noise

Suppose two parties have to share a surplus of random size. Each of the two can either commit to a demand prior to the realization of the surplus – as in the Nash demand game – or remain silent and wait until the surplus was observed. Adding the strategy to wait results in two strict equilibria, in each of which one player takes almost the whole surplus, provided uncertainty is small. If commitments concern only who makes the first offer, the more balanced Nash bargaining solution is approximately restored. In all cases, commitment occurs in equilibrium – despite the risk of breakdown of negotiations.

Rule Evolution and Equilibrium Selection

This paper studies rule evolution and its effect on selection between strict equilibria. Two rules are investigated; the myopic best response and naive imitation. If agents cannot change rules, equilibrium selection is determined by the relative frequency of agents using the two rules. When agents can only change rules through mutation, the efficient equilibrium weakly dominates the risk-dominant one. If agents can change rules by updating, then the efficient equilibrium strictly dominates the risk-dominant one. Journal of Economic Literature Classification Numbers: C72, C73, D81.

Evolutionary dynamics on infinite strategy spaces

Summary. The study of evolutionary dynamics was so far mainly restricted to finite strategy spaces. In this paper we show that this unsatisfying restriction is unnecessary. We specify a simple condition under which the continuous time replicator dynamics are well defined for the case of infinite strategy spaces. Furthermore, we provide new conditions for the stability of rest points and show that even strict equilibria may be unstable. Finally, we apply this general theory to a number of applications like the Nash demand game, the War of Attrition, linear-quadratic games, the harvest preemption game, and games with mixed strategies.

The Robustness of Equilibria to Incomplete Information

A number of papers have shown that a strict Nash equilibrium action profile of a game may never be played if there is a small amount of incomplete information (see, for example, Carlsson and van Damme (1993a)). We present a general approach to analyzing the robustness of equilibria to a small amount of incomplete information. A Nash equilibrium of a complete information game is said to be robust to incomplete information if every incomplete information game with payoffs almost always given by the complete information game has an equilibrium which generates behavior close to the Nash equilibrium. We show that many games with strict equilibria have no robust equilibrium and examine why we get such different results from existing refinements. If a game has a unique correlated equilibrium, it is robust. A natural many-player many-action generalization of risk dominance is a sufficient condition for robustness.