Payoff cellular automata and reflexive games

Abstract

In the paper I introduce the notion of payoff cellular automata instead of payoff matrices. By using these automata we can formalize contextbased reflexive games for k players on different finite or infinite levels of reflexion. Each player possesses the own decision rule defined by a Boolean function on his/her payoffs within a context which continuously changes. They are rules of the zero level of reflexion. If player 1 follows a decision rule which is a Boolean combination with player 2’s decision rule of the zero level of reflexion, then we say that player 1’s decision rule is of the first level. Meanwhile, if player 2 follows a decision rule which is a Boolean combination with player 1’s decision rule of the first level of reflexion, then we say that player 2’s decision rule is of the second level, etc. I suppose that at different time step t, players can change their decision rules, as well. So, under these conditions reflexive games can be very sophisticated, but payoff cellular automata allow us to formalize them.