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Unformatted text preview: Notes on evolutionary game dynamics Patrick De Leenheer * February 16, 2009 These notes describe some basic concepts from game theory (Nash and strict Nash equilib- rium and evolutionary stable strategies), and how these lead to a remarkable equation called the replicator equation which has been used as a model for evolution. They are based largely on . Example : The hawk-dove game. When animals engage in a conflict (over mates, land ,food etc) they may pick one of two strategies. Either they behave as hawks in which case they fight until one of them gets injured or the opponent flees. Or they behave as doves in which case they may display aggressive behavior, but they retreat as soon as their opponent shows signs of escalating the conflict into a fight. Assume that the winner of the contest gains G > 0 (a mate, some land, some food), and injury leads to a loss of I > 0 (no mate, no food, scratches, a decreased level of self-confidence etc). We assume that the cost of an injury is larger than the value of the gain: G- I < . Let us consider the outcome of the 4 possible conflict situations: 1. When two hawks meet, there will be a fight with one winner and one loser, and the expected payoff for each hawk is ( G- I ) / 2. 2. When a hawk meets a dove, the dove bails out, and the hawk receives G > 0. 3. When a dove meets a hawk, the dove flees, and gets nothing, but also does not get harmed, so his payoff is 0. 4. When a dove meets a dove, he might or might not flee, and his expected gain will be G/ 2. We summarize these outcomes in the following expected payoff matrix : A = G- I 2 G G 2 =- ++ + The first row contains the expected payoffs of a hawk against a hawk and dove respectively. Similarly, the second row contains the payoffs of a dove against a hawk and dove respectively. Now suppose you are the first player (the row player), and you wish to decide which strategy to pick. Of course, your pick should be such that you maximize your expected payoff. If you knew that your opponent was a hawk strategist, you would obviously pick the dove strategy since > ( G- I ) / 2. On the other hand, if you knew that your opponent was a dove strategist, you would pick the hawk strategy since G > G/ 2. But in reality you often don’t know your opponent’s strategy, and then it is not clear which strategy you should pick. Moreover, the game might take place repeatedly, and you could learn your opponent’s strategy by observing his or her behavior. It would not be wise of your opponent to fix his strategy and stick to it forever after. Smarter would be to behave as a hawk or a dove with a certain probability. Hawk and dove strategies are examples of pure strategies . Someone playing hawk 50% of the time and dove 50% is an individual who is playing a mixed strategy which can be represented as a probability vector p = . 5 . 5 . There are many other possible strategies of course. In fact we * Email: [email protected] Department of Mathematics, University of Florida....
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- Summer '06
- Game Theory, Nash equilibria, John Nash, Replicator equation