# Game Theory and Agents

Abstract: A fundamental problem in multi agent systems is conflict resolution. A conflict occurs in general when the agents have to deal with inconsistent goals, such as a demand for shared resources. We investigate how a game theoretic approach may be a helpful and efficient tool in examining a class of conflicts in multi agent systems. We consider the hawk-and-dove game both from an evolutionary and from an iterated point of view. An iterated hawk-and-dove game is not the same as an infinitely repeated evolutionary game because in an iterated game the agents are supposed to know what happened in the previous moves. In an evolutionary game evolutionary stable strategies will be most successful but not necessarily be a unique solution. An iterated game can be modeled as a mixture of a prisoner's dilemma game and a chicken game. These kinds of games are generally supposed to have successful cooperating strategies. We also discuss situations where a chicken game (CG) is a more appropriate model than a prisoner's dilemma (PD) game and describe a simulation of iterated prisoner's dilemma (IPD) and iterated chicken (ICG) games. We study a parameterized class of cooperative games, with these classical games as end cases, and we show that chicken games to a higher extent reward cooperative strategies than defecting strategies. We then introduce generous, even-matched, and greedy strategies as concepts for analyzing games. A two person PD game is described by the four outcomes (C,D), (C,C), (D,C), and (D,D), where the outcome (X,Y) is the probability of that the opponent acts Y, when the own player acts X. In a generous strategy the proportion of (C,D) is larger than that of (D,C), i.e. the probability of facing a defecting agent is larger than the probability of defecting. An even-matched strategy has the (C,D) proportion approximately equal to that of (D,C). A greedy strategy is an inverted generous strategy. The basis of the partition is that it is a zero-sum game given that the sum of the proportions of strategies (C,D) must equal that of (D,C). In a population simulation, we compare the PD game with the CG, given complete as well as partial knowledge of the rules for moves in the other strategies. In a traffic intersection example, we expected a co-operating generous strategy to be successful when the cost for collision was hig h in addition to the presence of uncertainty. The simulation indeed showed that a generous strategy was successful in the CG part, in which agents faced uncertainty about the outcome. If the resulting zero-sum game is changed from a PD game to a CG, or if the noise level is increased, it will favor generous strategies rather than an even-matched or greedy strategies. Four different PD like games were studied by running a round robin as well as a population tournament, using 15 different strategies. The results were analyzed in terms of definitions of generous, even-matched, and greedy strategies. In the round robin, PD favored greedy strategies. CG and coordinate game were favoring generous strategies, and compromise dilemma the unstably even-matched strategy Anti Tit-for-Tat. These results were not surprising because all strategies used were fully dependent on the mutual encounters, not the actual payoff values of the game. A population tournament is a zero-sum game balancing generous and greedy strategies. When strategies disappear, the population will form a new balance between the remaining strategies. A winning strategy in a population tournament has to do well against itself because there will be numerous copies of that strategy. A winning strategy must also be good at resisting invasion from other competing strategies. These restrictions make it natural to look for winning strategies among originally generous or even-matched strategies. For three of the games, this was found true, with original generous strategies being most successful. The most diverging result was that compromise dilemma, despite its close relationship to PD, had two greedy strategies almost entirely dominating the population tournament. In game theory, iterated strategic games are considered harder to analyze than repeated games (for which the theory of mixed strategies is a suitable tool). However, iterated games are in many cases more fit to describe the situation of computerized agents, since it take into account previous moves of the opponents, rather than just assigning each possible action a certain probability. We introduce the notion of characteristic distributions and discuss how it can be used to simplify and structure the analysis of strategies in order to provide a good basis for choosing strategies in games to come and formulate a No free lunch theorem for game theory.

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