Josef Hofbauer

Evolutionary Games and Population Dynamics: Contents

Every form of behavior is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realized how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centered not on the concept of rational players but on the population dynamics of behavioral programs. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behavior, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms. Comprehensive treatment of ecological and game theoretic dynamics Invasion dynamics and permanence as key concepts Explanation in terms of games of things like competition between species

Evolutionary Games and Population Dynamics: Dynamical Systems and Lotka–Volterra Equations

Every form of behavior is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realized how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centered not on the concept of rational players but on the population dynamics of behavioral programs. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behavior, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms. Comprehensive treatment of ecological and game theoretic dynamics Invasion dynamics and permanence as key concepts Explanation in terms of games of things like competition between species

q-Catalan numbers

Abstract q-analogs of the Catalan numbers C n = ( 1 (n + 1) )( n 2n ) are studied from the view-point of Lagrange inversion. The first, due to Carlitz, corresponds to the Andrews-Gessel-Garsia q-Lagrange inversion theory, satisfies a nice recurrence relation and counts inversions of Catalan words. The second, tracing back to Mac Mahon, arise from Krattenthalers and Gessel and Stantons q-Lagrange inversion formula, have a nice explicit formula and enumerate the major index. Finally a joint generalization is given which includes also the Polya-Gessel q-Catalan numbers.

Adaptive dynamics and evolutionary stability

A dynamics for frequency dependent selection is proposed and applied to several biological examples. The relation with game dynamics and evolutionary stability is analyzed.

Stochastic Approximations and Differential Inclusions

The dynamical systems approach to stochastic approximation is generalized to the case where the mean differential equation is replaced by a differential inclusion. The limit set theorem of Benaim and Hirsch is extended to this situation. Internally chain transitive sets and attractors are studied in detail for set-valued dynamical systems. Applications to game theory are given, in particular to Blackwells approachability theorem and the convergence of fictitious play.

Coexistence for systems governed by difference equations of Lotka-Volterra type.

The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.

Uniform persistence and repellors for maps

We establish conditions for an isolated invariant set M of a map to be a repellor. The conditions are first formulated in terms of the stable set of M . They are then refined in two ways by considering (i) a Morse decomposition for M, and (ii) the invariantly connected components of the chain recurrent set of M . These results generalize and unify earlier persistence results.

A general cooperation theorem for hypercycles

We derive a condition for a closed invariant subset of a compact dynamical system to be an attractor (resp. repellor) combining the usual Ljapunov function methods with time averages. Applications are given to concrete systems endowed with some cyclic symmetry. In particular, cooperation of the inhomogeneous hypercycle is shown.

On the occurrence of limit cycles in the Volterra-Lotka equation

defined on [w; = {X = (xi, . . . ,x,) E [w”: Xi 3 0 for all i}. It is a classical result (see [l, p, 2131 or [2, p. 3001) that for n = 2 isolated periodic orbits are not possible. We will show that Hopf bifurcations and hence stable limit cycles occur for dimensions n 2 3. This will be done by showing in Section 2 that (1.1) is equivalent to a certain differential equation on the simplex &+i, the “replicator equation”

Evolution in games with randomly disturbed payoffs

We consider a simple model of stochastic evolution in population games. In our model, each agent occasionally receives opportunities to update his choice of strategy. When such an opportunity arises, the agent selects a strategy that is currently optimal, but only after his payoffs have been randomly perturbed. We prove that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games. We conclude by contrasting the evolutionary process studied here with stochastic fictitious play.