Convergent transfer schemes for 𝑁-person games

Abstract

Introduction. The object of this paper is to describe a transfer scheme that converges to the kernel of a game and another scheme that converges to the bargaining set. Both the bargaining set and the kernel can be described as payoff vectors in which every objection (appropriately defined for each case) by one player against another player has a counter-objection. If, for a given payoff vector, a player i can sustain an objection against another playerj, a reasonable negotiation step might be for j to transfer enough of his payoff to i so that i can no longer sustain his objection. This results in a new payoff vector subject to further negotiation. We will show that any infinite sequence of such transfers causes the payoff vector to converge to a vector in the appropriate bargaining set provided only that maximal transfers are frequently made. Thus negotiation can approach the bargaining set without the players solving the defining conjunctive-disjunctive system of linear inequalities. Also, this scheme can be used as a computational technique to find one and possibly more bargaining points. The desired convergence is established by proving the convergence of a general transfer scheme of which the kernel and bargaining set schemes are special cases. For the purpose of this paper, a game is given by an ordered pair (N, v) where N is an n-element set of players and v, the characteristic function, is a zero-one normalized set function over the subsets of N (i.e. v satisfies v({i}) = 0 for all i in N and v(N)= 1). Let E be the space of real n-vectors indexed by N and let A be the set of all group and individually rational payoff vectors. Under the zero-one normalization, A is the set of x in E such that

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