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

Abstract

A new solution concept for two-player zero-sum matrix games with multidimensional payoff is introduced. It is based on extensions of the vector order in R K to order relations in the power set of R K , 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, i.e. minimax and maximin strategies, for games with multi-dimensional payoffs. It is shown that-in contrast to games with one-dimensional payoffs-minimax/maximin strategies are independent from Shapley's notion of equilibrium strategies for such games. Therefore, the two concepts are combined into new equilibrium notions for which existence theorems are given. By means of examples, relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax/maximin solutions and Pareto optimal security strategies are clarified. A algorithm for computing optimal strategies is presented.

Characterizing and Computing the Set of Nash Equilibria via Vector Optimization

“…As the strategies consist of choosing the probabilities x i , the shared constraint set is X := [0, 1] 2 with real-valued strategies X 1 = X 2 = R. The cones C and C i are given in as in (9) with Z = R 4 . Note that since the objective functions are not convex, the subproblems C i -arg min{(x, f (x)) | x ∈ X}, despite having a convex ordering cone C i for i = 1, 2, are not convex vector optimization problems.…”

“…in Figure 1a and Figure 1b for Example 2.11) are the projection of the efficient frontiers onto the x component. However, these Pareto optimal points are not the projection of the upper images onto the x component; this idea is tempting, as the argument x is also part of the objective function (x, f (x)), but as the ordering cone C i (given explicitly in (9) Example 2.13. We now consider a modification to the zero-sum game presented in Example 2.11 to make it a general-sum game and such that the set of Nash equilibria will no longer be a singleton.…”

“…The Nash equilibrium in vector-valued games is also called, e.g., the Shapley equilibrium in[9], the Pareto equilibrium in[24, Chapter 11], and the Pareto Nash equilibrium[6].…”

Choosing sets: preface to the special issue on set optimization and applications

“…A complete duality theory for set-(and also vector-valued) functions could be established including satisfying duality results for multicriteria linear optimization Heyde et al (2009). New applications came within reach (see Hamel et al (2015) for an overview): for example, the complete lattice-valued duality approach produced consistent pricing processes as dual variables for financial market models with frictions Hamel et al (2011)-exactly the "right" dual variables which were used in finance before Jouini et al (2004), Kabanov (1999); new solution concepts for games with multi-dimensional payoffs Hamel and Löhne (2018) as well as quantiles for multivariate random variables Hamel and Kostner (2018) could be defined in ways which parallel the one-dimensional case much more than all previous approaches.…”

“…On the other hand, one can find examples where the supremum can be generated by dominated sets which shows that minimality/maximality (= not dominated with respect to minimization and maximization, respectively) on the one hand and attainment of the infimum/supremum on the other hand become different concepts in complete lattice-valued optimization. For games with multi-dimensional payoffs, these concepts correspond to worst-case (= maximin or minimax) approaches and are different from more traditional Nash equilibrium type concepts Hamel and Löhne (2018), Maeda (2015). It turned out that this approach is especially useful if the underlying "base" preference/order relation is not total-in contrast to Kreps' assumption. Surprisingly, for a long time mathematics has contributed very little to the theory of set relations and its applications.…”