Existence and essential stability of Nash equilibria for biform games with Shapley allocation functions

Chenwei Liu; Shuwen Xiang; Yanlong Yang; Chenwei Liu; Shuwen Xiang; Yanlong Yang

doi:10.3934/math.2022432

AIMS Mathematics

2022, Volume 7, Issue 5: 7706-7719. doi: 10.3934/math.2022432

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Existence and essential stability of Nash equilibria for biform games with Shapley allocation functions

1.
School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China
2.
College of Mathematical and Information Science, Guiyang University, Guiyang, Guizhou 550005, China

Received: 26 August 2021 Revised: 02 February 2022 Accepted: 10 February 2022 Published: 16 February 2022
MSC : 46T20, 49J53, 91A10, 91A12, 91A40

We define the Shapley allocation function (SAF) based on the characteristic function on a set of strategy profiles composed of infinite strategies to establish an n-person biform game model. It is the extension of biform games with finite strategies and scalar strategies. We prove the existence of Nash equilibria for this biform game with SAF, provided that the characteristic function satisfies the linear and semicontinuous conditions. We investigate the essential stability of Nash equilibria for biform games when characteristic functions are perturbed. We identify a residual dense subclass of the biform games whose Nash equilibria are all essential and deduce the existence of essential components of the Nash equilibrium set by proving the connectivity of its minimal essential set.
- biform games,
- Nash equilibrium point,
- essential connected component,
- generic stability,
- Shapley allocation function
Citation: Chenwei Liu, Shuwen Xiang, Yanlong Yang. Existence and essential stability of Nash equilibria for biform games with Shapley allocation functions[J]. AIMS Mathematics, 2022, 7(5): 7706-7719. doi: 10.3934/math.2022432

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Abstract

We define the Shapley allocation function (SAF) based on the characteristic function on a set of strategy profiles composed of infinite strategies to establish an n-person biform game model. It is the extension of biform games with finite strategies and scalar strategies. We prove the existence of Nash equilibria for this biform game with SAF, provided that the characteristic function satisfies the linear and semicontinuous conditions. We investigate the essential stability of Nash equilibria for biform games when characteristic functions are perturbed. We identify a residual dense subclass of the biform games whose Nash equilibria are all essential and deduce the existence of essential components of the Nash equilibrium set by proving the connectivity of its minimal essential set.

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