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Convergence of behaviour rules in iterated matrix games Patrick, Jonathan

Abstract

This master's thesis reports on a foray into Game Theory, focusing solely on the twoperson (not necessarily zero-sum) game. Primarily, I am interested in the convergence properties of different behaviour rules and how one might proceed to introduce some form of learning into the strategies of the players involved in the game. Therefore, I begin with the introduction of some key equilibrium sets - namely the set of Nash Equilibria (NE), the set of correlated equilibria (CE) and the marginal best-response set (MBR). I briefly discuss the relationship between these three sets before moving on to describe some desirable properties of behaviour rules. From there, I introduce six behaviour rules (four from the literature, two original) that attempt to incorporate some form of learning into the game. The four from the literature are Fictitious Play, Exponential Fictitious play, Regrets 1 and Regrets 2. I have named the two original behaviour rules Past Response and Modified Regrets. I then move on to describe the convergence properties of each. This thesis was originally motivated by a talk given by Andreu Mas-Collel on the properties of the two Regrets-based behaviour rules. Thus, a fair amount of time is spent reworking the convergence proofs of both Regretsl and Regrets2 as they were developed by Mas-Collel and Sergiu Hart. I then suggest an alternative proof of the Regretsl convergence properties. I close off the paper with some numerical results from three games - a zero-sum game, a game developed by Lloyd Shapley (called the Shapley game) and a game called Battle of the Buddies. They are designed to give some numerical confirmation of the convergence theorems stated earlier in the paper as well as some indication as to where further study might be useful.

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