A rule by which each arbitration game (cf. Cooperative game) is put into correspondence with a unique outcome of the game is called an arbitration solution. The first arbitration scheme was considered by J. Nash  for the case of a two-person game. Let be the set of outcomes, let be the status quo point, i.e. the point corresponding to the situation in which no cooperative outcome is realized, let be the corresponding arbitration game and let be an arbitration solution of it. An outcome is called a Nash solution if
Only a Nash solution satisfies the following axioms: 1) if is a linear non-decreasing mapping then is an arbitration solution of the game (invariance with respect to utility transformations); 2) , and there is no such that (Pareto optimality); 3) if , , , then (independence of irrelevant alternatives); and 4) if , , and is symmetric, then , (symmetry).
Another arbitration scheme for an -person game with characteristic function and player set was given by L.S. Shapley . The Shapley solution , where
and is the number of elements of the set , also satisfies the axiom of symmetry, but, moreover, , and for any two games and the equality holds. Arbitration schemes with interpersonal utility comparisons have also been considered .
The arbitration schemes of Nash and Shapley were generalized by J.C. Harsanyi . A Harsanyi solution satisfies, apart from the four axioms of Nash, the two axioms: 1) the solution depends monotonically on the initial demands of the players; and 2) if and are two solutions, then , defined by
is also a solution if and only if belongs to the boundary of the set .
Under suitable conditions an arbitration scheme depends continuously on the parameters of the game.
|||J. Nash, "The bargaining problem" Econometrica , 18 : 2 (1950) pp. 155–162|
|||L.S. Shapley, "A value for -person games" H.W. Kuhn (ed.) A.W. Tucker (ed.) M. Dresher (ed.) , Contributions to the theory of games , 2 , Princeton Univ. Press (1953) pp. 307–317|
|||H. Raiffa, "Arbitration schemes for generalized two-person games" H.W. Kuhn (ed.) A.W. Tucker (ed.) M. Dresher (ed.) , Contributions to the theory of games , 2 , Princeton Univ. Press (1953) pp. 361–387|
|||J.C. Harsanyi, "A bargaining model for the cooperative -person game" H.W. Kuhn (ed.) A.W. Tucker (ed.) M. Dresher (ed.) , Contributions to the theory of games , 4 , Princeton Univ. Press (1959) pp. 325–355|
Arbitration schemes are also called bargaining schemes and a Nash solution is also called a bargaining solution. The Shapley solution vector is also called the Shapley value. For other, more recent, bargaining schemes, such as the Kalai–Smorodinsky solution and Szidarovsky's generalization of the concept of a Nash solution, the reader is referred to [a1], [a2], respectively [a6]. For further developments concerning Harsanyi solutions, cf. [a3]. Some authors distinguish between bargaining schemes and arbitration schemes. Then the Nash scheme is a bargaining scheme and the Shapley one an arbitration scheme, [a5].
|[a1]||E. Kalai, M. Smorodinsky, "Other solutions to Nash's bargaining problems" Econometrica , 43 (1975) pp. 513–518|
|[a2]||A.E. Roth, "Axiomatic models of bargaining" , Lect. notes econom. and math. systems , 170 , Springer (1979)|
|[a3]||J.C. Harsanyi, "Papers in game theory" , Reidel (1982)|
|[a4]||R.J. Aumann, L.S. Shapley, "Values of non-atomic games" , Princeton Univ. Press (1974)|
|[a5]||A. Rapoport, "-person game theory: Concepts and applications" , Univ. Michigan Press (1970) pp. 168|
|[a6]||J. Szép, F. Forgó, "Introduction to the theory of games" , Reidel (1985) pp. 171; 199|
|[a7]||N.N. Vorob'ev, "Game theory. Lectures for economists and system scientists" , Springer (1977) (Translated from Russian)|
Arbitration scheme. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Arbitration_scheme&oldid=42564