Existence of Generalized Nash Equilibrium in n-Person Noncooperative Games under Incomplete Preference

Abstract

To prove the existence of Nash equilibrium by traditional ways, a common condition that the preference of players must be complete has to be considered. This paper presents a new method to improve it. Based on the incomplete preference corresponding to equivalence class set being a partial order set, we translate the incomplete preference problems into the partial order problems. Using the famous Zorn lemma, we get the existence theorems of fixed point for noncontinuous operators in incomplete preference sets. These new fixed point theorems provide a new way to break through the limitation. Finally, the existence of generalized Nash equilibrium is strictly proved in the n-person noncooperative games under incomplete preference.

Optimizing the strategy of activities using numerical methods for determining equilibrium

“…An analysis of the non-local convergence of regulators of the extremum search applied to general convex systems showed that potential surfaces of the payoff function with its associated response curves, which lie in a set of actions and determined extremal that lie outside U, overlap each other (16), (17), for non-quadratic payoff functions.…”

“…Paper [17] showed that the problem of achievement of the Nash equilibrium, as a kind of combination of strategies, relates to many important problems of mathematics, economics, and engineering technologies. A generalized game model plays an important role in tasks of management of the economy for proof of the existence of a general equilibrium.…”

The Guidance of an Enterprise Economy, by MartinShubik and EricSmith (MIT Press, Cambridge, MA, 2016), pp. 592

“…To date, the abstract economy framework has mainly been used to establish equilibrium existence theorems for economies having various non‐Arrow–Debreu features. Such features include: interdependent preferences, price‐dependent preferences, and preferences which may be both non‐transitive and non‐complete (Shafer & Sonnenschein, 1975; Li, 2018); externalities and public goods (Shafer & Sonnenschein, 1976; Florenzano, 2009); private information and a countable set of actions (Patriche, 2013); and economies with unbounded attainable sets in which consumers may have non‐complete non‐transitive preferences (Ounaies et al . , 2019), to name a few.…”