Journal of the Operations Research Society of Japan

In this paper, non-cooperative n-person games with set payoffs are considered, where the payoffs are represented as non-empty subsets of ℓ-dimensional Euclidean space ℝ^ℓ. First, several types of set relations on the set of all non-empty subsets of ℝ^ℓ are defined. Next, several kinds of concepts of Nash equilibrium strategies for the games are defined by using the set relations. Then, the Nash equilibrium strategies are characterized by Nash equilibrium strategies for non-cooperative n-person games with scalar payoffs. The proposed theory described above can be interpreted as uncertain multi-objective game theory. We also formulate an uncertain multi-objective game.

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In this study, we examine the various extensions of the doubly nonnegative (DNN) cone, frequently used in completely positive programming (CPP) to achieve a tighter relaxation than the positive semidefinite cone. To provide tighter relaxation for generalized CPP (GCPP) than the positive semidefinite cone, inner-approximation hierarchies of the generalized copositive cone are exploited to obtain two generalized DNN (GDNN) cones from the DNN cone. This study conducts theoretical and numerical comparisons to assess the relaxation strengths of the two GDNN cones over the direct products of a nonnegative orthant and second-order or positive semidefinite cones. These comparisons also include an analysis of the existing GDNN cone proposed by Burer and Dong. The findings from solving several GDNN programming relaxation problems for a GCPP problem demonstrate that the three GDNN cones provide significantly tighter bounds for GCPP than the positive semidefinite cone.

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