3 Conflict resolution in ecology and evolution: war and peace
🚧 This chapter is under construction. Content may change.
In the previous chapter we looked at the decisions of individual animals assuming that the the decisions where taken by each forager alone. For instance, for the sit-and-wait predator we assumed that the territory could be as large as the predator wanted, and that the optimal territory size was the one where the energy per unit time was maximized. But what happens if the territory is so large that it starts intruding in the territory of the neighbors? Or in other words, what happens if the optimal territory size is for instance 1 \(m^2\) , but the density of the predators is so high that there is more than 1 predator per square meter? Then there is a potential conflict between individuals and it may not be possible to have territory sizes that are optimal. More generally, one can say that the decision of an individual now affects the decisions of its conspecifics.
In evolutionary contexts, where decisions or phenotypes are determined by genes, we enter the so called realm of frequency-dependent selection. In contrasts with frequency independent selection, where the fitness of a genotype depends only on the environment, in frequence-dependent selection the fitness of a genotype depends also on the frequency of the different genotypes in the population.
The analysis of this kind of behavior conflict or evolutionary dynamics is the topic of game theory, particularly non-cooperative game theory. Some of the key concepts of game theory for animal behavior were developed by the biologist John Maynard Smith in the 1970’s and beautiful explained in his book “Evolution and the Theory of Games” Maynard Smith (1982). We now know that similar ideas were developed independently by the mathematician John Nash, twenty years earlier in the 1950’s, ideas that would lead to Nash being awarded the Nobel Prize of Economics in 1994.
In this chapter we examine three non-cooperative games that model situations of behavior or evolutionary conflict: the prisoner’s dillema, the Hawk-Dove game, and the war of attrition. They are called non-cooperative because each “player” does not know what the other “player” is going to do in the game, and the players cannot discuss the strategy that they are going to take in any cooperative game. They are also sometimes described as closed envelope games. This is, each player should write what “strategy” it adopts in each play without knowing the choice of the other player.
3.1 The prisoner’s dillema and the evolution of cooperation
A lot of the empahsis on evolutionary biology and ecology is place on competition, on the idea of “survival of the fittest” and that selection operates at the level of individuals favoring selfish behaviors. However, cooperation is super common in nature, from praire dogs taking turn as sentinels, informing the group of the arrival of a predator, to back scratching in primates. One beautiful explanation for such cooperation is based on the principles of kin selection, developed by Bill Hamilton, or the idea that it makes evolutionary sense to help others if they are genetically related to you, as this increases the probability of the same genes you have to be passed into the next generation. Hamilton’s contribution to this problem was seminal because he noted that genetic relatedness can be particularly high between females in haploid-diploid systems such as bees and ants, leading to a widespread of cooperating in rearing offspring and even the abdication of many individuals from reproducing in favor of assisting the queen. But can cooperation also evolve in the absence of genetic relatedness? This is the problem that can be studied with the prisoner’s dilemma.
In the prisionner’s dilemma game there are two players. it is inspired by a situation where two suspects of being accomplices in a crime are arrested and are being questioned in separate rooms without contact between them. They are faced with the dilemma of cooperating with each other by claimng inocence, or cutting a deal with the investigators by recognizing the responsibility for the crime and by denouncing the other player. If both cooperate they can potetinally get both away, but if one accomplice cooperates and the other defects, then the one that cooperated gets the maximum penalty.
We can formalize the prisoner’s dilemma game by stating that each player can choose one of two strategies in each round of the game, defect or cooperate. The pay-off matrix of the Prisionner’s Dilemma, from the pespective of player 1 (rows), playing against player 2 (columns) is
\[ \begin{array}{c|cc}\text{Player 1} \backslash \text{Player 2} & C & D \\ \hline C & 3 & 0 \\D & 5 & 1\end{array} \]
You can think of the pay-off points as a fitness gain in relation to the fitness before the play. The exact payoffs are not so important, and there are other variants of the game with different pay-offs. Instead, it is the relationship between the different pay-offs that determines the best strategy for the game. If the players knew what the other player would do, the best collective strategy would be to cooperate, as they both would receive 3 points for a total of 6 points. If one of the players defects while the other cooperate they accrue collectively only 5 points, while if both defect they would accrue together 2 points. But they don’t know what the other player is going to do (non-cooperative game) and need to make a decision that maximizes their individual fitness. What should they do?
The general solution that both Maynard Smith and Nash came up for these games is that one needs to identify which strategy is the best response to itself. This is called an Evolutionary Stable Strategy (ESS, also known as Nash Equilibrium1) because it is the strategy that when dominant in the population cannot be invaded by any other strategy. Mathematically speaking, the ESS is the strategy \(S^*\) that has the higher payoff when played against itself,
\[ \forall S_i ,\quad E(S^*,S^*) \geq E(S_i,S^*) \]
3.2 The Hawk-Dove game and animal contests
3.3 The war of atttrition
There’s a small difference between the two, see (Dugatkin and Reeve 1998).↩︎