Bicriterion differential games with qualitative outcomes

Combat games are studied as bicriterion differential games with qualitative outcomes determined by threshold values on the criterion functions. Survival and capture strategies of the players are defined using the notion of security levels. Closest approach survival strategies (CASS) and minimum risk capture strategies (MRCS) are important strategies for the players identified as solutions to four optimization problems involving security levels. These are used, in combination with the preference orderings of the qualitative outcomes by the players, to delineate the win regions and the secured draw and mutual kill regions for the players. It is shown that the secured draw regions and the secured mutual kill regions for the two players are not necessarily the same. Simple illustrative examples are given.This paper is based partially on research supported by the Council of Scientific and Industrial Research, India, through a Research Associateship Grant to the second author.     

Solution concepts in continuous-kernel multicriteria games

This paper considers nonzero-sum multicriteria games with continuous kernels. Solution concepts based on the notions of Pareto optimality, equilibrium, and security are extended to these games. Separate necessary and sufficient conditions and existence results are presented for equilibrium, Pareto-optimal response, and Pareto-optimal security strategies of the players.This paper is based partially on research supported by the Council of Scientific and Industrial Research, India, through a Research Associateship Grant to the first author.The authors are grateful to two anonymous referees for suggesting useful changes and pointing out some errors in a previous draft.     

Games with Ordered Outcomes

We present a brief review of the most important concepts and results concerning games in which the goal structure is formalized by binary relations called preference relations. The main part of the work is devoted to games with ordered outcomes, i.e., game-theoretic models in which preference relations of players are given by partial orders on the set of outcomes. We discuss both antagonistic games and n-person games with ordered outcomes. Optimal solutions in games with ordered outcomes are strategies of players, situations, or outcomes of the game. In the paper, we consider noncooperative and certain cooperative solutions. Special attention is paid to an extension of the order on the set of probabilistic measures since this question is substantial for constructing the mixed extension of the game with ordered outcomes. The review covers works published from 1953 until now.     

Vector linear programming in zero-sum multicriteria matrix games

In this paper, a multiple-objective linear problem is derived from a zero-sum multicriteria matrix game. It is shown that the set of efficient solutions of this problem coincides with the set of Paretooptimal security strategies (POSS) for one of the players in the original game. This approach emphasizes the existing similarities between the scalar and multicriteria matrix games, because in both cases linear programming can be used to solve the problems. It also leads to different scalarizations which are alternative ways to obtain the set of all POSS. The concept of ideal strategy for a player is introduced, and it is established that a pair of Pareto saddle-point strategies exists if both players have ideal strategies. Several examples are included to illustrate the results in the paper.     

Nondominated Nash points: application of biobjective mixed integer programming

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.     

A differential game without a qualitative saddle point

Saddle points are defined for two-person differential games in which the players have opposing preference orderings over lotteries on a set of qualitative objectives, rather than numerical payoff functions. A simple example is then given of a game without such a qualitative saddle point.     

Qualitative analysis of secured outcome regions for two-target games

Qualitative (game of kind) outcomes of two-target games are analyzed in this paper, under both the zero-sum and nonzero-sum preference ordering of outcomes by the players. The outcome regions of each player are defined from a security standpoint. The secured draw and mutual-kill regions of a player depend explicitly on his preference ordering of outcomes and should be constructed separately for each player, especially in a nonzero-sum game. General guidelines are presented for identifying the secured outcome regions of players in a class of two-target games that satisfy an Isaacs-like condition, in terms of the qualitative solutions of the two underlying single-target pursuit-evasion games. A construction has been proposed for obtaining the qualitative solution of a large class of two-target games. Illustrative examples are included.This work was done while the first author was a Research Associate in the Department of Electrical Engineering at the Indian Institute of Science, Bangalore, and was financially supported by the Council of Scientific and Industrial Research, Delhi, India.     

A sufficiency condition for coalitive pareto-optimal solutions

In ak-player, nonzero-sum differential game, there exists the possibility that a group of players will form a coalition and work together. If allk players form the coalition, the criterion usually chosen is Pareto optimality whereas, if the coalition consists of only one player, a minmax or Nash equilibrium solution is sought.In this paper, games with coalitions of more than one but less thank players are considered. Coalitive Pareto optimality is chosen as the criterion. Sufficient conditions are presented for coalitive Pareto-optimal solutions, and the results are illustrated with an example.     

Multicriteria Goal Games

In this paper, we deal with multicriteria matrix games. Different solution concepts have been proposed to cope with these games. Recently, the concept of Pareto-optimal security strategy which assures the property of security in the individual criteria against an opponent's deviation in strategy has been introduced. However, the idea of security behind this concept is based on expected values, so that this security might be violated by mixed strategies when replications are not allowed. To avoid this inconvenience, we propose in this paper a new concept of solution for these games: the G-goal security strategy, which includes as part of the solution the probability of obtaining prespecified values in the payoff functions. Thus, attitude toward risk together with payoff values are considered jointly in the solution analysis.     

Linear-quadratic,two-person,zero-sum differential games: Necessary and sufficient conditions

We consider linear-quadratic, two-person, zero-sum perfect information differential games, possibly with a linear target set. We show a necessary and sufficient condition for the existence of a saddle point, within a wide class of causal strategies (including, but not restricted to, pure state feedbacks). The main result is that, when they exist, the optimal strategies are pure feedbacks, given by the classical formulas suitably extended, and that existence may be obtained even in the presence of a conjugate point within the time interval, provided it is of a special type that we calleven.The partial support of the Trieste Unit of the GNAS, Italian CNR, is gratefully acknowledged.     

Common resources, optimality and taxes in dynamic games with increasing number of players

In the paper we examine the problem of exploitation of a common renewable resource. We use two kinds of models of this problem: games with finitely many players and games with a continuum of players. Equilibria in both cases are calculated and the payoffs are compared with payoffs in the situation in which each player behaves as a single owner of the resource. Various concepts of optimality are considered: optimality in the sense of property, Pareto optimality, maximization of a social welfare function of specific type and environmental sustainability. Another issue is the problem of enforcement of assumed profiles by so-called “linear” tax systems. Special attention is paid to the comparison of games with finitely many players and their continuum-of-players limit game.     

Nondominated equilibrium solutions of a multiobjective two-person nonzero-sum game in extensive form and corresponding mathematical programming problem

In most of studies on multiobjective noncooperative games, games are represented in normal form and a solution concept of Pareto equilibrium solutions which is an extension of Nash equilibrium solutions has been focused on. However, for analyzing economic situations and modeling real world applications, we often see cases where the extensive form representation of games is more appropriate than the normal form representation. In this paper, in a multiobjective two-person nonzero-sum game in extensive form, we employ the sequence form of strategy representation to define a nondominated equilibrium solution which is an extension of a Pareto equilibrium solution, and provide a necessary and sufficient condition that a pair of realization plans, which are strategies of players in sequence form, is a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. Finally, giving a numerical example, we demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

A new solution concept for two-player zero-sum matrix games with multi-dimensional payoffs is introduced. It is based on extensions of the vector order in \(\mathbb {R}^d\) to order relations in the power set of \(\mathbb {R}^d\), so-called set relations, and strictly motivated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates for games with multi-dimensional payoffs. It is shown that–in contrast to games with one-dimensional payoffs–the corresponding strategies are different from equilibrium strategies for games with multi-dimensional payoffs. The two concepts are combined into new equilibrium notions for which existence theorems are given. Relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax and maximin solutions as well as Pareto optimal security strategies are clarified.     

Formulation and analysis of combat problems as zero-sum bicriterion differential games

In this paper, we present a formulation and analysis of a combat game between two players as a zero-sum bicriterion differential game. Each player's twin objectives of terminating the game on his own target set, while simultaneously avoiding his opponent's target set, are quantified in this approach. The solution in open-loop pure strategies is sought from among the Pareto-optimal security strategies of the players. A specific preference ordering on the outcomes is used to classify initial events in the assured win, draw, and mutual kill regions for the players. The method is compared with the event-constrained differential game approach, recently proposed by others. Finally, a simple example of the turret game is solved to illustrate the use of this method.     

Existence of minmax points in discontinuous strategic games

In this note we prove the existence of minmax points for strategic form games where the sets of strategies are topological spaces and the payoff functions satisfy conditions weaker than continuity. The employed tools are the class of transfer weakly upper continuous functions and the class of weakly lower pseudocontinuous functions. An example shows that our result is of minimal character.     

A dynamical method for the calculation of Nash-equilibria in n-person games

In this paper, we are concerned with the calculation of Nash-equilibria in non-cooperative n-person games. For this purpose, we construct a continuous mapping of the Cartesian product of the strategy sets of the players into itself such that the fixed points of this mapping are Nash-equilibria. This gives rise to an iteration method for the calculation of fixed points of this mapping which leads to Nash-equilibria, if it converges. As important special cases Bi-matrix games and evolution matrix games are considered.     

A general solution concept for two-person,zero-sum games

In this paper, finitely additive mixed extensions of two-person zero-sum games are studied, where the players use probability contents as mixed strategies. It is well-known that symmetric finitely additive mixed extensions always have a saddle point, a finitely additive solution. We generalize this solution concept and assign a finitely additive solution to every bounded two-person, zero-sum game.