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A Generalised $λ$-Core Concept for Normal Form Games

Subhadip Chakrabarti, Robert P Gilles, Lina Mallozzi

Abstract

In this note we develop a generalisation of the $λ$-Core solution for non-cooperative games in normal form. We show that this generalised $λ$-Core is non-empty for the class of separable games that admit a socially optimal Nash equilibrium. Examples are provided that indicate that non-emptiness of the generalised $λ$-Core cannot be expected for large classes of normal form games.

A Generalised $λ$-Core Concept for Normal Form Games

Abstract

In this note we develop a generalisation of the -Core solution for non-cooperative games in normal form. We show that this generalised -Core is non-empty for the class of separable games that admit a socially optimal Nash equilibrium. Examples are provided that indicate that non-emptiness of the generalised -Core cannot be expected for large classes of normal form games.
Paper Structure (13 sections, 3 theorems, 29 equations)

This paper contains 13 sections, 3 theorems, 29 equations.

Table of Contents

  1. Introduction: Cores of non-cooperative games
  2. Some preliminaries
  3. Best response structures
  4. Characteristic functions
  5. The $\lambda$-characteristic function and its generalisation
  6. The Strong Reduction property
  7. A generalised $\lambda$-Core of a normal form game
  8. The generalised $\lambda$-Core of separable games
  9. Application: A status game
  10. Proof of Theorem \ref{['thm:Main']}
  11. Some concluding remarks
  12. Acknowledgements
  13. Declaration on conflict of interest

Key Result

Lemma 3.2

Let $\Gamma$ be a normal form game that satisfies the Strong Reduction Property. Then the following holds:

Theorems & Definitions (9)

  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Example 3.6
  • Example 3.7
  • Definition 4.1
  • Example 4.2