Evolutionary Poisson Games for Controlling Large Population Behaviors

Emerging applications in engineering such as crowd-sourcing and (mis)information propagation involve a large population of heterogeneous users or agents in a complex network who strategically make dynamic decisions. In this work, we establish an evolutionary Poisson game framework to capture the random, dynamic and heterogeneous interactions of agents in a holistic fashion, and design mechanisms to control their behaviors to achieve a system-wide objective. We use the antivirus protection challenge in cyber security to motivate the framework, where each user in the network can choose whether or not to adopt the software. We introduce the notion of evolutionary Poisson stable equilibrium for the game, and show its existence and uniqueness. Online algorithms are developed using the techniques of stochastic approximation coupled with the population dynamics, and they are shown to converge to the optimal solution of the controller problem. Numerical examples are used to illustrate and corroborate our results

Complementarities and Macroeconomics: Poisson Games

In many situations in macroeconomics strategic complementarities arise, and agents face a coordination problem. An important issue, from both a theoretical and a policy perspective, is equilibrium uniqueness. We contribute to this literature by focusing on the macroeconomic aspect of the problem: the number of potential innovators, speculators e.t.c. is large. In particular, we follow Myerson (1998, 2000) that in large games “a more realistic model should admit some uncertainty about the number of players in the game”. In more detail, we model the coordination problem as a Poisson game, and investigate the conditions under which unique equilibrium selection is obtained.Strategic Complementarities, Coordination Games, Poisson Games, Currency Crises, Innovation.

A study of Approval voting on Large Poisson Games

Approval voting features are analysed in a context of large elections with strategic voters: Myerson's Large Poisson Games. I first establish the Magnitude Equiva- lence Theorem (MET) which substantially reduces the complexity of computing the magnitudes of pivotal events. I also show that the Winner of the election coincides with the Profile Condorcet Winner at equilibrium when preferences are restricted to be single-peaked. This is a positive result that strengthens the positive conclusions some scholars have previously drawn over this voting rule. I finally show that, with- out the previous restriction over preferences, both concepts do not generally coincide anymore.

Population Uncertainty and Poisson Games

A general class of models is developed for analyzing games with population uncertainty. Within this general class, a special class of Poisson games is defined. It is shown that Poisson games are uniquely characterized by properties of independent actions and enviornmental equivalence. The general definition of equilibrium for games with population uncertainty is formulated, and it is shown that the equilibria of Poisson games are invariant under payoff-irrelevant type splitting. An example of a large voting game is discussed, to illustrate the advantages of using a Poisson game model for large games.

Convergence of Large Atomic Congestion Games

We consider the question of whether, and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider a sequence of games with an increasing number of players where each player's weight tends to zero. We prove that all (mixed) Nash equilibria of the finite games converge to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the second setting, we consider again an increasing number of players but now each player has a unit weight and participates in the game with a probability tending to zero. In this case, the Nash equilibria converge to the set of Wardrop equilibria of a different nonatomic game with suitably defined costs. The latter can also be seen as a Poisson game in the sense of Myerson (1998), establishing a precise connection between the Wardrop model and the empirical flows observed in real traffic networks that exhibit stochastic fluctuations well described by Poisson distributions. In both settings we give explicit upper bounds on the rates of convergence, from which we also derive the convergence of the price of anarchy. Beyond the case of congestion games, we establish a general result on the convergence of large games with random players towards Poisson games.Comment: 34 pages, 3 figure

A cognitive hierarchy theory of one-shot games: Some preliminary results

Strategic thinking, best-response, and mutual consistency (equilibrium) are three key modelling principles in noncooperative game theory. This paper relaxes mutual consistency to predict how players are likely to behave in in one-shot games before they can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model to predict behavior in one-shot games, and initial conditions in repeated games. The CH approach assumes that players use k steps of reasoning with frequency f (k). Zero-step players randomize. Players using k (≥ 1) steps best respond given partially rational expectations about what players doing 0 through k - 1 steps actually choose. A simple axiom which expresses the intuition that steps of thinking are increasingly constrained by working memory, implies that f (k) has a Poisson distribution (characterized by a mean number of thinking steps τ ). The CH model converges to dominance-solvable equilibria when τ is large, predicts monotonic entry in binary entry games for τ < 1:25, and predicts effects of group size which are not predicted by Nash equilibrium. Best-fitting values of τ have an interquartile range of (.98,2.40) and a median of 1.65 across 80 experimental samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable p-beauty contests. The CH model also has economic value because subjects would have raised their earnings substantially if they had best-responded to model forecasts instead of making the choices they did

Monotone methods for equilibrium selection under perfect foresight dynamics

This paper studies equilibrium selection in supermodular games based on perfect foresight dynamics. A normal form game is played repeatedly in a large society of rational agents. There are frictions: opportunities to revise actions follow independent Poisson processes. Each agent forms his belief about the future evolution of action distribution in the society to take an action that maximizes his expected discounted payo�. A perfect foresight path is de�ned to be a feasible path of the action distribution along which every agent with a revision opportunity takes a best response to this path itself. A Nash equilibrium is said to be absorbing if there exists no perfect foresight path escaping from a neighborhood of this equilibrium; a Nash equilibrium is said to be globally accessible if for each initial distribution, there exists a perfect foresight path converging to this equilibrium. By exploiting the monotone structure of the dynamics, a unique Nash equilibrium that is absorbing and globally accessible for any small degree of friction is identi�ed for certain classes of supermodular games. For games with monotone potentials, the selection of the monotone potential maximizer is obtained. Complete characterizations of absorbing equilibrium and globally accessible equilibrium are given for binary supermodular games. An example demonstrates that unanimity games may have multiple globally accessible equilibria for a small friction

A cognitive hierarchy model of games

Players in a game are “in equilibrium” if they are rational, and accurately predict other players' strategies. In many experiments, however, players are not in equilibrium. An alternative is “cognitive hierarchy” (CH) theory, where each player assumes that his strategy is the most sophisticated. The CH model has inductively defined strategic categories: step 0 players randomize; and step k thinkers best-respond, assuming that other players are distributed over step 0 through step k − 1. This model fits empirical data, and explains why equilibrium theory predicts behavior well in some games and poorly in others. An average of 1.5 steps fits data from many games

Computing Equilibria in Anonymous Games

We present efficient approximation algorithms for finding Nash equilibria in anonymous games, that is, games in which the players utilities, though different, do not differentiate between other players. Our results pertain to such games with many players but few strategies. We show that any such game has an approximate pure Nash equilibrium, computable in polynomial time, with approximation O(s^2 L), where s is the number of strategies and L is the Lipschitz constant of the utilities. Finally, we show that there is a PTAS for finding an epsilo