# Evolutionary Game Theory

## The Hawk-Dove Game

### By Alex Chan Cosmos: Cluster 4 (Animal Behavior and Game Theory)

Standard game theory assumes that all players are rational beings and have all the knowledge they need to make strategically informed choices. Standard game theory also assumes that the game will be played only once. However, this does not describe the world and its population very realistically. Evolutionary game theory assumes that the players have limited knowledge about the game and play it many times. This new theory can help model biological systems and human economic systems.

In game theory, games can often times be represented as a matrix of payoffs. A simple two-dimensional matrix represents the strategies and payoffs for two competing players. Each column represents a specific strategy the column player may choose to play, and each row, a specific strategy the row player may choose to play. The ordered pair of numbers in a certain column and row correspond to the payoffs to each player if both were to play the specified strategies. In non-zero sum games, the first number in the ordered pair is the payoff to player one while the second is the payoff to player 2. In zero sum games, where the payoffs for each column and row add to zero, equilibriums can be found in the form of saddle points involving pure strategies or mixed strategies. However, most evolutionary games are not zero-sum games and don't necessarily have a defined equilibrium point.

Evolutionary game theory has two major processes that standard game theory does not include. They are the selection process in replicator dynamics, and the mutation process. The selection process is the process in which a certain strategy grows faster than all of the other strategies. There are three different types of dynamics that lead to a positive growth rate. Payoff-positive dynamics are dynamics where all pure strategies that earn above average have a positive growth rate. Convex-monotone dynamics are where a pure or mixed strategy has a higher growth rate than a pure strategy if it earns a higher payoff than the second. Weakly payoff-positive dynamics are where at least some of the strategies that earn above average have positive growth rates. There are also different ways that the selection of a strategy can take place. The three strategies mentioned by Weibull (1998) are belief-based learning, reinforcement learning, and imitation.

The Hawk-Dove game is used often by evolutionary game theorists to model the behavior of a species that contains both aggressive and non-aggressive behaviors. The rest of this website will be dedicated to explaining this game and showing different variations that can be introduced into this game. I have created a rough simulation of the Hawk-Dove game using Visual Basic that can rapidly perform the game with both the 2 basic strategies (hawk and dove), and 2 other variations of strategies (bully and retaliator).