A strategy for a payoff-switching differential game based on fuzzy reasoning

In this paper, a new concept of a payoff-switching differential game is introduced. In this new game, any one player at any time may have several choices of payoffs for the future. Moreover, the payoff-switching process, including the time of payoff switching and the outcome payoff, of any one player is unknown to the other. Indeed, the overall payoff, which is a sequence of several payoffs, is unknown until the game ends. An algorithm for determining a reasoning strategy based on fuzzy reasoning is proposed. In this algorithm, the fuzzy theory is used to estimate the behavior of one player during a past time interval. By deriving two fuzzy matrices GSM, game similarity matrix, and VGSM, variation of GSM, the behavior of the player can be quantified. Two weighting vectors are selected to weight the relative importance of the player's behavior at each past time instant. Finally a simple fuzzy inference rule is adopted to generate a linear reasoning strategy. The advantage of this algorithm is that it provides a flexible way for differential game specialists to convert their knowledge into a "reasonable" strategy. A practical example of guarding three territories is given to illustrate our main ideas.