Structured strategies in games on graphs

1 Summary We discuss strategies in non-zero sum games of perfect information on graphs. The study of non-zero sum games on graphs is motivated by the advent of computational tasks on the worldwide web and related security requirements which have thrown up many interesting areas of interaction between game theory and computer science. For example, signing contracts on the web requires interaction between principals who do not know each other and typically distrust each other. Protocols of this kind which involve selfish agents can be easily viewed as strategic games of imperfect information. These are complex interactive processes which critically involve players reasoning about each others' strategies to decide on how to act. In the case of interactng web services, these games involve infinite plays as well. Developing a game theoretic computational study of such interactions is an interesting challenge. Admittedly, these are games of partial information, but a theoretical analysis is interesting even in the more restricted case of perfect information. On one hand, zero sum games on graphs have been extensively studied in logic and automata theory ([GTW02]), and on the other, a rich theory of non-zero sum matrix form games has been developed by game theorists ([OR94]). We call graph games large, to indicate that plays consist of (long) sequences of moves, whereas matrix form games are termed small, in the sense that a play is typically one simultaneous move. We can have matrix form presentations for sequential plays as well, but not very usefully for analysis. While one talks of winning strategies in win / loss games, when players have overlapping objectives, we consider the best response each player can offer to moves of other players. In a small game which consists of both players deciding on a move simultaneously, it is best analyzed by considering pairs of moves. When we have a pair