Part I. Introduction

# I-2. Introduction to evolutionary game theory

The goal of this chapter is to provide a brief introduction to Evolutionary Game Theory (EGT). Nonetheless, since EGT is a branch of a more general discipline called game theory, we believe it is useful to get familiar with the basic ideas underlying game theory first.

# 1. What is game theory?

Game theory is a discipline devoted to studying social interactions where individuals’ decisions are interdependent, i.e. situations where the outcome of the interaction for any individual generally depends not only on her own choices, but also on the choices made by every other individual. Thus, several scholars have pointed out that game theory could well be defined as ‘interactive decision theory’ (Myerson, 1997, p. 1). Some examples of interactive decisions for which game theory is useful are:

- Choosing the side of the road on which to drive.
- Choosing WhatsApp or Facebook messenger as your default text messaging app.
- Choosing which restaurant to go in the following situation:
Imagine that over breakfast your partner and you decided to have lunch together, but you did not specify where. Right before lunch time you discover that you left your cellphone at home, and there is no other way to contact your partner quickly. Fortunately, you are not completely lost: there are only two restaurants that you both love in town. Which one should you go to?

Social interactions like the ones outlined above are modeled in game theory as *games*. A game is an abstract representation of a social interaction which is meant to capture its most basic properties. In particular, a game typically comprises:

- the set of individuals who interact (called
*players*), - the different choices available to each of the individuals when they are called upon to act (named actions or
*pure strategies*), - the
*information*individuals have at the time of making their decisions, - and a
*payoff*function that assigns a value to each individual for each possible combination of choices made by every individual (Fig. 1). In most cases, payoffs represent the preferences of each individual over each possible outcome of the social interaction,^{[1]}though there are some evolutionary models where payoffs represent Darwinian fitness.

Player 2 |
|||

Player 2 chooses A |
Player 2 chooses B |
||

Player 1 |
Player 1 chooses A |
1 , 1 |
0 , 0 |

Player 1 chooses B |
0 , 0 |
2 , 2 |

Note that the payoff matrix shown in Fig. 1 could well be used for the three examples outlined above, assuming that there is a preferred side of the road, text messaging app or restaurant in town. To be sure, let us model the example about choosing a restaurant as a game, identifying each of its elements:

- The players would be you and your partner.
- Each of you may choose restaurant A or restaurant B.
- Neither of you have any information about the other’s choice at the time of making your decision.
- Both of you prefer eating together rather than being alone, no matter where, and you both prefer restaurant B over restaurant A. Thus, the best possible outcome for both would be to meet at restaurant B; second best would be meeting at restaurant A; and any lack of coordination would be equally awful.
^{[2]}

The three examples above are very different in many aspects, but they all could be modeled using the same game. This is so because games are abstractions that are meant to capture the bare essentials of the original social interaction and, at least to some extent, the three examples above share the same strategic backbone.

Having seen this, it may not come as a surprise that the sort of issues for which game theory can be useful is impressively broad and diverse, including applications in international relations, resource management, network routing (of vehicles or information packages), voting systems, linguistics, law, distributed control, evolutionary biology, design of incentive systems, and business and environmental regulations.

# 2. Traditional game theory

Game theory has nowadays various branches. Historically, the first branch to be developed was Traditional Game Theory (TGT) (von Neumann and Morgenstern (1944), Nash (1950), Selten (1965, 1975), Harsanyi (1967, 1968a, 1968b)). TGT is also the branch where most of the work has been focused, and the one with the largest representation in most game theory textbooks and academic courses.^{[3]}

In TGT, payoffs reflect preferences and players are assumed to be rational, meaning that they act as if they have consistent preferences and unlimited computational capacity to achieve their well-defined objectives. The aim of the discipline is to study how these instrumentally rational players would behave in order to obtain the maximum possible payoff in the formal game.

A key problem in TGT is that, in general, assuming rational behavior for any one player rules out very few actions –and consequently very few outcomes– in the absence of strong assumptions about what players know about others’ rationality, knowledge and actions. Hence, in order to derive specific predictions about how rational players would behave, it is often necessary to make very stringent assumptions about everyone’s beliefs and their reciprocal consistency. If one assumes common knowledge of rationality and consistency of beliefs, then the outcome of the game is a *Nash equilibrium*, which is a set of strategies, one for each player, such that no player, knowing the other players’ strategies in that set, could improve her expected payoff by unilaterally changing her own strategy (see Samuelson (1997, pp. 10-12) and Holt and Roth (2004) for several interpretations). An equivalent definition is the following: A Nash equilibrium is a strategy profile (i.e. one strategy for each player in the game) where every player is best responding to the strategies of the others.

Oftentimes, games have several Nash equilibria. As an example, the game depicted in Fig. 1 has three different Nash equilibria: the two strategy profiles where both players choose the same action (i.e. {A, A} and {B, B}), and a third equilibrium in *mixed* strategies, which means that players choose each action with a certain probability. In this third Nash equilibrium, both players choose action A with probability ⅔ (and action B with probability ⅓), a strategy that we denote (⅔, ⅓).

In this example, the equilibrium in mixed strategies is unsatisfactory for a number of reasons. First, since both actions can be chosen with strictly positive probability by each player, any observation of the actions actually taken by the two players would be consistent with this Nash equilibrium. Therefore, this equilibrium cannot be falsified by observing the outcome of the game. Another disappointing property of the Nash equilibrium in mixed strategies is the low payoff that players are expected to receive when they play it. In that equilibrium, outcome {B, B} would occur with probability ⅓ × ⅓, outcome {A, A} would occur with probability ⅔ × ⅔, and players would not coordinate with probability 2 × ⅓ × ⅔ , yielding a total expected payoff of ⅔ for each of them. Thus, this equilibrium is worse than any of the other two Nash equilibria for *both* players. Finally, the mixed-strategy equilibrium does not seem to be very robust. Imagine that one of the players deviates from this equilibrium only slightly, by choosing action B with a probability marginally greater than ⅓. Then the other player’s best response would be to choose action B with probability 1, and the deviator’s best response to that reaction would be to play B with probability 1, too. Thus, this mixed-strategy equilibrium does not seem to be very stable.^{[4]}

One could think that this diversity of equilibria, and the existence of the mixed-strategy equilibrium, may be partly an artifact of the fact that the game is played just once. It seems intuitive to think that if the game was played repeatedly, rational individuals would manage to coordinate in the (unique) *Pareto optimal*^{[5]} outcome {B, B}, and the other suboptimal outcomes would not be observed in any Nash equilibrium. However, that natural intuition turns out to be wrong. 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:

- At initial round
*t*= 1, play B. - At round
*t*> 1, play B if the other player chose B at time*t*– 1. Otherwise play A.

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.^{[6]}

The approach followed to model repeated interactions in Evolutionary Game Theory is rather different, as we explain below.

# 3. Evolutionary game theory

## 3.1. The beginnings

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, 1967). The initial development of the evolutionary approach to game theory came with important changes on how the main elements of a game (i.e. players, strategies, information and payoffs) were interpreted and used:

- Players (who most often represented non-human animals) were assumed to be pre-programmed to play one given strategy, i.e. players were seen as mere carriers of a particular fixed strategy that had been genetically endowed to them and could not be changed during the course of the player’s lifetime. As for the number of players, the main interest in early EGT was to study
*large populations*of animals, where the actions of one single individual could not significantly affect the overall success of any strategy in the population. - Strategies, therefore, were not assumed to be selected by players, but rather hardwired in the animals’ genetic make-up. Strategies were, basically, phenotypes.

The concept is couched in terms of a ‘strategy’ because it arose in the context of animal behaviour. 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 (1982, p. 10)

- Since strategies are not consciously chosen by players, but they are simply hardwired, information at the time of making the decision plays no significant role.
- Payoffs did not represent any order of preference, but Darwinian fitness, i.e. the expected reproductive contribution to future generations.

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 a matter of fact, early EGT models embraced a fairly direct interpretation of the essence of Wallace and Darwin’s idea of evolution by 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 itvaryhowever slightly in any manner profitable to itself, under the complex and sometimes varying conditions of life, will have abetter chance of surviving, and thus benaturally selected. From the strongprinciple of inheritance, anyselected variety will tend to propagate its new and modified form. Darwin (1859, p. 5)

The essence of this simple and groundbreaking** **idea could be algorithmically summarized as follows:

**IF**:

- More offspring are produced than can survive and reproduce, and
- variation within populations:
- affects the fitness (i.e. the expected reproductive contribution to future generations) of individuals, and
- is heritable,

**THEN**:

- evolution by natural selection occurs.

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 (1973) 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. 1 both pure strategies are ESSs, but the mixed strategy (⅔ , ⅓), corresponding to the symmetric Nash equilibrium in mixed strategies, is not an ESS. The intuition for this is clear: a population where every agent is playing strategy **I** = (⅔ , ⅓) would be invadable by e.g. a small fraction of mutants playing action B (i.e. strategy (0,1)). This is so because the mutants would obtain the same payoff against the incumbents as the incumbents among themselves (i.e. ⅔ on average), but a strictly greater payoff whenever they met other mutants (i.e. 2 for certain). Thus, natural selection would gradually favor the mutants over the incumbents.^{[7]}

The basic ideas behind EGT –i.e. that strategies with greater payoffs tend to spread more, and that fitness is frequency-dependent– soon transcended the borders of biology and started to permeate many other disciplines. 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 (1982), Boyd and Richerson (1985)).

## 3.2. An interpretation of evolutionary game theory where strategies are *explicitly selected* by individuals

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.^{[8]} In many contexts, it seems more natural to assume that players are capable of adapting their behavior within their lifetime, occasionally revising their strategy in a way that tends to favor strategies leading to higher payoffs over strategies leading to lower payoffs. 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:

**IF**:

- Players using different strategies obtain different payoffs, and
- they occasionally revise their strategies (by e.g. imitation or direct reasoning over gathered information), preferentially switching to strategies that provide greater payoffs,

**THEN**:

- the frequency of strategies with greater payoffs will tend to increase (and this change in strategy frequencies may alter the future relative success of strategies).

In this interpretation, the canonical evolutionary model typically comprises the following elements:

- A population of agents,
- a game that is recurrently played by the agents,
- a procedure by which revision opportunities are assigned to agents, and
- a decision rule, which dictates how individual agents choose their strategy when they are given the opportunity to revise.

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 chapter, let us revisit our coordination example (with payoff matrix shown in Fig. 1) in a population context. We will analyze two decision rules that lead to different results: *imitative pairwise-difference* and *best experienced payoff*.

- Under the
*imitative pairwise-difference rule*(Helbing (1992), Schlag (1998)), a revising agent looks at another individual at random and considers imitating her strategy only if that strategy yields a higher expected payoff than his current strategy; in this case he switches with probability proportional to the payoff difference. It can be proved that the dynamics of this rule in large populations will tend to approach the state where every agent plays action B if the initial proportion of B-players is greater than ⅓, and will tend to approach the state where every agent plays action A if the initial proportion of B-players is less than ⅓ (Fig. 2).^{[9]}

The following video shows some NetLogo simulations that illustrate these dynamics. In this book, we will learn to implement and analyze this model.

^{[10]}

- Under the
*best experienced payoff rule*(Osborne and Rubinstein (1998), Sethi (2000, 2021), Sandholm et al. (2019, 2020)), a revising agent tests each of the two strategies against a random agent, with each play of each strategy being against a newly drawn opponent. The revising agent then selects the strategy that obtained the greater payoff in the test, with ties resolved at random. It can be proved that the dynamics of this rule in large populations will tend to approach the state where every agent plays action B from any initial condition (other than the state where everyone plays A; see Fig. 3).^{[11]}

The following video shows some NetLogo simulations that illustrate these dynamics. In this book, we will learn to implement and analyze this model.

^{[12]}

The example above shows that different decision rules can lead to very different dynamics in non-trivial ways. Both rules above tend to favor best-performing strategies, and in both the mixed-strategy Nash equilibrium is unstable. However, given an initial state where 80% of the population is playing strategy A, one of the rules will almost certainly lead the population to the state where everyone plays A, while the other rule will lead the population to the state where everyone plays B. In this book, we will learn a range of different concepts and techniques that will help us understand these differences.

Evolutionary Game Theory and Engineering

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., 2017). 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, 2015).

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, 2015). In EGT terms, the goal is to identify decision rules 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* rule is more likely to lead to the most efficient outcome than the *imitative pairwise-difference* rule.

## 3.3. Take-home message

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. The aim is to identify which strategies are most likely to thrive in this “evolving ecosystem of strategies” and which will be wiped out, under different evolutionary dynamics. 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 decision rule.

## 3.4. Relation with other branches

The differences between TGT and EGT are quite clear and rather obvious. TGT players are rational and forward-looking, while EGT players adapt in a fairly gradual and myopic fashion. TGT is a theory stated in terms of a one-time interaction: even if the interaction is a repeated game, this long game is played just once. In stark contrast, dynamics are at the core of EGT: the outcomes of the game shape the distribution of strategies in the population, and this change in distribution modifies the relative success of different strategies when the game is played again. Finally, TGT is mainly focused on the study of end-states and possible equilibria, paying hardly any attention to how such equilibria might be reached. By contrast, EGT is concerned with the evolution of the strategy composition in a population, which in some cases may never settle down on an equilibrium.

The branch of game theory that is closest to EGT is the Theory of Learning in Games (TLG). Like EGT, TLG abandons the demanding assumptions of TGT on players’ rationality and beliefs, and assumes instead that players learn over time about the game and about the behavior of others (e.g. through reinforcement, imitation, or belief updating).

The process of learning in TLG can take many different forms, depending on the available information and feedback, and the way these are used to modify behavior. The assumptions made in these regards give rise to different models of learning. In most models of TLG, players use the history of the game to decide what action to take. In the simplest forms of learning (e.g. reinforcement or imitation) this link between acquired information and action is direct (e.g. in a stimulus-response fashion); in more sophisticated learning, players use the history of the game to form expectations or beliefs about the other players’ behavior, and they then react optimally to these inferred expectations.^{[13]}

The interpretation of EGT which assumes that players can revise their strategy is very similar to TLG. The main differences between these two branches are:

- EGT tends to study
*large*populations of*small*agents, who interact*anonymously*. This implies that players’ payoffs only depend on the distribution of strategies in the population, and also that any one player’s actions have little or no effect on the aggregate success of any strategy at the population level. In contrast, TLG is mainly concerned with the analysis of small groups of players who repeatedly play a game among them, each of them in her particular role. - The decision rules analyzed in EGT tend to be fairly simple and use information about the current state of the population only. By contrast, the sort of algorithms analyzed in TLG tend to be more sophisticated, and make use of the history of the game to decide what action to take.

# 4. How can I learn game theory?

To learn more about game theory, we recommend the following material:

- Overviews:
- Introductory: Colman (1995).
- Advanced: Vega-Redondo (2003).

- Traditional game theory:
- Introductory: Dixit and Nalebuff (2008).
- Intermediate: Osborne (2004).
- Advanced: Fudenberg and Tirole (1991), Myerson (1997), Binmore (2007).

- Evolutionary game theory:
- Introductory: Maynard Smith (1982), Gintis (2009).
- Intermediate: Hofbauer and Sigmund (2003), Sandholm (2009), Alexander (2023).
- Advanced: Hofbauer and Sigmund (1988), Weibull (1995), Samuelson (1997), Sandholm (2010a).
- Recent literature review: Newton (2018).

- The theory of learning in games:
- Introductory: Vega-Redondo (2003, chapters 11 and 12).
- Advanced: Fudenberg and Levine (1998), Young (2004).

- A common misconception about game theory relates to the roots of players’ preferences. There is no assumption in game theory that dictates that players’ preferences are formed in complete disregard of each other’s interests. On the contrary, preferences in game theory are assumed to account for anything, i.e. they may include altruistic motivations, moral principles, and social constraints (see e.g. Colman (1995, p. 301), Vega-Redondo (2003, p. 7), Binmore and Shaked (2010, p. 88), Binmore (2011, p. 8) or Gintis (2014, p. 7)). ↵
- Note that there is no inconsistency in being indifferent about outcomes {A, B} and {B, A}, even if you prefer restaurant B. It is sufficient to assume that you care about your partner as much as about yourself. ↵
- TGT can be divided further into cooperative and non-cooperative game theory. In
*cooperative*game theory, it is assumed that players may negotiate binding agreements that can be externally enforced (by e.g. contract law). In*non-cooperative*game theory, such agreements cannot be enforced externally, so they are relevant only to the extent that abiding by them is in each individual's interest. ↵ - The same logic applies if one assumes that the deviation consists in choosing action B with a probability marginally
*less*than ⅓. In this case, the other player's best response would be to choose action A with probability 1. ↵ - An outcome is Pareto optimal if it is impossible to make one player better off without making at least one other player worse off. ↵
- In general, any strategy profile which at every round prescribes the play of a Nash equilibrium of the stage game regardless of history is a (subgame perfect) Nash equilibrium of the repeated game. This can be easily proved using the one-shot deviation principle. ↵
- Note that mutants playing action A (i.e. strategy (1,0)) would also be able to invade the incumbent population. ↵
- As an example, note that payoffs interpreted as Darwinian fitness are added across different players to determine the relative frequency of different types of players (i.e. strategies) in succeeding generations. These interpersonal comparisons are inherent to the notion of biological evolution by natural selection, and pose no problems if payoffs reflect Darwinian fitness. However, if evolution is interpreted in cultural terms, presuming the ability to conduct interpersonal comparisons of payoffs across players may be controversial. In this link, you can watch a video that shows how unconvinced John Maynard Smith was by direct applications of the principles of
*Darwinian*natural selection in Economics. ↵ - To prove this statement, note that the mean dynamic of this decision rule is the well-known replicator dynamic (Taylor and Jonker, 1978). ↵
- See exercise 5 in chapter II-1 and exercise 3 in chapter II-5. ↵
- This statement is a direct application of Proposition 5.11 in Sandholm et al. (2020). Izquierdo et al. (2022) prove that this dynamic leads to the efficient state in a larger class of games called single-optimum coordination games. ↵
- See exercise 6 in chapter II-1 and exercise 4 in chapter II-5. ↵
- Izquierdo et al. (2012) provide a succinct overview of some of the learning models that have been studied in TLG. For a more detailed account, see chapters 11 and 12 in Vega-Redondo (2003). ↵