A Biform Game Model with the Shapley Allocation Functions

Chenwei Liu,Shuwen Xiang,Yanlong Yang

doi:10.3390/math9161872

Abstract

We define the mixed strategy form of the characteristic function of the biform games and build the Shapley allocation function (SAF) on each mixed strategy profile in the second stage of the biform games. SAF provides a more detailed and accurate picture of the fairness of the strategic contribution and reflects the degree of the players’ further choices of strategies. SAF can guarantee the existence of Nash equilibrium in the first stage of the non-cooperative games. The existence and uniqueness of SAF on each mixed strategy profile overcome the defect that the core may be an empty set and provide a fair allocation method when the core element is not unique. Moreover, SAF can be used as an important reference or substitute for the core with the confidence index.

Highlights

IntroductionWe define a mixed strategy form of the Shapley value and build an allocation function (SAF) on each mixed strategy profile in the second stage
The Shapley allocation function (SAF) can be an important reference for the value of the confidence index or a substitute for the core
SAF can be an important reference for “residual” bargaining problem with confidence index, or it can be used as a substitute for core allocation with confidence index

Summary

Introduction

We define a mixed strategy form of the Shapley value and build an allocation function (SAF) on each mixed strategy profile in the second stage This result can ensure the existence of Nash equilibria in the non-cooperative game of the first stage. To ensure the existence of Nash equilibrium of the induced non-cooperative game, in the first stage of a biform game, we consider that players are each trying to use mixed strategies (pure strategy is a special form of mixed strategy) to select the best game for themselves, where by “game” is meant the subsequent (second-stage) game of value.

The Examples of Mixed Strategy

Definitions and Existences

Examples

The Properties of SAF

Comparisons

Conclusions