Some dynamics of a strategic market game with a large number of agents

This paper is designed to combine the game-theoretic investigation of the static or equilibrium properties of large strategic market games together with the investigation of some very simple dynamics, which nevertheless are sufficient to show differences between two related games, one with only trade and one in which both borrowing from an outside bank and trade take place. The role of banking reserves emerges as relevant and sensitive to the transient state dynamics.Several 100,000 player games are simulated and the behavior is compared with the analytical prediction for the games with a continuum of agents.The dynamics considered here is so simple that it does not show adaptive learning. A natural extension calls for updating via a learning program such as a genetic algorithm.     

Higher order game dynamics

Continuous-time game dynamics are typically first order systems where payoffs determine the growth rate of the players? strategy shares. In this paper, we investigate what happens beyond first order by viewing payoffs as higher order forces of change, specifying e.g. the acceleration of the players? evolution instead of its velocity (a viewpoint which emerges naturally when it comes to aggregating empirical data of past instances of play). To that end, we derive a wide class of higher order game dynamics, generalizing first order imitative dynamics, and, in particular, the replicator dynamics. We show that strictly dominated strategies become extinct in n-th order payoff-monotonic dynamics n   orders as fast as in the corresponding first order dynamics; furthermore, in stark contrast to first order, weakly dominated strategies also become extinct for n?2$n?2$. All in all, higher order payoff-monotonic dynamics lead to the elimination of weakly dominated strategies, followed by the iterated deletion of strictly dominated strategies, thus providing a dynamic justification of the well-known epistemic rationalizability process of Dekel and Fudenberg [7]. Finally, we also establish a higher order analogue of the folk theorem of evolutionary game theory, and we show that convergence to strict equilibria in n-th order dynamics is n orders as fast as in first order.     

Evolutionary game dynamics and distributed recency-weighted learning

This paper presents a relationship between evolutionary game dynamics and distributed recency-weighted Monte Carlo learning. After reviewing some existing theories of replicator dynamics and agent-based Monte Carlo learning, we provide proofs of the formulation-level equivalence between these two models. The relationship will be revealed not only from a theoretical viewpoint, but also by computational simulations of the models. As a consequence, macro dynamic patterns generated by distributed micro-decisions can be explained by parameters defined at an individual level. In particular, given the equivalent formulations, we investigate how the rate of agents’ recency weighting in learning affects the emergent evolutionary game dynamic patterns. An increase in this rate negatively affects the inertia, making the evolutionary stability condition more strict, and positively affecting the evolutionary speed toward equilibrium.JEL Classification: C63, C73Supervisions and advice given by Arthur J. Caplan have greatly contributed to this paper. I am also grateful to the anonymous reviewers for their valuable comments redirecting my presentation to a more appropriate one.     

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence of play (the period-by-period behavior as well as the long-run frequency) to Nash equilibria of the one-shot stage game, and present a number of possibility and impossibility results. Basically, we show that if in addition to random experimentation some recall, or memory, is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it suffices to recall the last two periods of play.     

Stochastic uncoupled dynamics and Nash equilibrium

In this paper we consider dynamic processes, in repeated games, that are subject to the natural informational restriction of uncoupledness. We study the almost sure convergence of play (the period-by-period behavior as well as the long-run frequency) to Nash equilibria of the one-shot stage game, and present a number of possibility and impossibility results. Basically, we show that if in addition to random experimentation some recall, or memory, is introduced, then successful search procedures that are uncoupled can be devised. In particular, to get almost sure convergence to pure Nash equilibria when these exist, it suffices to recall the last two periods of play.     

Stochastic better-reply dynamics in finite games

In this paper, we analyze a model where individuals from finite populations are repeatedly drawn to play a finite game and in every period choose a weakly better reply to a sample distribution from a finite history of past play. For all finite games and sufficiently incomplete information, we prove convergence to minimal sets closed under better replies. This result complements previous findings in a deterministic continuous-time framework and implies convergence to strict Nash equilibria in many well-known classes of games.     

Analysis on the dynamics of a Cournot investment game with bounded rationality

In this work, a dynamic system of investment game played by two firms with bounded rationality is proposed. It is assumed that each firm in any period makes a strategy for investment and uses local knowledge to make investment strategy according to the marginal profit observed in the previous period. Theoretic work is done on the existence of equilibrium solutions, the instability of the boundary equilibriums and the stability conditions of the interior equilibrium. Numerical simulations are used to provide experimented evidence for the complicated behaviors of the system evolution. It is observed that the equilibrium of the system can loose stability via flip bifurcation or Neimark–Sacher bifurcation and time-delayed feedback control can be used to stabilize the chaotic behaviors of the system.     

Bargaining with a possibly committed seller

We consider negotiations with an open time horizon where a buyer has private information about his valuation and does not know whether the seller is committed to the advertised price. This setting combines two common specifications made in the non-cooperative bargaining literature: one side is privately informed about its valuation, which is drawn from a continuum, and the other side is possibly committed to a fixed offer. We analyze the game both in discrete and in continuous time and show convergence of the two settings, which extends results from Abreu and Gul [2000. Bargaining and reputation. Econometrica 68, 85–117]. One interesting result is that as time proceeds, the non-committed seller becomes less likely to concede in a given period, i.e., it appears as if he becomes more “stubborn.” We further show that a seller may prefer to negotiate with a “worse” buyer as this enhances the value of his possible commitment.     

Infection and immunization: A new class of evolutionary game dynamics

Building upon a central paradigm of evolutionary game theory, namely the invasion barrier, we propose the new Infection and Immunization Dynamics (InfImmDyn), modelling a plausible adaptation process in a large population. For general games, this yields a novel refinement of the Nash equilibrium concept based on dynamical arguments, close in spirit to Nash's original “mass action” idea in his Ph.D. thesis. For partnership games, InfImmDyn exhibits a better asymptotic behavior compared to other popular procedures like Fictitious Play and Replicator Dynamics. We establish even support separation of InfImmDyn in finite time, which can never be achieved by any interior-point method like those mentioned above. In fact, this property has not yet been established for any other evolutionary game dynamics.     

Preemptive investment game with alternative projects

This paper investigates a duopoly with two alternative investment projects. We examine a situation in which a firm cannot invest in any project that has been taken by the rival firm. The first mover's advantage in project choice leads to an equilibrium quite different from that in previous studies. Specifically, we show that in equilibrium, the investment time and the option value are between those in a duopoly with one project and a monopoly with one project. Moreover, we show that a high correlation between project values, unlike in a monopoly with two projects, plays a positive role in mitigating preemptive competition. The results complement the literature of real options games and of max-options and entail new empirical implications.     

An investment game with third-party intervention

This paper explores the effect of the possibility of third-party intervention on behavior in a variant of the Berg et al. [Berg, J., Dickhaut, J., McCabe, K., 1995. Trust, reciprocity and social history. Games and Economic Behavior 10, 122–142] “Investment Game”. A third-party's material payoff is not affected by the decisions made by the other participants, but this person may choose to punish a responder who has been overly selfish. The concern over this possibility may serve to discipline potentially selfish responders. We also explore a treatment in which the third party may also choose to reward a sender who has received a low net payoff as a result of the responder's action. We find a strong and significant effect of third-party punishment in both punishment regimes, as the amount sent by the first mover is more than 60% higher when there is the possibility of third-party punishment. We also find that responders return a higher proportion of the amount sent to them when there is the possibility of punishment, with this proportion slightly higher when reward is not feasible. Finally, third parties punish less when reward is feasible, but nevertheless spend more on the combination of reward and punishment when these are both permitted than on punishment when this is the only choice for redressing material outcomes.     

A Blotto game with Incomplete Information

We consider a Blotto game with Incomplete Information. A pure-strategy symmetric monotonic Bayesian equilibrium is found and its properties are discussed.     

Stochastic games with information lag

Two-player zero-sum stochastic games with finite state and action spaces are known to have undiscounted values. We study such games under the assumption that one or both players observe the actions of their opponent after some time-dependent delay. We develop criteria for the rate of growth of the delay such that a player subject to such an information lag can still guarantee himself in the undiscounted game as much as he could have with perfect monitoring. We also demonstrate that the player in the Big Match with the absorbing action subject to information lags that grows too rapidly will not be able to guarantee as much as he could have in the game with perfect monitoring.     

Stochastic evolution with slow learning

Summary. This paper studies the extent to which diffusion approximations provide a reliable guide to equilibrium selection results in finite games. It is shown that they do for a class of finite games with weak learning provided that limits are taken in a certain order. The paper also shows that making mutation rates small does not in general select a unique equilibrium but making selection strong does. Received: January 19, 2000; revised version: September 25, 2000     

A public contracting evolutionary game with corruption

In this paper, firms are randomly matched from a continuous population to play a public contracting game (say, building a bridge). Price is given, so firms compete in quality; the higher the quality offer, the less the profit. According to the official rules of the contest, the firm bidding the highest quality wins; however, we consider the possibility that firms attempt to corrupt the public officer in charge in order to be sure to win even if bidding a lower quality (quality bids are not publicly observable). Whether or not firms attempt to corrupt depends on how rewarding corruption is w. r. t. being honest. Within a dynamic evolutionary framewark, we investigate how social conventions with or without corruption are established under various assumptions concerning the corruptability of the public officer and the possibility of an external monitoring of the officer's decisions by a super partes authority.     

The biodiversity conservation game with heterogeneous countries

Biodiversity is an essential resource, which we classify as conditionally-renewable. In order to achieve conservation and sustainable use of biodiversity virtually all nation states signed the United Nations Convention on Biological Diversity. In this paper we investigate how the heterogeneity of countries in regard to ecosystems and wealth influences the stability of international biodiversity conservation agreements both without and with transfers. We further examine the effect of different degrees of ecosystem substitutability. We model a coalition formation game with players that have a continuous conservation choice. The conservation benefit is dependent on wealth and ecosystem quality. Aggregation of global benefits respects differences in ecosystem substitutability. In case of transfers, a fund redistributes coalition benefits according to a sharing rule. The main finding is that in the absence of transfers, compared to the homogeneous situation, heterogeneity in ecosystems and wealth reduces the size of a stable coalition. The destabilising effect is stronger the higher the ecosystem substitutability. Optimal transfers facilitate a large stable coalition.     

A Blotto game with multi-dimensional incomplete information

We examine a multi-dimensional incomplete information Colonel Blotto game in which each player’s n-tuple of battlefield valuations is drawn from a common n-variate joint distribution function that is uniform on the non-negative orthant of the surface of a sphere.     

Stochastic Optimal Growth with Unbounded Shock

This paper considers a neoclassical optimal growth problem where the shock that perturbs the economy in each time period is potentially unbounded on the state space. Sufficient conditions for existence, uniqueness, and stability of equilibria are derived in terms of the primitives of the model using recent techniques from the field of perturbed dynamical systems. Journal of Economic Literature Classification Numbers: C61, C62, O41.     

Stochastic stability in networks with decay

This paper considers a stylized model of network formation in which relations among agents are subject to frictions, as described in Bala and Goyal [A non cooperative model of network formation, Econometrica 68 (2000) 1181-1231]. We analyze the process of network formation in a dynamic framework where self-interested individuals can form or delete links and, occasionally, make mistakes. Then, using stochastic stability, we identify the network structures to which the formation process will converge.