Lattice-valued matrix game with mixed strategies for intelligent decision support Yang Xu a,⇑ , Jun Liu b , Xiaomei Zhong a , Shuwei Chen b,c a School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, PR China b School of Computing and Mathematics, University of Ulster, Northern Ireland, UK c School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, Henan, PR China article info Article history: Available online 13 October 2011 Keywords: Lattice-valued matrix game Lattice Decision support system Multi-agent system Knowledge-based system l / -Module Mixed strategy abstract 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 / -mod- ule, 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 math- ematical 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. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Game theory is a very important branch of applied mathemat- ics, which is a mathematical tool for making rational decision in conﬂict situations. The conﬂict includes players who select various strategies from a set of available strategies. While the conﬂicting payoffs may put the players at cross purposes, there may also be room for cooperation between them. The interactive situation, speciﬁed by the set of participants, the possible courses of action of each player, and the set of all possible utility payoffs, is called a game [1]. Game theory provides general methods of dealing with interac- tive optimization problems in decision science; its methods and concepts, particularly the notion of strategy and strategic equilib- rium ﬁnd a vast number of applications throughout economics and a wide range of social and behavioral sciences (including biol- ogy) [24,27,37,39,43,45], e.g., in optimization or decision-making problem, whenever an optimizing player expects a reaction from other players to his own actions, his payoff is determined by other player’s actions as well, and he is playing a game. A player’s strategy is a complete plan of actions to be taken when the game is actually played; it must be completely speciﬁed before the actual play of the game starts, and it prescribes the course of play for each decision that a player might be called upon to take, for each possible piece of information that the player may have at each time where he might be called upon to act [37]. In summary, game theory attempts to abstract essential elements of such competitive situations, put them in mathematical models, and use these models for decision-making. The game theory has been served as a conceptual and mathematical foundation to ex- plore a rational side of human decision making [4]. With the increasing complexity of decision situations and in- creased requirements for advanced knowledge and new intelligent techniques to support and enable better decisions, advanced knowledge-based methods and intelligent models have become a necessary component in current advanced decision support sys- 0950-7051/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2011.08.019 ⇑ Corresponding author. E-mail addresses: xuyang@home.swjtu.edu.cn (Y. Xu), j.liu@ulster.ac.uk (J. Liu), zhongxm@126.com (X. Zhong), chensw915@gmail.com (S. Chen). Knowledge-Based Systems 32 (2012) 56–64 Contents lists available at SciVerse ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys