Sorry about the absence last week. I fell ill and was unable to write anything for that week. This post was originally supposed to appear last Monday.
Prisoner’s Dilemma. Battle of the Sexes. Chicken. Matching Pennies. Stag Hunt. Rock, Paper, Scissors. All the various types of auctions. I have just listed before me a series of games. Not games in the conventional sense of the word (except for Rock, Paper, Scissors) although all can be played as such. These games are instead classic examples of mathematical structures. And the mathematics used to analyze these structures is called game theory. Game theory is a way of understanding how individuals with a set strategies choose and use their strategies to come out with the highest payoff possible. It is a powerful tool used to understand social situations but also has a lot of relevance to evolutionary theory. After all, what is evolutionary theory than individuals of a species (or two different species) competing with their own strategies to come on top (in this case, a higher net reproduction rate)? Game theory has spawned an entire subdivision of the field of evolutionary biology. And one of the classic evolutionary games is called Hawk-Dove.
In 1973, John Maynard-Smith and George Price, an aeronautical engineer and chemist respectively before becoming evolutionary biologists, wrote a Nature paper which first described the Hawk-Dove game. In the Hawk-Dove game, two individuals of a certain species meet over a resource. When they meet, there are one of two possible strategies: Hawk or Dove. Dove are peaceful. Hawks are fighters. Depending on which two meet, then the resources are divided in a specific manner. When two doves meet, they share the resources in an equal manner. When a dove meets a hawk, the hawk gets all the resource while the dove gets nothing. And when a hawk meets a hawk, they fight, with a cost, and the resource goes to the winner. To make the analysis easier, let us assume all hawks are identical in their ferocity, and therefore the outcome for a hawk meeting another hawk is a one-half chance of winning the resource minus the cost of fighting (Further clarification: the losing hawk gets nothing but suffers no costs. Realistically, both would receive a cost but mathematically, it does not matter).Now, let us assume there is a population of this species, each with a randomly chosen mixture of hawks and doves. What will be the final ratio of hawks to doves in the population?
Evolutionary game theory has a unique aspect to it that makes it different from other game theory analyses: it is dynamic. Game theory is often taught in terms of static games; you choose a strategy and then get what you get. Maybe you might get a second chance to choose a different strategy but the rewards don’t carry. In evolutionary game theory though, the rewards from each game carry. The rewards are built up and then translated into offspring. Therefore, in a evolutionary context, the worst strategy to be used in a game will quickly disappear from the population while the best strategy will soon take over the population, i.e. natural selection. So between the hawk and the dove, which is the worst strategy? Well, let us think about this logically. Let us assume that I am a special individual in the population. I don’t have a fixed strategy; I can choose whichever strategy I want. And let us also assume that I can know an individual’s strategy just by looking at them, so that I can choose my strategy tailored to my opponent before we even make a move. If I meet a dove, would I choose to be a hawk or a dove? Well, if I choose to be a dove, I share the resource with my opponent equally; if I choose to be a hawk, then I get the entire resource. Clearly, when meeting a dove, I always want to be a hawk. What about when meeting a hawk? What do I choose then? If I choose to be a dove, I get nothing, and if a choose to be a hawk, then it is a coin flip whether I get nothing or the resource minus the cost of fighting. On average, that comes out to on half of the resource minus the cost. “Hawk,” you may say. “It is always better to be a hawk! You always come out with something.” That is true. And certainly a population of nothing but hawks can be a final state. But only if the benefits of the resource outweigh the cost of fighting. When the costs outweigh the resource benefits, it is better to be a dove when meeting a hawk. Why would one start a bar fight over a nickle?
So what is the final population state? Well, that depends. If the resource being fought over is greater than the costs of fighting, then the hawks of the population get higher rewards and have more offspring than the doves. In this scenario, the percentages of hawks in the population increases until all members are hawks. No dove can do better than any hawk and therefore cannot establish themselves in this population (Hawks are said to have a unique strategy known as an evolutionary stable strategy, or ESS. I’ll talk about this in another post later on). What if the costs are larger than the resource? Then a really special solution occurs. To analyze this case, let’s further subdivide it into two. First, assume that we have a population of nearly all hawks but only a few doves. Both doves and hawks interact with other hawks nearly 100% of the time. The hawks will come out with negative payoff in each interaction, while the doves will come out with nothing in each interaction. The payoffs refer to the change in offspring number from the previous generation. Negative means less offspring, 0 means the same amount of offspring, and positive means more offspring. Therefore, while the number of doves remains stable, the number of hawks decline and the percentage of doves rises. Now the second scenario, the mirror of the first with all doves and few hawks. In this case, the payoff for each dove become half the resource while the payoff for the hawk is the entire resource. Doves grow in number, but hawks grow even faster. The percentage of hawks in this population will climb. So, in a population of all hawks, doves grow in percentage. In a population of all doves, hawks grow in percentage. If neither state is an absolute, then the final state must be some mixture of hawks and doves. And using mathematics, we can show that the final proportion of hawks will be the value of the resource divided by the cost of fighting (the proportion of doves is just 1 minus that). So if the resource is worth 10 but the cost of fighting is 20, then one-half or 50% of the population will be hawks and the other 50% will be doves. If the population deviates from these percentages in any way, then it will always return to this specific population structure in due time. In this case, how well you do depends on who your neighbours are.
The Hawk-Dove is perhaps the most widely recognized evolutionary game in all of game theory. When it first came out, it introduced concepts to evolution, such as the ESS, which would have radical implications for the field of biology. It have been included further to include a whole host of new and varied strategies such as retaliation and bourgeois behavior (acting like a dove outside but a hawk at home) which have continued to expand our knowledge of the field. And it started the entire field of evolutionary game theory, which offers a system of analysis that allows us to understand which strategies may be important and why. Hawk-Dove truly is the ultimate evolutionary game.