[Paper Review] Fixed point theorems for nonconvex valued correspondences and applications in game theory

Monica Patriche|arXiv (Cornell University)|Apr 2, 2013

Optimization and Variational Analysis13 references7 citations

TL;DR

This paper introduces new classes of nonconvex, noncontinuous correspondences—weakly naturally quasiconvex, *-weakly naturally quasiconvex, weakly biconvex, and those with *–weakly convex graph—and establishes fixed point theorems for them. Using a version of W. K. Kim’s quasi-point theorem and continuous selection techniques, it proves the existence of equilibria in generalized quasi-games under relaxed convexity and continuity assumptions, extending equilibrium existence results to broader game-theoretic models.

ABSTRACT

In this paper, we introduce several types of correspondences: weakly naturally quasiconvex, *-weakly naturally quasiconvex, weakly biconvex and correspondences with *--weakly convex graph and we prove some fixed point theorems for these kinds of correspondences. As a consequence, using a version of W. K. Kim's quasi-point theorem, we obtain the existence of equilibria for a quasi-game.

Motivation & Objective

• To extend fixed point theory to correspondences that are neither continuous nor convex-valued, addressing a long-standing open problem in fixed point theory.
• To introduce and analyze new classes of correspondences—weakly naturally quasiconvex, *-weakly naturally quasiconvex, weakly biconvex, and those with *–weakly convex graph—offering more flexible conditions than prior results.
• To apply these fixed point theorems to prove the existence of equilibria in generalized quasi-games, particularly in settings with infinite agents, nonconvex strategy sets, and non-continuous preferences.
• To generalize existing equilibrium existence theorems in game theory by weakening standard assumptions of convexity and continuity on constraint and preference correspondences.

Proposed method

• Introduce four new types of correspondences: weakly naturally quasiconvex, *-weakly naturally quasiconvex, weakly biconvex, and those with *–weakly convex graph, each relaxing classical convexity or continuity conditions.
• Prove fixed point theorems for these correspondences using topological and set-theoretic tools, including adherence of graphs and upper semicontinuity.
• Apply a version of W. K. Kim’s quasi-point theorem to derive equilibrium existence in generalized quasi-games.
• Use continuous selection techniques from Yannelis and Prahbakar (1995) to construct selections for correspondences with *–weakly convex graph and weakly naturally quasiconvex structure.
• Define the generalized quasi-game model with non-empty, convex-valued constraint correspondences $ B_i $, where $ A_i \subset B_i $, and $ \text{cl } B_i $ is upper semicontinuous.
• Establish equilibrium conditions by showing that for each agent $ i $, $ y_i^* \in \text{cl } B_i(x^*,y^*) $ and $ A_i(x^*,y^*) \cap P_i(x^*,y^*) = \emptyset $, using contradiction from selection and closure properties.

Experimental results

Research questions

• RQ1 Can fixed point theorems be established for correspondences that are neither continuous nor convex-valued, under weaker structural assumptions?
• RQ2 Do the new classes of correspondences—weakly naturally quasiconvex, *-weakly naturally quasiconvex, weakly biconvex, and those with *–weakly convex graph—suffice to guarantee fixed points without full convexity or continuity?
• RQ3 Can these fixed point results be applied to prove the existence of equilibria in generalized quasi-games with nonconvex strategy sets and non-continuous preferences?
• RQ4 Is the continuous selection technique effective for correspondences with *–weakly convex graph and weakly naturally quasiconvex structure in equilibrium models?
• RQ5 Under what conditions does the equilibrium condition $ y_i^* \in \text{cl } B_i(x^*,y^*) $ and $ A_i(x^*,y^*) \cap P_i(x^*,y^*) = \emptyset $ hold in a generalized quasi-game with uncountable agent sets?

Key findings

• A fixed point theorem is established for correspondences with *–weakly convex graph, proving the existence of a fixed point under relaxed graph structure conditions.
• The paper proves the existence of an equilibrium point $ (x^*, y^*) \in X \times X $ for a generalized quasi-game $ \Gamma $, where $ y_i^* \in \text{cl } B_i(x^*, y^*) $ and $ A_i(x^*, y^*) \cap P_i(x^*, y^*) = \emptyset $, under weakly naturally quasiconvex and open set assumptions.
• For the case where $ A_i \cap P_i $ is weakly biconvex and $ W_i $ is the interior of a biconvex hull, the equilibrium existence is guaranteed via continuous selection and fixed point arguments.
• The proof shows that if $ x_i^* \in \Phi_i(x^*, y^*) $, a contradiction arises with the assumption $ x_i \notin P_i(x,y) $, so $ \Phi_i(x^*, y^*) = \emptyset $, implying $ (x^*, y^*) \notin W_i $, which ensures the equilibrium condition.
• The results generalize prior equilibrium existence theorems by removing the need for full convexity or continuity, allowing for nonconvex and noncontinuous preference and constraint correspondences.
• The framework supports uncountable agent sets $ I $, with $ X_i $ compact convex in Hausdorff locally convex spaces, extending applicability to infinite-dimensional and large-scale game models.