Lattice-valued matrix game with mixed strategies for intelligent decision support

Game theory has been applied extensively to interpret and solve the complex and interrelated practical decision problems. The solution for these problems depends on the goals pursued by different interested parties, i.e., problems as conflict situations. Decision making approaches based on game theory have been an important and promising research direction in decision science, as well as in real-world practice. Many research approaches within this direction have been developed, but most are limited to the real-valued domain. A great amount of non-real valued domain practical game decision problems, especially the lattice-valued game, remain largely unexplored. This paper investigates the lattice-valued matrix game (including the real-valued matrix game as a special case). For decision purposes, it is an essential and indispensable step in theoretical game decision approaches to find the solutions for a matrix game; hence this work focuses on how to determine solutions of lattice-valued matrix game for decision purposes. Firstly, based on the work on lattice-valued matrix game with pure strategy, a concept of multi-dimension lattice-valued-level strategy is introduced based on a new algebra structure called the l∗-module, i.e., a lattice-ordered module with two lattice-ordered structures. Next, a concept of a mixed strategy lattice-valued matrix game is introduced and its basic properties are discussed. Finally, the necessary and sufficient condition for the existence of a solution for a mixed strategy lattice-valued matrix game is discussed, along with basic properties for the solution. The approaches and results discussed are mathematical in nature and entail fundamental research in the field of intelligent decision support. They will provide important and fundamental support for the application of a theoretical game approach in rational decisions for conflict situations, and also introduce a new branch of game-theory based decision approaches by extending real-valued game theory into lattice-value game theory.