ACCEPTABLE-AND-ATTRACTIVE FINITE APPROXIMATION OF CONTINUOUS NONCOOPERATIVE TWO-PERSON GAMES ON A PRODUCT OF SINUSOIDAL STRATEGY FUNCTIONAL SPACES

Анотація

A noncooperative two-person game is a model of an economic interaction process considered on a short interval. A whole process, which is a stack of a multitude of the short intervals, is modeled as a series of such noncooperative games. The series of the games is required to be solved without delay, so a method of the finite approximation of continuous noncooperative two-person games is presented. The method is based on sampling the functional spaces, which serve as the sets of pure strategies of the players. The pure strategy is a sinusoidal function of time, in which the phase lag is variable. The spaces of the players’ pure strategies are sampled uniformly so that the resulting finite game is a bimatrix game whose payoff matrices are square. The approximation procedure starts with not a great number of intervals. Then this number is gradually increased, and new, bigger, bimatrix games are solved until an acceptable solution of the bimatrix game becomes sufficiently close to the same-type solutions at the preceding iterations. The closeness is expressed in terms of the respective functional spaces, in which the player’s strategies at the succeeding iterations should be not farther from each other than at the preceding iterations. These requirements are transformed into the relaxed conditions which allow sampling the players’ sets of pure strategies: the respective distance polylines are required to be decreasing on average once they are smoothed with respective polynomials of degree 2, where the parabolas must be having negative coefficients at the squared variable. The acceptable solution is not only a situation, but rather a sub-rectangle of situations, associated with just the phase lags, defined by its center, which is the most attractive situation for the players.