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Properties of the core and other solution concepts of Bel coalitional games in the ex-ante scenario

Michel Grabisch, Silvia Lorenzini

TL;DR

The paper extends cooperative game theory to Bel coalitional games under ex-ante uncertainty, introducing contracts, belief-function uncertainty via Dempster-Shafer theory, and Choquet-based preferences. It proves that the ex-ante core forms an affine space when priors are identical and reduces to the core of the expected game, while defining ex-ante versions of the nucleolus, kernel, and bargaining set and showing that classical inclusion relations persist. In convex Bel games, the ex-ante bargaining set coincides with the ex-ante core under a strengthened bargaining notion, and the Shapley contract lies in the ex-ante core for such games. The work provides a rigorous framework for stability and negotiation under uncertainty, with future directions on ambiguous priors and voting applications.

Abstract

We study the properties of the core and other solution concepts of Bel coalitional games, that generalize classical coalitional games by introducing uncertainty in the framework. In this uncertain environment, we work with contracts, that specify how agents divide the values of the coalitions in the different states of the world. Every agent can have different a priori knowledge on the true state of the world, which is modeled through the Dempster-Shafer theory, while agents' preferences between contracts are modeled by the Choquet integral. We focus on the "ex-ante" scenario, when the contract is evaluated before uncertainty is resolved. We investigate the geometrical structure of the ex-ante core when agents have the same a priori knowledge which is a probability distribution. Finally, we define the (pre)nucleolus, the kernel and the bargaining set (a la Mas-Colell) in the ex-ante situation and we study their properties. It is found that the inclusion relations among these solution concepts are the same as in the classical case. Coincidence of the ex-ante core and the ex-ante bargaining set holds for convex Bel coalitional games, at the price of strengthening the definition of bargaining sets.

Properties of the core and other solution concepts of Bel coalitional games in the ex-ante scenario

TL;DR

The paper extends cooperative game theory to Bel coalitional games under ex-ante uncertainty, introducing contracts, belief-function uncertainty via Dempster-Shafer theory, and Choquet-based preferences. It proves that the ex-ante core forms an affine space when priors are identical and reduces to the core of the expected game, while defining ex-ante versions of the nucleolus, kernel, and bargaining set and showing that classical inclusion relations persist. In convex Bel games, the ex-ante bargaining set coincides with the ex-ante core under a strengthened bargaining notion, and the Shapley contract lies in the ex-ante core for such games. The work provides a rigorous framework for stability and negotiation under uncertainty, with future directions on ambiguous priors and voting applications.

Abstract

We study the properties of the core and other solution concepts of Bel coalitional games, that generalize classical coalitional games by introducing uncertainty in the framework. In this uncertain environment, we work with contracts, that specify how agents divide the values of the coalitions in the different states of the world. Every agent can have different a priori knowledge on the true state of the world, which is modeled through the Dempster-Shafer theory, while agents' preferences between contracts are modeled by the Choquet integral. We focus on the "ex-ante" scenario, when the contract is evaluated before uncertainty is resolved. We investigate the geometrical structure of the ex-ante core when agents have the same a priori knowledge which is a probability distribution. Finally, we define the (pre)nucleolus, the kernel and the bargaining set (a la Mas-Colell) in the ex-ante situation and we study their properties. It is found that the inclusion relations among these solution concepts are the same as in the classical case. Coincidence of the ex-ante core and the ex-ante bargaining set holds for convex Bel coalitional games, at the price of strengthening the definition of bargaining sets.
Paper Structure (13 sections, 10 theorems, 73 equations)

This paper contains 13 sections, 10 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dempster-Shafer Theory
  4. Coalitional Games
  5. Bel coalitional games
  6. Coalitional games with uncertainty
  7. Bel coalitional games
  8. Structure of the ex-ante core
  9. The ex-ante bargaining set and related solution concepts
  10. The ex-ante bargaining set
  11. The ex-ante nucleolus and the ex-ante kernel
  12. Convex Bel coalitional games
  13. Conclusion

Key Result

Theorem 1

Given a Bel coalitional game $(N,\Omega,m_\nu,(v_\omega)_{\omega\in\Omega})$ with all identical priors, and a grand contract $\mathbf{c}^N$, if for all $S\subseteq N$ it holds then the grand contract $\mathbf{c}^N$ is in the ex-ante-core of the Bel coalitional game.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 1
  • Definition 6
  • Definition 7
  • Theorem 1
  • Theorem 2
  • ...and 23 more