The Hawks and Doves game is one of the simplest models of game theory that describes competitive relationships in a certain animal population and the development of an evolutionarily stable strategy.
Game Rules
Imagine a population of animals in which individual individuals compete with each other for a certain resource. In the simplest case, it can be male mating tournaments for the right to mate with a female. Since two males participate in the mating tournament, the tournament can be represented as a game of two participants. Suppose that, by temperament, the males fall into two groups - we will conditionally call them “Pigeons” and “Hawks”. These names are not related to the specific type of animal, but are understood in a figurative sense: hawks as a symbol of aggressiveness, and pigeons as a symbol of peacefulness. In reality, these names have nothing to do with reality: in nature, pigeons (like any other animals) are quite aggressive.
Individuals of each group have the following features. Hawks always fight to victory and retreat only if they receive serious injuries. Pigeons confine themselves to threats and a demonstration of aggressiveness, trying to psychologically crush the opponent, but if it comes to a real fight, they back down.
Thus, if a pigeon fights with a hawk, the hawk wins, but the retreating pigeon does not receive any damage in the battle and, in principle, does not lose anything. If two pigeons fight, then one of them (the one who has stronger nerves) wins, no one gets injured, but both spend a certain amount of energy on a long psychological confrontation. If two hawks fight, then one of them wins, and for the other, the battle ends in severe injuries.
Mathematical wording
To translate the game into the language of mathematics, we evaluate the results of the tournament in the form of arbitrary units (points) gained or lost by the participants. We will evaluate the victory in the tournament (the ability to leave offspring) at V = 50 points, the loss at L = 0 points, the receipt of a severe injury in W = –100 points, and the energy expenditure for a long standoff at E = –10 points.
Then, in the battle of two pigeons, one of them receives 50 points of victory and, in addition, both spend 10 points in the process of a long confrontation. Assuming that the probability of victory is the same for everyone (i.e. 0.5), we get that the average pigeon gain in the battle with another pigeon is S (G, G) = 50 ∙ 0.5 - 10 = 15 points.
In the battle between two hawks, each with a probability of 0.5 receives a gain of 50 points and with the same probability - a mutilation, which we rated at –100 points. The average gain will be S (I, I) = (50–100) ∙ 0.5 = –25 points.
In the fight of a dove with a hawk, the dove loses and gets S (G, I) = 0 points, the hawk wins and gets S (I, G) = 50 points.
The results of the tournament can be visualized in the form of the so-called payment matrix:
| Dove | Hawk | |
|---|---|---|
| Dove | 15 | 0 |
| Hawk | 50 | –25 |
Denote the fraction of hawks in the population by z, then the proportion of pigeons will be 1 – z. If two males randomly participate in the fight, then with probability z 2 they are two hawks, with probability (1 – z) 2 they are two pigeons and with probability 2z (1-z) they are pigeons against the hawk.
Find the average number of points that opponents get as a result of the battle.
A hawk with probability z fights with another hawk and receives an average of –25 points and with a probability of 1 – z with a pigeon and gets 50 points. This will average
S I (z) = –25 ∙ z + 50 ∙ (1 – z) = –25z + 50 - 50z = 50 - 75z.
Similarly for the pigeon we get
S Г (z) = 0 ∙ z + 15 ∙ (1 – z) = 15 - 15z.
We plot these equations in the coordinate axes S - z.
As can be seen from the graph, the paylines for pigeons and hawks intersect at some point defined by the ratio: 50 - 75z = 15 - 15z 60z = 35
z = 35/60 = 0.583 ...
To the right of this point (i.e., with an increase in the proportion of hawks), pigeons have an advantage, so their relative number will increase, thereby decreasing z. To the left of this point (with a decrease in the number of hawks) hawks have an advantage, so their number will increase, thereby increasing z. Thus, any shift of z from the point of equality of the winnings of pigeons and hawks causes processes that tend to return the population to the point of equilibrium. The state of the population corresponding to the equilibrium point is called an evolutionarily stable strategy.
General wording
We denote the win in case of victory in the tournament V, loss L, damage from a severe injury W, and energy costs for a long confrontation E.
Then the elements of the payment matrix can be expressed by the following relationships:
The payment matrix will look like:
| Dove | Hawk | |
|---|---|---|
| Dove | ||
| Hawk |
The average gain of hawks with their share in the population z will be
and the average win of pigeons
The population equilibrium point will be reached with the following hawk fraction: