On the solutions of lattice-valued matrix game with fuzziness

Game theory is a very important branch of applied mathematics. There have been a lot of excellent results within eighty years of its history. Many research areas have been developed, but most of them are limited to the real domain. A great amount of non-real practical game problems, especially the lattice-valued game, remains unexplored. The paper focuses on the lattice-valued matrix game. Firstly, we propose the concept of lattice-valued matrix game with pure strategy and discuss the sufficient and necessary conditions for the existence of the solution of the lattice-valued matrix game with pure strategy. Then, considering the real situation that the strategy set of the players are often a fuzzy set and the matrix are often described by fuzzy sets, we investigate the lattice-valued matrix game with fuzziness. In particular, we investigate the solution of a lattice-valued matrix game with pure strategy and fuzzy value matrix, with fuzzy strategy and classical game matrix, as well as with fuzzy strategy and the function value game matrix. It is assumed that L={L, v, ∧} is a lattice, "⩽" is the partial order in L