Noncooperative Games in a Fuzzy Environment
Pp. 238-260 (23)
Andre A. Keller
Noncooperative games correspond to situations where two players or more
are in conflict. Each player has mixed strategies, given payoffs and an objective
function. Data of the game may be either crisp numbers or fuzzy numbers. The game is
equivalent to a minimax bilinear programming problem (BLP). A Nash equilibrium
solution is achieved if the objectives of the two players are fulfilled simultaneously.
Different techniques can be used to achieve equilibrium solutions, such as with the
Mangasarian-Stone optimality conditions, or with the Lemke-Howson pivot algorithm.
A single-objective bimatrix game is first explored in a crisp environment. These nonzero
sum games are such that the sum of payoffs differs from zero for each pair of pure
strategies. Thereafter, a fuzzy context can be considered as possible imprecise data and
vague statement of the players. Equilibrium is defined w.r.t. the degree of attainment of
fuzzy goals. Numerical examples are solved in both crisp and fuzzy environments.
Multi-objective matrix games (or zero-sum games) describe situations where the players
have several objectives. Such problems use equivalent programming problems. These
multi-objective games can also be placed in a fuzzy environment where the goals are
fuzzy. Thereafter, more general multi-objective bimatrix games are developed in both
crisp and fuzzy environment. Numerical examples illustrate different game situations.
Aggregation method, Bilinear programming, Bimatrix game,
Complementary pivot algorithm, Degree of attainment, Equilibrium solution,
Fuzzy decision rule, Expected payoff, KKT optimality conditions, Minimax
Theorem, Mixed strategies, Multi-objective bimatrix game, Multi-objective
matrix game, Nash equilibrium, Noncooperative game, Normal form, Payoff
matrix, QP equivalent problem, Value of a game.