Dynamic Coalitional TU Games: Distributed Bargaining among Players' Neighbors

TL;DR

This paper introduces a distributed bargaining protocol for dynamic, stochastic TU games, proving convergence to core allocations under mild conditions and extending to cases with empty cores via averaging.

Contribution

It develops a novel decentralized bargaining algorithm for dynamic TU games and proves its convergence to core allocations, even in complex scenarios with empty cores.

Findings

Convergence of bargaining process to the core with probability 1.

Extension of the protocol to average games with ergodic characteristic functions.

Applicability to dynamic, stochastic, and interval-valued TU games.

Abstract

We consider a sequence of transferable utility (TU) games where, at each time, the characteristic function is a random vector with realizations restricted to some set of values. The game differs from other ones in the literature on dynamic, stochastic or interval valued TU games as it combines dynamics of the game with an allocation protocol for the players that dynamically interact with each other. The protocol is an iterative and decentralized algorithm that offers a paradigmatic mathematical description of negotiation and bargaining processes. The first part of the paper contributes to the definition of a robust (coalitional) TU game and the development of a distributed bargaining protocol. We prove the convergence with probability 1 of the bargaining process to a random allocation that lies in the core of the robust game under some mild conditions on the underlying communication graphs. The second part of the paper addresses the more general case where the robust game may have empty core. In this case, with the dynamic game we associate a dynamic average game by averaging over time the sequence of characteristic functions. Then, we consider an accordingly modified bargaining protocol. Assuming that the sequence of characteristic functions is ergodic and the core of the average game has a nonempty relative interior, we show that the modified bargaining protocol converges with probability 1 to a random allocation that lies in the core of the average game.