Game Theory and Animal Behaviour

Edited by:
  • Lee Alan Dugatkin &
  • Hudson Kern Reeve
Oxford University Press: 1998. 320pp £55, $78.

Evolutionary Games and Population Dynamics

  • Josef Hofbauer &
  • Karl Sigmund
Cambridge University Press: 1993 323 pp. £50, $69.95 (hbk); £16.95, $27.95 (pbk)

Mathematical models of all sorts have contributed deeply to our understanding of biology. These two volumes deal with game theory, which is the branch of optimization models (formalized by von Neuman and Morgenstern in 1944) in which the behaviour of an individual depends on the frequency of certain actions of other individuals in the population. Although it was originally envisioned as an economic theory, John Maynard Smith recognized in the early 1970s that formal game theory had much to offer to biology, particularly because some of the concepts that had only vague definition in economics, such as utility, had very clear definitions and implications in biology, such as fitness. The ideas of Maynard Smith, based on the notion of an evolutionarily stable strategy (roughly, a strategy that, when adopted by all the members of a population, prevents the invasion through natural selection of any alternative strategy), form the foundation of evolutionary game theory.

Mathematical applications in biology are as diverse as biology itself, ranging from knot theory through partial differential equations to optimization methods. Donald Ludwig once wrote, “the misleading term ‘Mathematical Biology’ suggests that some part of Biology is being taken over by Mathematics. Biological Mathematics would be a better term. Linear thinking and methods still dominate the applications of mathematics. Biological requirements deflect us from that easy path”. Both books leave the easy path behind.

Dugatkin and Reeve's edited collection explores various mathematical methods and models in evolutionary game theory, including the Prisoner's Dilemma, the Hawk-Dove game, the Sequential Assessment game, and the War of Attrition Model. But it also describes many interesting examples of real-life animal behaviour, including games between foraging producers and scroungers, reciprocal grooming in impala, territorial defence by birds and spiders, animal communication, parent-offspring conflict, and colony founding by ants. There are many accounts of experimental tests of game theory models, along with clear discussions of the limitations of the game theory approach.

The quality of writing (often a problem in edited volumes) is uniformly good. The chapter by R. Gomulkiewicz is especially important, because it connects game theory, other optimization methods, and quantitative genetics with a focus on an empirical strategy for detecting adaptation and constraint. These different mathematical methods are often described in opposition to each other, when they should be viewed as different ways of seeing the same problem.

The starting point of Hofbauer and Sigmund's book is the replicator equation, which recognizes that the rate of change of a certain type in a population depends on its frequency and its expected reproduction, relative to the other types in the population. Many mathematical results from the past decade are included, so the book is up to date and could be used for a course in mathematical aspects of theoretical ecology. The book has two introductions: one for game theorists and one for biologists. Even so, most biologists will find it hard going. The first part could be used in an advanced course on population dynamics or ecological modelling; but even here there is some sophisticated mathematics and heavy-duty notation. For example, the fourth chapter begins with the Poincaré-Bendixson theorem and continues to Hopf's bifurcation theorem; these are elegant, important and beautiful theorems, but difficult.

In the middle two sections of the book, the authors provide an elegant tour of evolutionary game theory and its connections to the classical equations of population dynamics. They provide a few specific examples, such as the rock-scissors-paper game and the repeated Prisoner's Dilemma, which may help bring the mathematics into clearer focus.

The last part of the book connects dynamical systems and frequency dependent genetics, playing much the same role as the article by Gomulkiewicz in showing the connections between dynamical systems and genetics.

Reading these two books together is interesting because it focuses the difference between theoretical biology (Dugatkin and Reeve) and mathematical biology (Hofbauer and Sigmund) in much the same way that theoretical and mathematical physics differ. Together, they have something for anyone interested in game models in organismal biology.