The Existence Of Equilibrums In Several Kinds Of Games

In order to study the existence of Nash equilibria,the common method is to transform the game model into some multivalued mapping,then give appropriate assumptions to satisfy the existing fixed point theorem.In this paper,we study the existence of Nash equilibrium points by applying fixed point theory in three types of non-cooperative games,namely,n-players non-cooperative non-monetized games,uncertain games and games with infinite players.In the n-players non-cooperative non-monetized game,the order relation is firstly introduced in the strategy set(topological space)and outcome space,then a maximal element theorem in the product order space is proved by using the existing fixed point theorem in the order space.Finally we present an existence theorem of extended Nash equilibria by using the maximal element theorem.In uncertain game,the strategy set and uncertain parameter set are assigned corresponding order relation.Then we construct the set-valued mapping and prove that it satisfies the existing fixed point theorem under the assumptions.Moreover,we obtain an existence theorem of Nash equilibrium points in uncertain games.In the game with infinite players,the strategy set is endowed with a topological structure and an order structure,and the corresponding existence of Nash equilibrium points is obtained by using the properties of compact sets in the topological space.Due to the introduction of order relation,the existence theorem obtained in this paper is more extented and convenient to be applied.Several real game models in Chapter4 also check the practicability of the theorem.In general,compared with the previous conclusions,the conditions we obtained are easier to verify and more practical,which is an exploration with theoretical significance and application value.