Existence of nash equilibrium in generalized b-metric spaces via multi-valued fixed-point theorems

In this paper, the existence problem for Nash equilibrium is restated as a fixed point problem for a multivalued mapping in a generalized b-metric strategy profile space with metric contractivity rather than the traditional topological properties of compactness, convexity, and continuity. The strategy space is endowed with a complete b-metric with constant s≥1, and the set-valued best response mapping Bis closed and bounded-valued with Pompeiu-Hausdorff contraction property H_d (B,(x)B(y) )≤λd(x,y). The central result establishes the compatibility threshold s√λ<1, at which Bhas a unique fixed point identical to a Nash equilibrium. The bound provides the trade-off between the level of contraction and the magnification caused by relaxing the triangle inequality, reverting to the well-known condition λ<1 when s=1. Based on this existence result, the article provides a geometric rate of convergence for best response iterations, a perturbation stability result for best response correspondences that are uniformly close, and a Tykhonov-type well-posedness result with extra assumptions. A two-person game with true set-valued best responses is solved analytically as an example for validating the assumptions, with some sensitivity analysis of the threshold s√λ<1. Contribution The contribution is methodological in that it finds sufficient conditions for equilibrium existence using metric completeness and contractivity without depending mainly on convexity and upper semicontinuity, and incorporates best response sensitivity explicitly.