Some Applications Of The Representation Of Symmetric Groups In Cooperative Games

A solution of cooperative games is a mapping Φ from the game space G to itself,where Φ(v)provides an allocation scheme for an arbitrary game v∈E G.One of the defining challenges in cooperative game theory is to find useful and meaningful solutions.Moreover,linear symmetric solutions are of particular importance,for instance,the Shapley value.As far as we know,up to now researchers focus on the solutions that assign payoffs to individual players.More precisely,such a solution satisfies the additive axiom,i.e.,the payoff of a coalition equals to the sum of payoffs of individual players who belong to the coalition.However,there are rare studies on the solutions that assign payoffs to multi-element coalitions instead of individual players.This motivates us to study the linear symmetric solutions that assign payoffs to k-element coalitions with k>1,by the representation theory of symmetric groups.The main content of this dissertation is as follows.In Chapter one,a survey of the related background is presented and the organization and the main results of this thesis are also briefly introduced.In Chapter two,some representations of finite groups,especially,of the symmetric groups,are summarized.Here there is also a self-contained procedure of studying the Shapley value by using the representation theory of symmetric groups.In Chapter three,we study the linear symmetric solutions that assign payoffs to 2-element coalitions.The explicit linear symmetric solutions are showed when the big coalition containing 4 or 5 players.For general case,an implicit structural characterization of the linear symmetric solutions is presented here.In Chapter four,several problems needing a further study are listed.