Tijs Value

The Tijs value (T-value) is a solution concept in cooperative game theory. It presents a reasonable expectation of the share-out of the global winning among players and a fair division in normative contexts.

S. H. Tijs proposed this value in 1981 for a particular class of cooperative games called quasibalanced games along with a particularization of that value for simple games. In 1984, in joint research with T. S. H. Driessen, Tijs extended his [Page 668]-value to all cooperative games. This extension does not possess the properties of individual rationality and dummy players, having instead the T-value defined for quasibalanced games.

Some preliminary definitions follow. The T-value and T-index will be presented and illustrated with two examples. Certain properties will be given.

Let N = {1,2,…, n} be a set of players. A cooperative game on N is expressed in characteristic function form where v: 2N → R assigns a value v(S) to every coalition S ⫅ N under the condition v(Ø)= 0. A game is superadditive if v(S ∪ T) ≥ v(S)+v (T) for all disjoint coalitions S, T of N. A game v is said to be simple if its characteristic function assumes only the values 0 and 1; in the first case the coalition is said to be losing, in the second case, winning. A simple game is monotonic if v(N) = 1 and v(S) ≤ v(T) whenever S ⫅ T ⫅ N. Subsequent references to simple games assume monotonicity.

The upper vector bv = bv1, bv2, …, bvn ɛ Rn and gap function gv: 2N → R of the game v are defined respectively as:

Each component of the upper vector is called the marginal contribution of the player with respect to the N. If a game v is superadditive, bvi ≥ v({i}), so any player prefers his marginal contribution to the amount v({i}) which he can obtain playing alone. Observe that for superadditive games gv (N) ≥ 0.

Tijs focuses his attention on those superadditive games where , so it is impossible to give everyone at least a marginal contribution. In these cases the vector of marginal contributions bv (upper vector) is preferred by the players, but it is not a feasible solution for the game. So, in all cases where gv(N)> 0 and gv({i}) ≥ 0, the quantity bv is called the utopic payment vector. With respect to bv, the grand coalition N has to make a concession of the quantity in order to obtain an feasible solution for the game. Tijs suggested that the concession of each ith player, and then his or her contribution to gv(N), should be determined by minimizing the value of the gap function with respect to all coalitions containing i. The vector pv=(pv1, pv2, …, pvn) ɛ Rn such that for all i ɛ N is called the concession vector. The class QvB of quasibalanced n-person games is defined as:

The Tijs value (T-value) of every quasibalanced game v is defined as:

If gv(N)= 0, each player receives his or her marginal contribution. If gv(N)< 0 each player forgoes his or hershare: .

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