Long-run behavior in games

This paper has two purposes. The first is to describe the existing theory of long-run behavior of populations playing a normal-form game. In this paper the emphasis is on symmetric 2x2 games which is the most analytical convienent. The methods here relies on that agents are not fully rational, they can make mistakes when playing. The reason for these possible mistakes or perturbations are not specified but it can be random experementation or just ignorance. When studying these processes the theory of Markov chains becomes useful. When assuming that there is one population playing, the dynamics of the distribution of strategies is a one-dimensional Markov chain. By using standard theory or the Markov chain tree theorem we can deduce a limiting distribution for the process(we let time go towards infinity). This limiting distribution will be a function of the perturbations mentioned earlier. Then, if we let the pertubations tend to zero, it will often be the case that the probability of the process being in a specific state is much higher than for all the other states. This idea leads to the concept of stochastic stability which was introduced by Foster and Young,1993. This concept gives a prediction of how the behavior of the population will be in the long-run i.e which strategies of the game agents are most likely to play. In 2x2 games there is a link between the risk dominant equlibrium and the stochastically stable state and this is used to verify the results when examples of the use of the theory is presented. The second purpose of this paper is to extend the existing theory. The extension here is that we let two populations play against each other in the game. We assume that one population operates the rows and the other the columns. This calls for a different theoretical approach but still the theory of Markov chains is important. The best-response dynamics is affected by the distribution of strategies in the other population. When we let the time horizon tend to infinity we can again compare the probabilities of being in the different states as the perturbations go to zero. The stochastically stable state, which will be pairs of distributions in the two populations, shows which strategy both populations will play in the long-run. This state corresponds to the predicton we got in the one population case. The approach here is totally theoretical. The methods used is game theory,mathematics and statistics. When new concepts or theorems needed to find the