What Is an Evolutionary Game Theory ?

What Is an Evolutionary Game Theory ? 

by : Aria Ratmandanu 

       When Charles Darwin developed his theory of natural selection, he created a picture of the evolutionary process in which organismic adaptation was ultimately caused by competition for survival and reproduction. This biological "struggle for existence" bears considerable resemblance to the human struggle between businessmen who are striving for economic success in competitive markets. Long before Darwin published his work, social scientist Adam Smith had already considered that in business life, competition is the driving force behind economic efficiency and adaptation. It is indeed very striking how similar the ideas are on which the founders of modern theory in evolutionary biology and economics have based their main thoughts.

          Ideally, this similarity of ideas could have led to a permanent interchange between disciplines after Darwin (1859) wrote his The Origin of Species. It seems, however, as if the theories of evolution and of economics first needed to mature independently before such an interdisciplinary dialogue could become very fruitful. Biologists, on the one hand, had to explore the mechanisms of inheritance and to achieve their own synthesis of theories about phenotypic and genetic evolution. Economists, on the other hand, had to develop the mathematical backbone of their classical theory of competition. This backbone surely is the theory of games which came into existence with a famous book written by von Neumann and Morgenstern (1944). The intense dialogue between biology and economics started a few decades later. Both disciplines have since tried to systematically explore what their concepts have in common and how biologists and economists can share the effort of further theory development. The field of evolutionary game theory emerged as the major result of this exploration.

        What initiated the biological interest in game theory ? The important event was a change of paradigm regarding the level of aggregation (i.e., species, population, group, or individual) at which natural selection shows its strongest effects. Until the early 1960s, many biologists had held the view that the evolution of an organismic trait can be explained by identifying the trait's benefit to the species, or to other unitsabove the level of the individual. This view was then deeply shaken (Williams 1966, Maynard Smith 1976). It neither represented Darwin's original thoughts properly, nor did it stand up to scrutiny in the updated theory of evolution (but see Wilson and Dugatkin, both in this volume, for alternative views of this subject).

       We are now used to the idea that natural selection tends to act more effectively at the level of individuals than at higher levels of aggregated entities. Therefore, we have a strong inclination to look at natural selection "through the eyes" of the individuals that carry out the Darwinian struggle for existence. This helps us to understand the conceptual link between evolution and the theory of games. Similar to the theory of evolutionary adaptation, the latter theory is also deeply rooted in methodological individualism. After all, we expect businessmen to strive for their own success. Depending on the circumstances, this may or may not increase the well-being of society, very much like natural selection may have a positive or negative effect on the overall performance of animal groups or populations (Riechert & Hammerstein 1983). 

What is an evolutionary game ?

         A classical game is a model in economic decision theory describing the potential interactions of two or more individuals whose interests do not entirely coincide. The term "game" is chosen because whenever we specify such a model, this resembles the process of creating a new parlor game. We have to make precise (a) who is involved, (b) what are the possible actions, and (c) how individual success depends on the behavior of all participants. Obviously, even a biologist who is not dealing with decision theory, but with functional analysis of animal or plant interactions, needs exactly these three ingredients in order to describe the phenotypic scenario of competition. Therefore, the structure of a game arises naturally in evolutionary studies.

        As far as the mathematical representation of a biological game is concerned, the modeler has a choice of several forms. For example, a very explicit description of the phenotypic scenario would be given by a game in extensive form (Selten 1983, 1988). Roughly speaking, this is a mathematical decision tree, the branching points (nodes) of which correspond to the players' objective decision situations, and branches stand for the alternative actions that are possible in such a situation. A superimposed structure describes the possible information states of the players, and a strategy is a "list of behavioral instructions" for all the different information states (subjective situations) which may arise during a game. By "behavioral instruction" it is not necessarily meant that in a given situation a single alternative has to be used with probability 1. Therefore, an instruction can be to use several alternatives, each with positive probability. A strategy is called "pure" if none of its instructions are of the latter type. In other words, in a pure strategy no randomization of action takes place.

       How can one simplify a game in extensive form? A more condensed way of describing such a game is the so-called normal form (also referred to as the strategic form). In this description, pure strategies are named by numbers and their instructions are not made explicit in the model. The normal form only contains information about how strategies and payoffs relate to each other. Suppose that there is a finite set of alternative pure strategies and that organisms interact pairwise in a symmetric game. Symmetry means here that both "players" have the same set of strategies and that payoff depends only on strategies and not on the question of who is player 1 and player 2. A payoff matrix a = (aij) then describes what a player would receive if he plays his rth pure strategy against another player who plays his y'th pure strategy. The matrix, a, is all one needs in order to specify a symmetric game in normal form. Implicitly, however, there are more strategies than the ones that define rows and columns of this matrix. A general strategy .s is a probability distribution over a play- er's pure strategies. Let E (s, r) denote the expected payoff for playing such a strategy s against another player's strategy r. Biologically speaking, this function describes how an individual's expected fitness is changed according to his performance in the game.

          We are now entering the discussion of the dynamic context in which an evolu- tionary game is imbedded. In order to analyze any kind of a game, one needs a background theory about the process that generates behavior. This is the point where biology and classical economics differ dramatically. Theoreticians in classical economics rely on the process of rational decision making. They idealize the human brain as an apparatus with incredibly powerful cognitive skills and with the dedication to make the best use of them. In contrast, evolutionary biologists tend to invoke natural selection as the principal "decision maker." In their picture, individual behavior is governed by less potent mental procedures which are passed on from generation to generation via genetic inheritance. The biological theory of games is about the evolution of these procedures (strategies). In this approach, sophisticated behavioral adaptations of animals are thought to reflect the calculation power of the evolutionary process, rather than cognitive skills of the individual brain.

         The theory of the evolutionary game can be based on fairly different assumptions about the mode of inheritance, and its picture of genetics can be either more or less explicit. Initially, evolutionary game theory was considered to be a phenotypic approach to frequency-dependent selection in which genetics had to be approximated very crudely by the assumption of exact asexual inheritance. Let us have a brief look at such a selection model for a population with discrete nonoverlapping generations. Suppose that n different strategies s^,...,sn are initially present in the population. Let jc, denote the relative frequency of strategy st in the population, and let x= (A:,, . . . , xn) be the population frequency distribution of strategies. Suppose that the expected fitness wt of an individual "playing i" is frequency-dependent and that it can be defined as a function w,-(jc). Let w(x) denote the population mean fitness. Then, after one generation the new population state x' =(x',,..., x'n) is given by the fol- lowing difference equation, known as the discrete replicator equation with frequency- dependent fitness:

         In order to link this replicator equation with a phenotypic game, one has to be more explicit about the nature of the fitness function w,-(jc). Suppose that in every generation, animals interact pairwise in a game-like situation, that pair formation is random with respect to strategies, and that a payoff matrix describes how an individual's expected fitness is changed by the course of actions in the game. The fitness of strategy st can then be defined as

where £(s(,.j;) is the game payoff for playing strategy j; against strategy .v;., and w(x) is the basic fitness expectation an organism would have if it could avoid playing the game at all.

         The first model in evolutionary game theory (Maynard Smith & Price 1973) left it to the reader's intuition to imagine the dynamic context of the evolutionary game. However, it is obvious that either the discrete replicator equation (1) was what Maynard Smith and Price (1973) had in mind, or else a smoother version of this model (Taylor & Jonker 1978), in which the difference equation is replaced by a closely related differential equation (see also Hofbauer & Sigmund 1988). Many biologists feel uneasy with these equations, because they only describe phenotypic change without keeping track of the underlying genetics. Indeed, once genetics is added to the replicator equation, the evolutionary game can strongly change its dynamic properties. Genetics then constrains the course of phenotypic evolution.

         Undoubtedly, theoreticians have to face a dilemma in this regard. Genetics is important, but if one studies evolutionary games together with the underlying genetics, this often becomes such a tedious task that the theory loses most of its heuristic power. The only tractable approaches seem to be one-locus models (reviewed in Cressman 1992) and models of quantitative genetics, where many genes with very small effects are considered. Both these approaches are based on strong assumptions and thereby circumvent at least part of the dilemma under discussion. Gomulkiewicz (this volume) gives a very nice review of how far one gets with evolutionary game theory in the framework of quantitative genetics. However, as we shall see later in section 1.4, there is also another philosophy of how to overcome the modeler's dilemma. This is the philosophy of the "streetcar," which works well even in the difficult context of general «-locus genetics, where genes are allowed to have strong effects on phenotypes. 

My next article is about John Nash's equilibrium.. hehe