Abstract
Abstract In this paper, we introduce the concept of order-clustered fixed point of set-valued mappings on preordered sets and give several generalizations of the extension of the Abian-Brown fixed point theorem provided in (Mas-Colell et al. in Microeconomic Theory, 1995), which is from chain-complete posets to chain-complete preordered sets. By using these generalizations and by applying the order-increasing upward property of set-valued mappings, we prove several existence theorems of the extended and generalized Nash equilibria of nonmonetized noncooperative games on chain-complete preordered sets. MSC:46B42, 47H10, 58J20, 91A06, 91A10.
Highlights
1 Introduction In traditional game theory, fixed point theory in topological spaces or metric spaces has been an essential tool for the proof of the existence of Nash equilibria of noncooperative games, in which the payoff functions of the players take real values
The existence of generalized and extended Nash equilibria of nonmonetized noncooperative games has been studied by applying fixed point theorems to ordered sets
We apply the extensions of the Abian-Brown fixed point theorem provided in the last section to prove some existence theorems for the extended Nash equilibrium of nonmonetized noncooperative games on chain-complete preordered sets, which can be considered as extensions of the results proved by Li in [ ], which are on chain-complete posets
Summary
In traditional game theory, fixed point theory in topological spaces or metric spaces has been an essential tool for the proof of the existence of Nash equilibria of noncooperative games, in which the payoff functions of the players take real values (see [ – ]). The existence of generalized and extended Nash equilibria of nonmonetized noncooperative games has been studied by applying fixed point theorems to ordered sets. From assumption A in this theorem, we can write the set of maximal element clusters of F(b) by {[u ], [u ], .
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