# Game theory of Evolution – Summer

Today, I am going to examine with you, through the lens of game theory, one of the most important scientific theories, Evolution!

As we have learnt in biology class, evolution is inheritable changes of characteristics within a population overtime. While mutations are random, certain mutations will give organisms an adaptive advantage in certain environments and become more common within a population overtime. Yet, which mutations will successfully causing an evolutionary change in the gene distribution of a population?

We can answer this question using game theory. More specifically, we will consider two different mutations (flight and cooperation) respectively within a population of squirrels that are all genetically programed to “fight” when another squirrel competes with it for resources.

First, what would happen for the gene distribution in the population overtime if there’s a random mutation for “flight”?

If we assume that if both squirrels fight or flight, they would waste the existing resourse. On the other hand, if one chose to fight and the other chose flight, they can each separately find 1 unit of resource. Thus, we have the following payoff matrix:

 Squirrel 1 (left)/ 2(Top) fight flight fight (0,0) (1,1) flight (1, 1) (0,0)

The strategy set (fight, fight) is obviously not a Nash equilibrium, thus, the gene pool will change over generation and will no longer be purely “fight”. To figure out how the gene pool is going to change, let’s calculate the expected payoff for strategy fight and flight. Assume chance of mutation fight happening is a relatively small number x, then we have: Expected payoff of fight = x; Expected payoff of flight = 1-x. Because x is relatively small in the beginning, “flight” has a higher payoff. Overtime, the gene distribution in this population would gradually change and “flight” gene will become more popular, until the point at which x=1-x; x=0.5 where neither of the strategy will have a higher payoff. At this point, the strategy to flight and to fight each takes up half of the population. Similarly, if we started with a population of pure “flight” gene, we would have the same gene distribution overtime if a “fight” mutant occurs.

Let’s consider another situation. Now a mutant of “cooperation” occurs with a chance of y in the population of pure “fight” genes. What would happen now?

If two squirrels cooperate in finding resources, they would find more resources and each gain 2 units of resources. Yet, if one of them decides to fight, no resource will be gained. Thus, we have a new payoff matrix:

 Squirrel 1 (left)/ 2(Top) fight cooperate fight (0,0) (0,0) cooperate (0, 0) (2,2)

This time, we see in this matrix both (fight, fight) and (cooperate, cooperate) are Nash equilibriums. Yet, towards which Nash equilibrium will the population move?

Let’s consider the expected payoff for each strategy: Expected payoff for fight: 0; expected payoff for cooperation: 2y. Thus, because y>0, 2y>0. Cooperation will always have an adaptive advantage over fight, will produce more offspring, and eventually replace the fight gene.

When we turn the problem around, a population of pure cooperate gene will not be invaded by a “fight” mutant, because an individual will do strictly better if it does not deviate from the (cooperate, cooperate) equilibrium. Thus, any “fight mutant” will quickly be eliminated in the population. In this case, we consider the “cooperate” strategy a evolutionary stable strategy, a strategy which if adopted by a population, cannot be invaded by an alternative strategy mutation that is initially rare.

In the same situation, “fight” is not an evolutionary stable strategy because it is invaded by the cooperation mutation.

Similarly, in the earlier case of “fight” and “flight”, neither “fight” or “flight” is a evolutionary stable strategy because the situation where both players are playing “flight” or “fight” is not a Nash equilibrium and therefore gene distribution will change when a rare mutation occurs.

From all these examples, we can conclude that a strategy S is only an evolutionary stable strategy when:

(S,S) is a Nash equilibrium AND

for a mutant strategy S’, (1) expected payoff (S, S)> expected payoff (S’, S) OR

(2) expected payoff (S, S)= expected payoff (S’, S) AND expected payoff (S,S’)> expected payoff (S’,S’).

Evolution and the idea of an evolutionary stable strategy is an extension and a more complicated application of the idea of Nash equilibrium, expected payoff, and strictly/weakly dominated strategies I’ve mentioned in my previous posts. At the same time, it’s a valuable tool in analyzing phenomena related to evolution in biology.