Strategic reasoning in compositional games

The central innovation introduced by game theory is its strategic dimension. A player’s environment is not neutral, and she expects that other players will try to out-guess her plans. Reasoning about such expectations and strategizing one’s own response accordingly constitutes the main logical challenge of game theory. Evaluation games have been long used by logicians to define the semantics of various logics. For instance, in first order logic, given a structure A and a formula α, verifier claims that A |= α whereas falsifier claims that this is not true. The game positions are subformulas of α. Every disjunct is associated with the verifier who picks one of the disjuncts whereas a conjunct is associated with the falsifier. Similarly, existential and universal quantifiers are associated verifier and falsifier respectively. For negation the players switch roles. It can be shown that in this two player zero sum game, the verifier has a winning strategy iff A |= α. To logically reason about games, one would like to present games in a compositional manner. Inspired by evaluation games, the natural game operators would be choice, sequential composition, dual and iteration of sequential composition. Game logic [Par85] is a logic to reason about such determined two person zero sum games. Semantics of the logic is defined in terms of neighbourhood functions which represents the outcomes players can enforce in the game. The logic makes assertions about composing neighbourhoods or abilities of players. The role of strategies in game logic is limited to just expressing the abilities of players and strategies themselves do not figure in the logical framework.