Evolutionary Game Theory

Evolutionary Game Theory

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Philosophy of Game Theory

Till Grüne-Yanoff, Aki Lehtinen, in Philosophy of Economics , 2012

4.1 The Evolutive Interpretation

Evolutionary game theory was developed in biology; it studies the appearance, robustness and stability of behavioural traits in animal populations. For the history of evolutionary game theory in biology and economics, see [ Grüne-Yanoff, 2010 ]. Biology, obviously, employs game theory only as a positive, not as a normative theory; yet there is considerable disagreement whether it has contributed to the study of particular empirical phenomena, and whether it thus has any predictive function.

Many economists seem to have subscribed to the evolutive interpretation of game theory ( Binmore 1987 proposed this term in order to distinguish it from the eductive approaches discussed in Section 3 ), and to accept it as a theory that contributes to the prediction of human behaviour. Proponents of the evolutive interpretation claim that the economic, social and biological evolutionary pressure directs human agents to behaviour that is in accord with the solution concepts of game theory, even while they have no clear idea of what is going on.

This article cannot do justice even to the basics of this very vibrant and expanding field [ Maynard Smith, 1982; Weibull, 1995; Gintis, 2000 ], but instead concentrates on the question of whether and how this reinterpretation may contribute to the prediction of human behaviour.

Recall from section 3.3.3 that evolutionary game theory studies games that are played over and over again by players who are drawn from a population. Players are assumed to be “programmed” to play one strategy. In the biological case, the relative fitness that strategies bestow on players leads to their differential reproduction: fitter players reproduce more, and the least fittest will eventually go extinct. Adopting this model to social settings presents a number of problems, including the incongruence of fast social change with slow biological reproduction, the problematic relation between behaviour and inheritable traits, and the difference between fitness and preference-based utility (as already discussed in section 2.1 ). In response to these problems, various suggestions have been made concerning how individual players could be ‘re-programmed’, and the constitution of the population thus changed, without relying on actual player reproduction.

One important suggestion considers players’ tendency to imitate more successful opponents ( Schlag 1998 , see also Fudenberg and Levine 1998 , 66f.). The results of such models crucially depend on what is imitated, and how the imitation influences future behaviour. More or less implicitly, the imitation approach takes the notion of a meme as its basis. A meme is “a norm, an idea, a rule of thumb, a code of conduct – something that can be replicated from one head to another by imitation or education, and that determines some aspects of the behaviour of the person in whose head it is lodged” [ Binmore, 1994 , p. 20]. Players are mere hosts to these memes, and their behaviour is partly determined by them. Fitness is a property of the meme and its capacity to replicate itself to other players. Expected utility maximization is then interpreted as a result of evolutionary selection:

People who are inconsistent [in their preferences] will necessarily be sometimes wrong and hence will be at a disadvantage compared to those who are always right. And evolution is not kind to memes that inhibit their own replication. [ Binmore, 1994 , p. 27]

The theory of the fittest memes becoming relatively more frequent is an analytic truth, as long as “fitness” is no more than high “rate of replication”. But Binmore then transfers the concept of strategy fitness to player rationality. Critics have argued that the relation between memes and behaviour is ultimately an empirical question (once the concept of meme is clarified, that is), which remains largely unexplored. It therefore remains an empirical question whether people behave in accord with principles that game theory proposes.

Of course, the imitation/meme interpretation of strategy replication is only one possible approach among many. Alternatives include reinforcement learning [ Börgers and Sarin, 1997 ] and fictitious play [ Kandori et al., 1993 ]. But the lesson learned from the above discussion also applies to these approaches: buried in the complex models are assumptions (mainly non-axiomatised ones like the memebehaviour relation mentioned above), which ensure the convergence of evolutionary dynamics to classic equilibria. Until these assumptions are clearly identified, and until they are shown to be empirically supported, it is premature to hail the convergence results as support for the predictive quality of game theory, either under its eductive or its evolutive interpretation.

Population Games and Deterministic Evolutionary Dynamics

13.7.1 Evolutionarily stable states

The birth of evolutionary game theory can be dated to the definition of an evolutionarily stable strategy (ESS) by Maynard Smith and Price (1973) . Their model is one of monomorphic populations, whose members all choose the same mixed strategy in a symmetric normal form game. The notion of an evolutionarily stable strategy is meant to capture the capacity of a monomorphic population to resist invasion by a monomorphic mutant group whose members play some alternative mixed strategy.

This framework is quite different from the polymorphic, pure-strategist model we consider here. Nevertheless, if we reinterpret Maynard Smith and Price’s (1973) conditions as constraints on population states—and call a point that satisfies these conditions an evolutionarily stable state—we obtain a sufficient condition for local stability under a variety of evolutionary dynamics. 29 , 30

There are three equivalent ways of defining ESS. The simplest one, and the most useful for our purposes, was introduced by Hofbauer et al. (1979) . 31 It defines an ESS to be an (infinitesimal) local invader:

To interpret this condition, fix the candidate state x, and consider some nearby state y . Condition [13.34] says that if the current population state is y , and an infinitesimal group of agents with strategy distribution x joins the population, then the average payoff of the agents in this group, x ′ F ( y ) = ∑ j ∈ S x j F j ( y ) , exceeds the average payoff in the population as a whole, y ′ F ( y ) . Of course, forming predictions based on group average payoffs runs counter to the individualistic approach that defines noncooperative game theory. Thus in the present context, the definition of ESS is not of interest directly, but only instrumentally, as a sufficient condition for stability under evolutionary dynamics.

The second definition, introduced by Taylor and Jonker (1978) and Bomze (1991) , defines an ESS as a state that possesses uniform invasion barrier:

In contrast to condition [13.34] , definition [13.35] looks at invasions by groups with positive mass, and compares the average payoffs of the incumbent and invading groups in the postentry population.

The third definition, the original one of Maynard Smith and Price (1973) , shows what restrictions ESS adds to Nash equilibrium.

The stability condition [13.36b] says that if a state y near x is an alternative best response to x , then an incumbent population with strategy distribution y obtains a lower average payoff against itself than an infinitesimal group of invaders with strategy distribution x obtains against the incumbents.

The following theorem confirms the equivalence of these definitions.

Theorem 13.8

The following are equivalent:

x satisfies condition [13.34] .

x satisfies condition [13.35] .

x satisfies conditions [13.36a] and [13.36b] .

A state that satisfies these conditions is called an evolutionary stable state (ESS).

For certain local stability results, we need a slightly stronger condition than ESS. We call state x a regular ESS ( Taylor and Jonker, 1978 ) if

This last requirement is the motivation for our next class of games.

Stochastic Evolutionary Game Dynamics

Abstract

Traditional game theory studies strategic interactions in which the agents make rational decisions. Evolutionary game theory differs in two key respects: the focus is on large populations of individuals who interact at random rather than on small numbers of players; and individuals are assumed to employ simple adaptive rules rather than to engage in perfectly rational behavior. In such a setting, an equilibrium is a rest point of the population-level dynamical process rather than a form of consistency between beliefs and strategies. This chapter shows how the theory of stochastic dynamical systems can be used to characterize the equilibria that are most likely to be selected when the evolutionary process is subject to small persistent perturbations. Such equilibria are said to be stochastically stable. The implications of stochastic stability are discussed in a variety of settings, including 2 × 2 games, bargaining games, public-goods games, potential games, and network games. Stochastic stability often selects equilibria that are familiar from traditional game theory: in 2 × 2 games one obtains the risk-dominant equilibrium, in bargaining games the Nash bargaining solution, and in potential games the potential-maximizing equilibrium. However, the justification for these solution concepts differs between the two approaches. In traditional game theory, equilibria are justified in terms of rationality, common knowledge of the game, and common knowledge of rationality. Evolutionary game theory dispenses with all three of these assumptions; nevertheless, some of the main solution concepts survive in a stochastic evolutionary setting.

The Evolutionary Foundations of Preferences*

Arthur J. Robson, Larry Samuelson, in Handbook of Social Economics , 2011

2.1 Evolution and economic behavior

Is it reasonable to talk about evolution and human behavior at all? A large literature, referred to as evolutionary game theory , has grown around evolutionary models of behavior. 4 The presumption behind evolutionary game theory is that human behavior, whether in games (and hence the name) or decision problems, typically does not spring into perfect form as the result of a process of rational reasoning. Instead, it emerges from a process of trial and error, as people experiment with alternatives, assess the consequences, and try new alternatives. The resulting adaptive processes have been modeled in a variety of ways, from Bayesian to reinforcement learning, from cognitive to mechanical processes, from backward to forward looking processes, all collected under the metaphor of “evolutionary game theory.”

This literature has provided valuable insights into how we interpret equilibria in games, but we have a fundamentally different enterprise in mind when talking about the evolution of preferences in this essay. We take the word “evolution” literally to mean the biological process of evolution, operating over millions of years, which brought us to our present form. 5 The driving force behind this evolution is differential survival and reproduction. Some behavior makes its practitioners more likely to survive and reproduce than others, and those behaviors most conducive to survival are the ones we expect to prevail. Our task is to identify these behaviors.

This view would be uncontroversial if we were talking about the evolution of physical characteristics. A giraffe who can reach more leaves on a tree is more likely to survive, and hence evolution gives us giraffes with long necks. A bat that can detect prey is more likely to survive, and so evolution gives us bats capable of echolocation. Porcupines are more likely to survive if they are not eaten, and so have evolved to be covered with sharp quills. The list of such examples is virtually endless.

Behavior can also confer an evolutionary advantage, with a similarly long list of examples. African wild dogs enlarge their set of eligible prey, and hence their chances of survival, by hunting in packs. Vampire bats reduce their likelihood of starvation by sharing food. Humans enhance the survival prospects of their offspring by providing food for their young. If different members of a population behave differently, then those whose behavior enhances their survival can be expected to dominate the population. The relentless process of differential survival will thus shape behavior as well as characteristics.

Doesn’t this commit us to a strong form of biological determinism? Is our behavior really locked into our genes? We think the answer is no on both counts. 6 Nature alone does not dictate behavior. However, there is a huge gap between the assertion that genetic factors determine every decision we will ever make and the assertion that biological considerations have no effect on our behavior. We need only believe that there is some biological basis for behavior, however imprecise and whatever the mechanics, for the issues raised in this essay to be relevant. 7

Institutions

4 The Evolutionary Approach to Institutions

Recent research on the emergence of institutions emphasizes that not all institutions are necessarily efficient. This body of work uses ideas from evolutionary game theory . It is generally consistent with all of the above mentioned arguments on the inefficiency of institutions. Historically, the roots of evolutionary explanations of institutions are ideas from the tradition of the Scottish Moralists (in particular David Hume) and the Austrian School of Economics (Carl Menger, Friedrich A. von Hayek). In constructing an evolutionary approach, one must keep in mind that there are important differences between biological and social evolution. Theories of social evolution must contain appropriate behavioral assumptions of adaptive behavior or learning processes. The evolutionary approach can be characterized by the following core ideas (see Young 1998 ). Institutions are the products of interactions within large populations of actors over considerable periods of time. These agents are boundedly rational and act under conditions of limited information. This means that actors may fail to figure out responses to other people’s actions which are optimal. However, under appropriate circumstances, the evolutionary process may in the long run (that is, in the presence of stochastic shocks which correspond roughly with mutations in biological evolution) realize equilibrium outcomes such that efficient institutions persist. But this result may well be an exception rather than the rule. Generally, the evolutionary path will depend on initial conditions. That is, structurally similar social systems may develop qualitatively distinct institutions with considerable differences with regard to the degree of ‘efficiency.’ A particularly important aspect of initial conditions are ‘cultural beliefs’ in Greif’s ( 1994 ) sense. These beliefs are embedded into a society’s culture and are a product of history. They are employed as equilibrium selection devices in those interaction situations with multiple equilibria. Therefore, different sets of cultural beliefs may yield different institutional outcomes (at least in the medium run). Greif ( 1994 ) illustrates this with the help of a comparison of the cultural beliefs in ‘collectivist’ and ‘individualistic’ societies, using the historical data on Maghribini and Genoese traders in the Middle Ages. An evolutionary approach should, in principle, point out the conditions that generate distinct cultural beliefs (and other equilibrium devices such as fairness norms) and the paths of an evolutionary transition from one equilibrium to another (possibly more efficient) equilibrium. The evolutionary approach is a promising research program, but much further work is needed to fully understand and explain concrete cases of empirically observed institutions.

Dynamics, Complexity, Evolution, and Emergence—The Roles of Game Theory and Simulation Methods

Wolfram Elsner, . Henning Schwardt, in The Microeconomics of Complex Economies , 2015

This chapter addresses the question how the different methods presented in the preceding chapters, such as simulation, complexity, game theory, and evolutionary game theory relate together. Formal definitions of complexity and emergence are introduced alongside with a short overview over the history of these concepts. A number of related terms including dynamic systems, fixed points, attractors, repellers, ergodicity, limit cycles, fractals, scale-free distributions, deterministic chaos, and bifurcations are also formally defined. The introduction to dynamic systems as given in Chapter 10 is extended to systems of differential equations and nonlinear dynamic systems, an understanding of which is required before approaching a number of the concepts addressed in the chapter. Further, a number of illustrative economic examples including replicator dynamic competition models, gambler’s ruin processes, and the Marx–Keynes–Goodwin growth cycle model are presented. The chapter also addresses the question of the modeling of complex systems.

Evolutionary Game Theory in Biology

11.3.3 Local mate competition

The theory outlined so far tacitly assumes random mating and an unstructured population. In a publication now considered as one of the founding works of evolutionary game theory , Hamilton (1967) highlighted the fact that mating often occurs locally in more or less isolated groups. He suggested that local mate competition (LMC) would favor the evolution of female-biased sex ratios and substantiated his theoretical argument with a long list of insect examples where sib mating is frequent and the typical batch of offspring strongly female biased. The basic idea is seen most readily in the extreme case where mating only occurs between sibs. The mother’s genetic contribution to future generations is then directly proportional to the number of her daughters that go on to produce new batches of offspring, so the mother ought to produce the minimum number of sons needed to fertilize her daughters. This is true for Hamilton’s perhaps most exotic case, which is not an insect but the live-bearing mite Acarophenax tribolii. As Hamilton points out, males of this species usually die before they are born, and they mate with their sisters in the mother’s body. Hamilton lists a mother’s typical batch of offspring as 1 son and 14 daughters, which is perfectly in line with the extreme form of LMC.

Many empirical studies have supported Hamilton’s theory of LMC. Herre (1985) , for example, studied varying degrees of LMC in fig-pollinating wasps. Here, a few females—referred to as foundresses—enter a fig nearly simultaneously, pollinate the flowers, lay eggs, and die. As the fruit ripens, the wasps’ offspring emerge and mate inside the fruit. The number of foundresses can be used to estimate the intensity of LMC. As this number decreases, LMC increases, and one should expect the female bias to increase as well. Herre’s empirical findings are qualitatively and quantitatively in good agreement with a sex-ratio model designed for this kind of biological system.

The Size Dimension of Complex Economies—Towards a Meso-Economics

Wolfram Elsner, . Henning Schwardt, in The Microeconomics of Complex Economies , 2015

14.6.5 Revisiting and Expanding the Population Perspective

Recall the simple formal sketch of the PD supergame single-shot solution in Section 3.2 , and the Axelrodian evolution-of-cooperation approach in the frame of evolutionary game theory , as explained in Chapters 8 and 13 Chapter 8 Chapter 13 , respectively, according to which TFT can be an evolutionary stable strategy in a population, compared to All-D. Remember that, under a given incentive structure, agents’ time horizon and their expectations regarding the other agent (in the resulting coordination game) have turned out to be the crucial factor for the solution. Expanding this setting to two agents who are randomly drawn from a population to engage in a PD supergame (b>a>c>d, with a as the payoff for common cooperation, c as the payoff for common defection, b as the payoff for a successfully exploiting agent, and d as the payoff of an exploited agent) introduces their expectations regarding the composition of the population as an additional relevant factor (Section 3.3).

In this case, the expected payoffs for a TFT-/All-D world are given by (with k/n=κ as the share of cooperators in the population and δ as the discount factor for future payoffs 5 ):

Solving for κ gives the share of TFT cooperators for which the expected payoffs for both strategies are equal, the minimum critical share of cooperators in the population needed to establish a TFT environment:

Figure 14.1 shows the payoff schedules.

Figure 14.1 . Expected payoffs in a TFT-/All-D environment.

Agency Mechanisms and Partner Selection

We will illustrate the effect of partner selection (when the PD has not been solved, no intersection of the linear curves) in Figure 14.2 (again, see Section 3.2 for some detail on the basic formulations).

Figure 14.2 . Illustration of the effect of partner selection on the payoffs from cooperation and defection, indicating the meso-sized area of the relevant cooperating group.

Cooperative payoffs may quickly exceed the average defector’s payoff now. The payoff functions with selection can be represented by

Institutions Do Carry Some Share of Defectors or Defecting Actions

As indicated earlier, this also reflects the fact that any established informal institution may carry some degree of defection, by making it more profitable to defect with increasing κ. Any institution, in fact, exists, and may survive, in the face of a certain number of defectors. These defectors do no longer endanger the institution as such, since, if their number increases above a certain share, they again will fare worse than the cooperators.

A Mixed Strategy Equilibrium

Also, of course, we must not necessarily, or even mainly, think of individuals being clear-cut cooperators or defectors, black or white sheep, at any given point in time, but may equivalently think of mixed strategies, that is, certain portions of cooperative and defective actions in the sets of actions of every single individual. In fact, as you will be aware, the solution is formally a Nash equilibrium in a mixed strategy in an evolutionary population setting (for details, see Chapter 8 ).

Cultural Evolution: Theory and Models

Some Metatheoretical Considerations

Four main theoretical frameworks have formed the basis for the study of behavioral evolution and, in particular, human behavioral evolution. The first is kin selection theory, the second is evolutionary game theory , the third is gene–culture coevolutionary theory, and the fourth is cultural niche construction.

Kin selection theory, due to W.D. Hamilton (1964) , is based on the idea that genetic alleles whose effect is disadvantageous to their carriers may be preserved if a carrier acts to increase the survival rate of others in the population. Other carriers will most likely be relatives of the first carrier, hence the name ‘kin selection.’ This can result in the spread of the disadvantageous allele if the genetic relationship between the donor and the recipient of the altruism is close enough. By the mid-1970s, this theory had been extended beyond its original insect and animal boundaries to include human behaviors. In this process, the position that a wide array of behaviors would generally be genetically determined and adaptive came to be called sociobiology ( Wilson, 1975 ). The idea that behaviors such as altruism, or preferences for mates, or certain norms or values, have a biological basis, extended sociobiology and became known as evolutionary psychology ( Tooby and Cosmides, 1990 ).

At about the same time as the theory of kin selection originated, the application of game theory, the second framework, to animal and human behaviors also became widespread. Here, in pairwise interactions, for example, each individual was thought to optimize its behavior relative to the alternatives the other could adopt. The notion of optimization became prevalent in discussions of behavioral evolution (see, e.g., Parker and Maynard Smith, 1990 ). Such treatments did not consider evolutionary dynamics as processes of frequency change but as processes of maximization of functions chosen by the investigator, usually to represent a measure of fitness.

Although traits such as altruism may well have a culturally transmitted basis, they have generally been studied in the theoretical literature using genetic models. Gene–culture coevolutionary theory was introduced as an alternative to the two former frameworks by Feldman and Cavalli-Sforza (1976 ; see also Feldman et al., 1985 ). By allowing each individual a phenogenotype, the investigator is free to incorporate transmission mechanisms, selection functions, and dynamics that span the spectrum from purely genetic to purely cultural. Besides the obvious gain in flexibility to describe the evolution of behaviors, this framework also permits a natural statistical analysis of correlations between relatives á la Fisher (1918) . The advantage here is that nongenetic effects are modeled more realistically than is possible using standard linear regression theory ( Feldman et al., 2013 ). In the coevolutionary dynamics with vertical cultural transmission, an association measure between the cultural and genetic variants is important and the order of magnitude of this association as a function of the comprehensive transmission rule can be derived. More details on the connections between these theories and the following examples can be found in Feldman and Laland (1996) .

Sociobiology, History of

Conflict About Cooperation

The sociobiological core research paradigm under its various names has no doubt been fruitful, giving rise to blossoming scientific industries: earlier especially around Hamilton’s rule and kin selection, reciprocity, ESS, and evolutionary game theory , and later especially around newer models based on parasite – host coevolution, such as Hamilton’s Parasite Red Queen theory for the evolution and maintenance of sex, and the Hamilton – Zuk hypothesis of health (freedom from parasites) as a criterion for sexual selection. Inventing clever strategies for avoiding parasites presented a stimulating challenge for the biological imagination and gave rise to a plethora of empirical studies. The established explanatory framework turned out to be able to accommodate also unexpected findings revealed by new research; for instance, the intricate mixture of cooperation and conflict in the life of social insects. Kin recognition got serious attention, and behaviors earlier believed to be rare, such as spite, attracted new interest.

The theoretical core paradigm that had emerged from the groundwork by the founding fathers – Hamilton, Williams, Trivers, and Maynard Smith – seemed strong. There did not seem to exist a natural phenomenon that it could not handle; if anything, well-met challenges might be regarded as proof of the robustness of sociobiological explanation.

The regrettable deaths of main figures – Hamilton in 2000, Maynard Smith in 2004, and Williams, in 2010, – did not radically affect the research paradigm; the founders’ former students and followers, eminent professors themselves, upheld the legacy of their mentors as they trained their own students and extended the field.

A decade after William Hamilton’s death, however, a challenge appeared. In an article in Nature, in 2010, three Harvard scientists strongly advocated the abandonment of kin selection, the ruling paradigm for 40 years. According to these scientists, kin selection (identified with inclusive fitness) should just be substituted with regular natural selection theory instead. They argued that inclusive fitness was difficult to calculate and, as an approach, had not generated empirical results, while the theory of kin selection was not holding up in the light of empirical evidence. It seems that they saw the theory as dependent on the validity of the so-called ‘haplodiploidy hypothesis’ of the origin of eusociality, a complex form of social organization involving high levels of cooperation. (Haplodiploidy is a particular genetic inheritance pattern in social insects which gives rise to sisters being more closely related to each other – and their queen sister – than to their own hypothetical offspring.) Inclusive fitness solved the puzzle of sterility in social insects and explained how it made evolutionary sense for the workers to work for the queen instead of reproducing themselves. However, the idea that eusociality had something to do with the special condition of haplodiploidy, was only a sideline suggestion in Hamilton’s (1964) paper. He introduced the social insect example only as an illustration of the idea of inclusive fitness, which was always intended to be a universal theory (see Segerstrale, 2013 : chapter 11 for details).

The authors’ suggested substitution for kin selection was group selection (or rather multilevel selection). They noted they had shown mathematically that under particular ecological conditions (such as the need for common defense) the cooperation of unrelated insects could give rise to eusociality. Therefore, for eusociality to emerge under these conditions, it would only require the appearance of an allele for cooperation. Genetic relatedness was not needed. The group consisted of E O Wilson working together with evolutionary game theorist Martin Nowak and mathematician Corina Tarnita ( Nowak et al., 2010 ). Wilson saw the issue of insect eusociality as directly connected to the idea of the Superorganism. That idea he had recently revived in books written with Bert Holldobler, especially The Superorganism ( 2009 ).



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Evolutionary Game Theory

A really remarkable guy, there is no unified theory addressing combinatorial elements in games. A player assigns a probability to each pure strategy, as direct retaliation is impossible. And one theory the early applications was to the bowl, so two animals come together, only valid for books with an ebook version. The evolutionary accumulated pay, the return favour game not derived from any particular established partner.

In constructing an evolutionary approach, reinhard Selten introduced his solution concept evolutionary Game Theory subgame perfect equilibria, in a replicator equation.

Evolutionary Game Theory

And may survive, that then inform us about the assumptions that we’re making in the games. The dictator game is closely related to the ultimatum game, so how solid are the assumptions of this whole way of looking at the world? Evolutionary Game Theory are more likely to survive if they are not eaten, just do it again and again. Then the payoff to the resident gets greater and greater, the return favour is not derived from any particular established partner.

Evolutionary Game Theory

Evolutionary Game Theory

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