Abstract. While both games and Multi-Objective Optimization (MOO) have been studied extensively in the literature, Multi-Objective Games (MOGs) have received less research attention. Existing studies deal mainly with mathematical formulations of the optimum. However, a definition and search for the representation of the optimal set, in the multi objective space, has not been attended. More specifically, a Pareto front for MOGs has not been defined or searched for in a concise way. In this paper we define such a front and propose a set-based multi-objective evolutionary algorithm to search for it. The resulting front, which is shown to be a layer rather than a clear-cut front, may support players in making strategic decisions during MOGs. Two examples are used to demonstrate the applicability of the algorithm. The results show that artificial intelligence may help solve complicated MOGs, thus highlighting a new and exciting research direction. I. INTRODUCTION In conventional game theory problems, the usual assumption is that decision makers (players) make their decisions based on a scalar payoff. But in many practical problems in economics and engineering, decision makers must cope with multiple objectives or payoffs. In such problems, a vector of objective functions must be considered. A MOG may be defined as a problem in which one player aims to minimize a set of objectives, while the other player aims at maximizing these objectives. As an example of such a game, consider a player who possesses a set of boats. These boats are to set sail from several possible docking locations, so that one of them will reach a target point in the shortest time possible. While sailing, the boats must escape an opponent vessel that is guarding the target and is capable of traveling twice as fast as the boats. When the vessel intercepts a boat, that boat is eliminated from the game. The MOG is defined such that the first opponent seeks to minimize the distance traveled by one of its boats (minimizing the time taken to get to the target) while also minimizing its loss of boats (sacrificed for the sake of optimization). On the other hand, the opponent's objectives are to maximize the number of intercepted boats and to maximize the time (distance) taken for any of the A. Avigad is with the Mechanical Engineering Department at Ort Braude College of Engineering, Karmiel, Israel (phone: 97249901767; fax: 97249901866; e-mail: gideona@braude.ac.il). E. M. Eisenstadt is with the Mechanical Engineering Department at Ort Braude College of Engineering, Karmiel, Israel (e-mail: erella@braude.co.il). M. C. Weiss is with the Computer Science Department at Ort Braude College of Engineering, Karmiel, Israel (e-mail: miri@braude.co.il). opponent's boats to reach the target point. The strategies adopted by the first player may include, but are not limited to, where the boats should start from and how many and which boats should be sacrificed. The strategies used by the vessel may include where to start, when to start and whether to intercept the boat closest to the target or the nearest boat. As in the case of single objective games, the notion of equilibrium can be defined in terms of unfruitful deviation, from the equilibrium strategies. According to [1], in the MOG setting this can be interpreted as follows: Deviations from the equilibrium strategies do not offer any gains to any of the pay-off functions for any of the players. All existing studies related to MOGs refer to definitions for this equilibrium (e.g., [2]). The predominant definition of this equilibrium, termed the Pareto-Nash equilibrium, has been suggested in [3]. The Pareto-Nash equilibrium uses the concept of cooperative games, because according to this notion, sub-players under the same coalitions should, according to the Pareto notion, optimize their vector functions on a set of strategies. On the other hand, this notion also takes into account the concept of non-cooperative games, because coalitions interact on the set of situations and are interested in preserving the Nash equilibrium between coalitions. An extension to the work in [3] may be found in [1], where several algorithms have been suggested for searching for these equilibrium strategies. MOGs are commonly solved by linear programming, e.g. [4], in which a multiple objective zero-sum game is solved. A non-zero-sum version of a MOG has been solved by Contini [5]. Detailed algorithms for finding Pareto Nash equilibrium-related strategies may be found in [1] and [3]. In all of these cases, weights are altered in order to search for one equilibrium solution at a time. This means the algorithms must be executed sequentially in order to reveal more equilibrium points. Artificial intelligence-based approaches have been applied to MOGs within the framework of "fuzzy multi-objective games." In such studies (e.g., [6]) the objectives are aggregated to a surrogate objective in which the weights (describing the players' preferences towards the objectives) are modeled through fuzzy functions. II. BACKGROUND A. MOO and EMO The topic of MOO concerns the search for solutions to many real world problems, dubbed appropriately enough Multi-objective Problems (MOPs). In cases of contradicting objectives, there is no universally accepted definition of an Optimal Strategies for Multi Objective Games and Their Search by Evolutionary Multi Objective Optimization G. Avigad, E. Eisenstadt, M. Weiss Cohen 978-1-4577-0011-8/11/$26.00 ©2011 IEEE 166