# Game Theory 13: Replicator Dynamics

REPLICATOR DYNAMICS

The Replicator equation is the first and most important game dynamic studied in connection with evolutionary game theory. The replicator equation and other deterministic game dynamics have become essential tools over the past 40 years in applying evolutionary game theory to behavioral models in the biological and social sciences.

REPLICATOR EQUATION

These models show the growth rate of the proportion of agents using a certain strategy. As we will illustrate, this growth rate is equal to the difference between the average payoff of that strategy and the average payoff of the population as a whole.

MODEL

There are three primary elements to a replicator model:

Firstly we have a set of agent types, each of which represents a particular strategy and each type of strategy has a payoff associated with it which is how well they are doing.

There is also a parameter associated with how many of each type there are in the overall population – each type represents a certain percentage of the overall population.

Now in deciding what they might do, people may adopt two approaches.

• They may simply copy what other people are doing, in such a case the likelihood of an agent adopting any given strategy would be relative to its existing proportion of that strategy within the population. So if lots of people are doing some strategy the agent would be more likely to adopt that strategy over some other strategy that few are doing.
• Alternatively, the agent might be more discerning and look to see which of other people’s strategies is doing well and adopt the one that is most successful, having the highest payoff.

The replicator dynamic model is going to try and balance these two potential approaches that agents might adopt, and hopefully, give us a more realistic model than one where agents simply adopt either strategy solely.

Given these rules, the replicator model is one way of trying to capture the dynamic of this evolutionary game, to see which strategies become more prevalent over time or how the percentage mix of strategies changes.

In a rational model, people will simply adopt the strategy that they see as doing the best amongst those present. But equally, people may simply adopt a strategy of simply copying what others are doing. If 10% are using strategy 1, 50% strategy 2, and 40% strategy 3, then the agent is more likely to adopt strategy 2 due to its prevalence.

So the weight that captures how likely an agent will adopt a certain strategy in the next round of the game is a function of the probability times the payoff.

If we wanted to think about this in a more intuitive way, we might think of having a bag of balls where the ball represents a strategy that will be played in the game. If a strategy has a better payoff then it will be a bigger ball and you will be more likely to pick that bigger ball.

Equally, if there are more agents using that strategy in the population, there will be more balls in the bag representing that strategy, meaning again you will be more likely to choose it. The replicator model is simply computing which balls will get selected and thus what strategies will become more prevalent.

One thing to note though is that the theory typically assumes large homogeneous populations with random interactions. The replicator equation differs from other equations used to model replication in that it allows the fitness function to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of selection. But unlike other models, the replicator equation does not incorporate mutation and so is not able to innovate new types or pure strategies.

FISHERS FUNDAMENTAL THEOREM

An interesting corollary to this is what is called Fisher’s Fundamental Theorem, which is a model that tries to capture the role that variation plays in adaptation. The basic intuition is that a higher variation in the population will give it greater capacity to evolve optimal strategies given the environment.

Thus given a population of agents trying to adapt to their environment, the rate of adaptation of a population is proportional to the variation of types within that population. Fisher’s Fundamental Theorem then works to incorporate this additional important parameter, of the degree of variation among the population, so as to better model the overall process of strategy evolution.

GAMES

Static game-theoretic solution concepts, such as Nash equilibrium, play a central role in predicting the evolutionary outcomes of game dynamics.

Conversely, game dynamics that arise naturally in analyzing behavioral evolution lead to a more thorough understanding of issues connected to the static concept of equilibrium. That is, both the classical and evolutionary approaches to game theory benefit through this interplay between them.

Replicator Dynamic models have become a primary method for studying the evolutionary dynamics in games both social, economic and ecological.
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