The projection dynamic and the replicator dynamic

We investigate a variety of connections between the projection dynamic and the replicator dynamic. At interior population states, the standard microfoundations for the replicator dynamic can be converted into foundations for the projection dynamic by replacing imitation of opponents with “revision driven by insecurity” and direct choice of alternative strategies. Both dynamics satisfy a condition called inflow–outflow symmetry, which causes them to select against strictly dominated strategies at interior states; still, because it is discontinuous at the boundary of the state space, the projection dynamic allows strictly dominated strategies to survive in perpetuity. The two dynamics exhibit qualitatively similar behavior in strictly stable and null stable games. Finally, the projection and replicator dynamics both can be viewed as gradient systems in potential games, the latter after an appropriate transformation of the state space.     

The projection dynamic and the replicator dynamic

We investigate a variety of connections between the projection dynamic and the replicator dynamic. At interior population states, the standard microfoundations for the replicator dynamic can be converted into foundations for the projection dynamic by replacing imitation of opponents with “revision driven by insecurity” and direct choice of alternative strategies. Both dynamics satisfy a condition called inflow–outflow symmetry, which causes them to select against strictly dominated strategies at interior states; still, because it is discontinuous at the boundary of the state space, the projection dynamic allows strictly dominated strategies to survive in perpetuity. The two dynamics exhibit qualitatively similar behavior in strictly stable and null stable games. Finally, the projection and replicator dynamics both can be viewed as gradient systems in potential games, the latter after an appropriate transformation of the state space.     

Excess payoff dynamics and other well-behaved evolutionary dynamics

We consider a model of evolution in games in which agents occasionally receive opportunities to switch strategies, choosing between them using a probabilistic rule. Both the rate at which revision opportunities arrive and the probabilities with which each strategy is chosen are functions of current normalized payoffs. We call the aggregate dynamics induced by this model excess payoff dynamics. We show that every excess payoff dynamic is well-behaved: regardless of the underlying game, each excess payoff dynamic admits unique solution trajectories that vary continuously with the initial state, identifies rest points with Nash equilibria, and respects a basic payoff monotonicity property. We show how excess payoff dynamics can be used to construct well-behaved modifications of imitative dynamics, and relate them to two other well-behaved dynamics based on projections.     

Brown–von Neumann–Nash dynamics: The continuous strategy case

Brown and von Neumann introduced a dynamical system that converges to saddle points of zero sum games with finitely many strategies. Nash used the mapping underlying these dynamics to prove existence of equilibria in general games. The resulting Brown–von Neumann–Nash dynamics are a benchmark example for myopic adjustment dynamics that, in contrast to replicator dynamics, allow for innovation, but require less rationality than the best response dynamics. This paper studies the BNN dynamics for games with infinitely many strategies. We establish Nash stationarity for continuous payoff functions. For negative semidefinite games (that include zero sum games), we generalize the results of Brown and von Neumann. In addition, we show that evolutionarily robust Nash equilibria are asymptotically stable. A complete stability analysis for doubly symmetric games is also obtained.     

Factors influencing the adoption of the internet of things in supply chains

This study investigates the application of the Internet of Things (IoT) in supply chains through evolutionary game theory. In the modeling process, the payoff matrices for the game between the core enterprises and the suppliers and the replicator dynamics equations for these two parties are constructed. The punishment mechanism is introduced to prevent the “free-ride” behavior of core enterprises. As the key technology of the IoT, the radio-frequency identification (RFID) implementation strategies of both parties are analyzed on the basis of the replicator dynamics. Both parties eventually achieve evolutionarily stable strategies by adopting the strategies of other participants. Furthermore, a sensitivity analysis is conducted to validate the effects of the parameters on the selection of RFID technology. Results show that the willingness of core enterprises and suppliers to implement RFID can be strengthened by adjusting relevant parameters. Thus, this study could contribute in generalizing the application of the IoT in supply chains.     

Does Neutral Stability Imply Lyapunov Stability?

A central concept in evolutionary game theory is that of an evolutionarily stable strategy. However, many games of interest for economics have only strategies which meet the weaker requirement of neutral stability. While neutral stability implies Lyapunov stability under the replicator dynamics in pairwise interactions, we give here an example in which an evolutionarily stable strategy is not Lyapunov stable in a more general context. We propose to use instead the notions of (strong) uninvadability and (strong) unbeatability, refinements of evolutionary and neutral stability, respectively. Strong uninvadability implies asymptotic stability, and strong unbeatability implies Lyapunov stability under the replicator dynamics. Journal of Economic Literature Classification Number: C72.     

The replicator dynamics does not lead to correlated equilibria

It is shown that, under the replicator dynamics, all strategies played in correlated equilibrium may be eliminated, so that only strategies with zero marginal probability in all correlated equilibria survive. This occurs in particular in a family of 4×4 games built by adding a strategy to a Rock-Paper-Scissors game.     

Dynamically stable sets in infinite strategy spaces

Evolutionary game theory has largely focused on finite games. Dynamic stability is harder to attain in infinite strategy spaces; Bomze [Bomze, I., 1990. Dynamical aspects of evolutionary stability. Monatsh. Math. 110, 189–206] and Oechssler and Riedel [Oechssler, J., Riedel, F., 2001. Evolutionary dynamics on infinite strategy spaces. Econ. Theory 17, 141–162] provide conditions for the stability of rest points under the replicator dynamics. Here, conditions are given for the stability of sets of strategies under this process.     

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. Received: June 25, 1999; revised version: January 31, 2000     

Evolution of harvesting strategies: replicator and resource dynamics

Abstract. Economic theories of managing renewable resources, such as fisheries and forestry, traditionally assume that individual harvesters are perfectly rational and thus able to compute the harvesting strategy that maximizes their discounted profits. The current paper presents an alternative approach based on bounded rationality and evolutionary mechanisms. It is assumed that individual harvesters face a choice between two harvesting strategies. The evolution of the distribution of strategies in the population is modeled through a replicator dynamics equation. The latter captures the idea that strategies yielding above average profits are demanded more than strategies yielding below average profits, so that the first type ends up accounting for a larger part in the population. From a mathematical perspective, the combination of resource and evolutionary processes leads to complex dynamics. The paper presents the existence and stability conditions for each steady-state of the system and analyzes dynamic paths to the equilibrium. In addition, effects of changes in prices are analyzed. A main result of the paper is that under certain conditions both strategies can survive in the long-run. Correspondence to: J. Noailly     

Is limit pricing evolutionarily stable?

We have analyzed the question whether or not limit-pricing behavior can be viewed as a strategy, which survives the replicator dynamics in a game in which firms change strategies through imitation. Our results show that, regardless of the proportion of low cost incumbents in the population, potential entrants that play the strategy "enter" when they observe limit output are wiped out. The unique ESS consists of all low cost incumbents playing the "limit output" strategy, and all high cost incumbents playing the "monopoly output". This ESS includes potential entrants playing "stay out" after observing the limit output, and playing "enter" if monopoly output is observed in the first stage. Hence, limit pricing survives the replicator dynamics and appears in the stable state of the population.JEL Classification: B25, C73, L11We would like to thank an anonymous referee for valuable comments on an earlier version of this paper. Correspondence to: U. Soytas     

Stochastic Replicator Dynamics

This article studies the replicator dynamics in the presence of shocks. I show that under these dynamics, strategies that do not survive the iterated deletion of strictly dominated strategies are eliminated in the long run, even in the presence of nonvanishing perturbations. I also give an example that shows that the stochastic dynamics in this article have equilibrium selection properties that differ from other dynamics in the literature.     

Cooperation in the sporadically repeated prisoners�� dilemma via reputation mechanisms

We consider the sporadically repeated Prisoners’ Dilemma, examine a class of two-state reputation mechanisms and find that only three have efficient robust perfect Nash equilibria in Markov strategies. Since the asymptotic stability properties are crucial to their plausibility as realistic solutions, we examine replicator dynamics for the simplest reputation mechanism that entails both punishment and penitence. The strategy to cooperate with Good opponents and defect against Bad opponents is a global attractor of replicator dynamics, and hence cooperation is successfully sustained in the long run.     

Learning by replicator and best-response: the importance of being indifferent

This paper compares two learning processes, namely those generated by replicator and best-response dynamics, from the point of view of the asymptotics of play. We base our study on the intersection of the basins of attraction of locally stable pure Nash equilibria for replicator and best-response dynamics. Local stability implies that the basin of attraction has positive measure but there are examples where the intersection of the basin of attraction for replicator and best-response dynamics is arbitrarily small. We provide conditions, involving the existence of an unstable interior Nash equilibrium, for the basins of attraction of any locally stable pure Nash equilibrium under replicator and best-response dynamics to intersect in a set of positive measure. Hence, for any choice of initial conditions in sets of positive measure, if a pure Nash equilibrium is locally stable, the outcome of learning under either procedure coincides. We provide examples illustrating the above, including some for which the basins of attraction exactly coincide for both learning dynamics. We explore the role that indifference sets play in the coincidence of the basins of attraction of the stable Nash equilibria.     

Aggregative games and best-reply potentials

This paper introduces quasi-aggregative games and establishes conditions under which such games admit a best-reply potential. This implies existence of a pure strategy Nash equilibrium without any convexity or quasi-concavity assumptions. It also implies convergence of best-reply dynamics under some additional assumptions. Most of the existing literature’s aggregation concepts are special cases of quasi-aggregative games, and many new situations are allowed for. An example is payoff functions that depend on own strategies as well as a linear combination of the mean and the variance of players’ strategies.     

THE ENDOWMENT EFFECT AS BLESSING

We study the idea that seemingly unrelated behavioral biases can coevolve if they jointly compensate for the errors that any one of them would give rise to in isolation. We suggest that the “endowment effect” and the “winner's curse” could have jointly survived natural selection together. We develop a new family of “hybrid‐replicator” dynamics. Under such dynamics, biases survive in the population for a long period of time even if they only partially compensate for each other and despite the fact that the rational type's payoff is strictly larger than the payoffs of all other types.     

Evolutionary Game Theory with Two Groups of Individuals

Static stability conditions are developed for evolutionary games that involve two groups of individuals such as the evolution of frequencies in a two-species biological system. These conditions are related to a local version of the Pareto optimal solution concept. Solutions that satisfy these conditions are shown to be dynamically stable for the replicator dynamic. The theory is applied to a mate desertion model where the two groups (male and female) have two possible desertion times. In particular, for biologically reasonable fitness parameters, both sexes adopt mixed mean strategies where the sex that has the least to lose by deserting is the first, on average, to desert. Journal of Economic Literature Classification Numbers: C70, C72.     

Rationalizable foresight dynamics

This paper proposes and studies the rationalizable foresight dynamics. A normal form game is repeatedly played in a random matching fashion by a continuum of agents who make decisions at stochastic points in time. A rationalizable foresight path is a feasible path of action distribution along which each agent takes an action that maximizes his expected discounted payoff against another path which is in turn a rationalizable foresight path. We consider a set-valued stability concept under this dynamics and compare it with the corresponding concept under the perfect foresight dynamics.     

A Behavioral Learning Process in Games

This paper studies the cumulative proportional reinforcement (CPR) rule, according to which an agent plays, at each period, an action with a probability proportional to the cumulative utility that the agent has obtained with that action. The asymptotic properties of this learning process are examined for a decision-maker under risk, where it converges almost surely toward the expected utility maximizing action(s). The process is further considered in a two-player game; it converges with positive probability toward any strict pure Nash equilibrium and converges with zero probability toward some mixed equilibria (which are characterized). The CPR rule is compared in its principles with other reinforcement rules and with replicator dynamics. Journal of Economic Literature Classification Number: C72.     

虚拟平台中智慧城市信息转移演化博弈分析

在新一代信息技术支撑下的智慧城市中,构建信息化资源共享的虚拟平台成为其信息转移和共享的有效途径。因此文章分析了信息作为生产要素在农业时代、工业时代、信息时代、智慧时代的表现形式及转移虚拟平台构建的必要性。在信息传递主体有限理性的情况下,将演化博弈理论引入到信息转移过程中形成政府与信息主体的收益矩阵,并研究虚拟平台在官管民营模式下,信息转移过程中资源主体和政府的复制动态方程及稳定状态点现实解释。用复制动态相位图表示政府和信息资源主体在转移共享和激励行为比例变化的复制动态关系,探讨四种情况下稳定状态的收敛性与抗扰动性。最后得出结论为不管任何状态下的复制动态关系和稳定性,政府通过调整战略和信息平台中的收益分配比重,既不影响信息主体的积极性,同时又能提高其收益,使博弈双方主体均有能力和动力推动信息转移共享,从而达到帕累托上策均衡。