Dynamic Coalitional TU Games: Distributed Bargaining Among Players' Neighbors

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.

Opinion dynamics in coalitional games with transferable utilities

“…The value of a coalition also reflects the monetary benefit demanded by that coalition to be a part of the grand coalition. In the context of coalitional TU games, robustness and dynamics naturally arise in all the situations where the coalition values are uncertain and time-varying, see e.g., [6], [7], [15]. We also link the approach to the set invariance theory [3] and stochastic stability theory [1], [10], [14], which provides us some useful tools for stability analysis.…”

Strategic thinking under social influence: Scalability, stability and robustness of allocations

“…The value of a coalition also reflects the monetary benefit demanded by that coalition to be part of the grand coalition. In the context of coalitional TU games, robustness and dynamics naturally arise in all the situations where the coalition values are uncertain and time-varying; see e.g., [10,11,25].…”

“…We recall that for a Brownian motion we have EdB t = 0 and EdB T t dB = t q dt and dropping the second-order terms (in dt 2 ) one obtains (25). Then the stochastic derivative of V (x) can be obtained by applying the infinitesimal generator to V (x), which yields…”

Distributed <inline-formula> <tex-math notation="TeX">$n$</tex-math></inline-formula>-Player Approachability and Consensus in Coalitional Games

“…Following [11] (see also [8]) we can make the following assumptions on the information structure. We let A(t) be the weight matrix with entries a i j (t).…”

“…The problem arises in the context of dynamic coalitional games with Transferable Utilities (TU games) [8]. A coalitional TU game consists in a set of players, who can form coalitions, and a characteristic function that provides a value for each coalition.…”