An existence result for farsighted stable sets of games in characteristic function form
This paper considers NTU games, that is, the class of cooperative games where the utility can be non transferable among the players. Interesting real situations, as for example market problems, can be modelled using NTU games. One important result in the literature of game theory is that there exist games that admit no von Neumann-Morgenstern solution (cf. [\textit{W. Lucas}, ``A game with no solution'', Bull. Am. Math. Soc. 74, 237--239 (1968; Zbl 0157.51101)]) with respect to the classical direct dominance relation. One significant development in this literature were the two notions of indirect dominance (cf. [\textit{J. C. Harsanyi}, ``An equilibrium-point interpretation of stable sets and a proposed alternative definition'', Manage. Sci., Theory 20, 1472--1495 (1974; Zbl 0335.90063)]) that take into account the fact that the players may be farsighted while making a move, they may take into account the further moves by other groups of players that may ensue. In this paper, the second of such notions is used. In stark contrast to that for the direct dominance relation, at least one stable set with respect to the indirect (or farsighted) dominance relation exists for every cooperative game where the utility is transferable among the agents (cf. [\textit{S. Béal, J. Durieu} and \textit{P. Solal}, ``Farsighted coalitional stability in TU-games'', Math. Soc. Sci. 56, No. 3, 303--313 (2008; Zbl 1155.91308)]). This paper generalizes that result to the setting of NTU games under an innocuous condition. Let \((N,V)\) be an NTU game. For any \(S\subset N\) and any \(x,y\in V(N)\) \(y\prec_Sx\) means that \(y_i<x_i\) for every \(i\in S\). A pay-off vector is an imputation if it is individually rational as well as efficient. \(I(N,V)\) denotes the set of imputations. A coalition \(S\) can enforce an imputation \(x\) from an (and any) imputation \(z\), denoted as \(z\to_S x\), if and only if \(x |_S \in V(S)\). The notion of indirect dominance used in the paper is the following. An imputation \(z\) is indirectly (or farsightedly) dominated by another imputation \(x\), denoted as \(x\gg z\), if there exist a finite collection of imputations \(z=z^0, z^1, z^2,\dots, z^m=x\) and a finite collection of coalitions \(S_0, S_1, S_2,\dots, S_{m-1}\) such that \(z^i\to_{S_i} z^{i+1}\) and \(z^i\prec_{S_i} x\) for \(i=0, 1, 2,\dots, m-1\). Note that, when initiating a chain of enforcements, each coalition active at any outcome of the chain compares the pay-offs at the final outcome of the chain to the immediate pay-offs and not the very next outcome in the chain. A subset \(K\) of the set of imputations of the game is a farsighted stable set if (a) for all \(x,z\in K\) neither \(x\gg z\) nor \(z\gg x\) (internal stability), and (b) for all \(z\in I(N,V)\backslash K\) there exists \(x\in K\) such that \(x\gg z\) (external stability). The main result of the paper establishes that, for any game \((N,V)\) where the set of individually rational pay-off vectors is bounded, a farsighted von Neumann-Morgenstern stable set exists. With respect to the assumption, note that the representation of any reasonable economy by NTU games should satisfy the boundedness of the set of individually rational pay-off vectors. On the other hand, the assumption is not a necessary condition for the existence of a farsighted stable set. Finally, they remark that a proper exploration of the structural properties of the farsighted stable sets in general NTU games remains to be done.
scientific article; zbMATH DE number 5913907