To understand this, let us briefly review how repeated games are modeled in TGT. In a repeated game , a certain basic game called stage game is played a number of rounds; the payoff obtained by each player in the repeated game is the sum of the potentially discounted payoffs obtained in each of the rounds. At any round, all the actions chosen by each of the players in previous rounds are known by everyone. A strategy in this repeated game is a complete plan of action for every possible contingency that may occur.
For instance, in our coordination game, a possible strategy for the 3-round repeated game would be:. Importantly, note that, even though the game is played repeatedly, the interaction only occurs once, since the strategies of the individuals dictate what to do in every possible history of the long game.
Players could send their strategies by mail, and robots could implement them. So, does repetition lead to sharper predictions about how rational players may interact? Not at all, rather the opposite. It turns out that when a game is repeated, the number of Nash equilibria generally multiplies, and there is a wide range of possible outcomes that can be supported by them.
As an example, in our coordination game, any sequence formed by combining the three Nash equilibria of the stage game is a Nash equilibrium of the repeated game, and there are many more. The approach followed to model repeated interactions in Evolutionary Game Theory is rather different, as we explain below.
Some time after the emergence of traditional game theory, biologists realized the potential of game theory to formally study adaptation and coevolution of biological populations, particularly in contexts where the fitness of a phenotype depends on the composition of the population Hamilton, The initial development of the evolutionary approach to game theory came with important changes on how the main elements of a game i.
The idea, however, can be applied equally well to any kind of phenotypic variation, and the word strategy could be replaced by the word phenotype; for example, a strategy could be the growth form of a plant, or the age at first reproduction, or the relative numbers of sons and daughters produced by a parent. Maynard Smith , p. The main assumption underlying evolutionary thinking was that strategies with greater payoffs at a particular time would tend to spread more and thus have better chances of being present in the future.
The first models in EGT, which were developed for biological contexts, assumed that this selection biased towards individuals with greater payoffs occurred at the population level, through a process of natural selection. As many more individuals of each species are born than can possibly survive ; and as, consequently, there is a frequently recurring struggle for existence, it follows that any being, if it vary however slightly in any manner profitable to itself, under the complex and sometimes varying conditions of life, will have a better chance of surviving , and thus be naturally selected.
From the strong principle of inheritance , any selected variety will tend to propagate its new and modified form. Darwin , p. The essence of this simple and groundbreaking idea could be algorithmically summarized as follows:.
The key insight that game theory contributed to evolutionary biology is that, once the strategy distribution changes as a result of the evolutionary process, the relative fitness of the remaining strategies may also change, so previously unsuccessful strategies may turn out to be successful in the new environment, and thus increase their prevalence.
In other words, the fitness landscape is not static, but it also evolves as the distribution of strategies changes. An important concept developed in this research programme was the notion of Evolutionarily Stable Strategy ESS , put forward by Maynard Smith and Price for 2-player symmetric games played by individuals belonging to the same population.
Informally, a strategy I for I ncumbent is an ESS if and only if, when adopted by all members of a population, it enjoys a uniform invasion barrier in the sense that any other strategy M for M utant that could enter the population in sufficiently low proportion would obtain a strictly lower expected payoff in the postentry population than the incumbent strategy I.
The ESS concept is a refinement of symmetric Nash equilibrium. As an example, in the coordination game depicted in Fig. This is so because the mutants would obtain the same payoff against the incumbents as the incumbents among themselves i.
Thus, natural selection would gradually favor the mutants over the incumbents. In economic contexts, it was understood that natural selection would derive from competition among entities for scarce resources or market shares. In other social contexts, evolution was often understood as cultural evolution, and it referred to dynamic changes in behavior or ideas over time Nelson and Winter , Boyd and Richerson Evolutionary ideas proved very useful to understand several phenomena in many disciplines, but —at the same time— it became increasingly clear that a direct application of the principles of Darwinian natural selection was not always appropriate for the study of non-Darwinian social evolution.
The key distinction is that, in this latter interpretation, strategies are selected at the individual level rather than at the population level. Also, in this view of selection taking place at the individual level, payoffs do not have to represent Darwinian fitness anymore, but can perfectly well represent a preference ordering, and interpersonal comparisons of payoffs may not be needed.
Following this interpretation, the algorithmic view of the process by which strategies with greater payoffs gradually displace strategies with lower payoffs would look as follows:.
In this interpretation, the canonical evolutionary model typically comprises the following elements:. Note that this approach to EGT can formally encompass the biological interpretation, since one can always interpret the revision of a strategy as a death and birth event, rather than as a conscious decision.
Having said that, it is clear that different interpretations may seem more natural in different contexts. The important point is that the framework behind the two interpretations is the same.
To conclude this section, let us revisit our coordination example with payoff matrix shown in Fig. We will analyze two revision protocols that lead to different results: imitative pairwise-difference protocol and best experienced payoff protocol.
The following video shows some NetLogo simulations that illustrate these dynamics. In this book, we will learn to implement and analyze this model. Simulation runs of the imitative pairwise-difference protocol in coordination game [[1 0][0 2]].
Simulation runs of the best experienced payoff protocol in coordination game [[1 0][0 2]]. The example above shows that different protocols can lead to very different dynamics in non-trivial ways. Both protocols above tend to favor best-performing strategies, and in both the mixed-strategy Nash equilibrium is unstable.
In this book, we will learn a range of different concepts and techniques that will help us understand these differences. Many engineering infrastructures are becoming increasingly complex to manage due to their large-scale distributed nature and the nonlinear interdependences between their components Quijano et al.
Examples include communication networks, transportation systems, sensor and data networks, wind farms, power grids, teams of autonomous vehicles, and urban drainage systems.
Controlling such large-scale distributed systems requires the implementation of decision rules for the interconnected components that guarantee the accomplishment of a collective objective in an environment that is often dynamic and uncertain. To achieve this goal, traditional control theory is often of little use, since distributed architectures generally lack a central entity with access or authority over all components Marden and Shamma, The concepts developed in EGT can be very useful in such situations.
The analogy between distributed systems in engineering and the social interactions analyzed in EGT has been formally established in various engineering contexts Marden and Shamma, In EGT terms, the goal is to identify revision protocols that will lead to desirable outcomes using limited local information only.
As an example, at least in the coordination game discussed above, the best experienced payoff protocol is more likely to lead to the most efficient outcome than the imitative pairwise-difference protocol.
EGT is devoted to the study of the evolution of strategies in a population context where individuals repeatedly interact to play a game. Strategies are subjected to evolutionary pressures in the sense that the relative frequency of strategies which obtain higher payoffs in the population will tend to increase at the expense of those which obtain relatively lower payoffs. In this sense, note that EGT is an inherently dynamic theory.
There are two ways of interpreting the process by which strategies are selected. In biological systems, players are typically assumed to be pre-programmed to play one given strategy throughout their whole lifetime, and strategy composition changes by natural selection.
By contrast, in socio-economic models, players are usually assumed capable of adapting their behavior within their lifetime, revising their strategy in a way that tends to favor strategies that provide greater payoffs at the time of revision. Whether strategies are selected by natural selection or by individual players is rather irrelevant for the formal analysis of the system, since in both cases the interest lies in studying the evolution of strategies. In this book, we will follow the approach which assumes that strategies are selected by individuals using a revision protocol.
TGT players are rational and forward-looking, while EGT players adapt in a fairly gradual and myopic fashion. This can be modelled mathematically. Doing so allows us to understand whether these strategies can coexist or if one of them prevails. Let V be the value of winning a contest, and C be the cost of injury in a contest.
Represent the frequency of hawks in the population as p , and the frequency of doves as 1-p. Now, define two functions F H and F D which define the expected fitness of playing the hawk and dove strategies, respectively. Playing as a hawk will mean engaging in a hawk-vs-hawk contest with frequency p.
The expected utility of doing so is understood as the average outcome. Half the time the hawk wins V , half the time it loses C. Playing as a dove will win nothing against hawks. But a dove will encounter another dove with frequency 1-p. This reveals the frequency p at which the hawk strategy confers no more or less fitness than the dove strategy.
At this frequency, there is no advantage to either strategy, so this is the equilibrium at which both strategies may coexist. The evolution of complex systems is a fascinating field of study. Understanding how natural forces and competitive pressures can shape individual-level traits to give rise to complex social behaviours has been one of the major areas of research in biological sciences in the last few decades.
The ability of relatively simple mathematical models to accurately predict outcomes of dynamic systems is also a key point to take away. In this case, it is the existence of a feedback loop that results in the two strategies reaching an equilibrium.
The advantage conferred by either strategy varies depending on how many others in the population are playing that strategy. In other words, when more individuals play "dove", there is an advantage to playing "hawk".
However, as more individuals play "hawk", the expected value of playing "dove" increases. Finally, the availability of programming and software tools makes it possible to test theoretical predictions through simulation. If you found this article interesting, you may also find How to Model an Epidemic With R worth checking out, too.
You can follow more of my writing at gleeson. If this article was helpful, tweet it. Learn to code for free. Get started. Forum Donate. Peter Gleeson. Each animal can play one of two strategies: Hawks are aggressive, and will fight for a resource at all costs.
Doves are passive, and will share instead of fight for a resource. There are three pairwise competitions that can exist: Hawk vs Hawk If two hawks compete, they will engage in a battle to win the resource.
This is a winner-takes-all scenario — the winner obtains the full value of the resource. The injured loser pays a price, and loses a certain amount of fitness. Hawk vs Dove If a hawk meets a dove, the dove will back down immediately.
0コメント