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The preferences of players in non-cooperative games represent their choice inthe set of available options, which meet the completeness property if playersare able to compare any pair of available options. In the existing literature,the variational reformulation of generalized Nash games relies on the numericalrepresentation of players' preferences into objective functions, which is onlypossible if preferences are transitive and complete. In this work, we firstcharacterize the jointly convex generalized Nash equilibrium problems in termsof a variational inequality without requiring any numerical representation ofpreferences. Furthermore, we provide the suitable conditions under which anysolution of a quasi-variational inequality becomes an equilibrium for thegeneralized Nash game with non-ordered (incomplete and non-transitive)non-convex inter-dependent preferences. We check the solvability of these gameswhen the strategy maps of players are (possibly) unbounded. As an application,we derive the occurrence of competitive equilibrium for Arrow-Debreu economyunder uncertainty.

Variational Reformulation of Generalized Nash Equilibrium Problems with Non-ordered Preferences