Equilibrium Points in Nonzero-Sum n-Person Submodular Games

A submodular game is a finite noncooperative game in which the set of feasible joint decisions is a sublattice and the cost function of each player has properties of submodularity and antitone differences. Examples of submodular games include 1) a game version of a system with complementary products; 2) an extension of the minimum cut problem to a situation where players choose from different sets of nodes and perceive different capacities, with special cases being a game with players choosing whether or not to participate in available economic activities and a game version of the selection problem; 3) the pricing problem of competitors producing substitute products; 4) a game version of the facility location problem; and 5) a game with players determining their optimal usage of available products. A fixed point approach establishes the existence of a pure equilibrium point for certain submodular games. Two algorithms which correspond to fictitious play in dynamic games generate sequences of feasible joint decisions converging monotonically to a pure equilibrium point. Bounds show these algorithms to be very efficient when the set of feasible decisions is finite. An optimal decision for each player is an isotope function of the decisions of other players.