# Arbitration scheme

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 $R= \{ u = ( u _ {1} \dots u _ {n} ) \}$ be the set of outcomes, let $d = ( d _ {1} \dots d _ {n} )$ be the status quo point, i.e. the point corresponding to the situation in which no cooperative outcome is realized, let $[ R, d ]$ be the corresponding arbitration game and let $\overline{u}$ be an arbitration solution of it. An outcome $u ^ {*}$ is called a Nash solution if

$$\prod _ { i } ( u _ {i} ^ {*} - d _ {i} ) = \max _ {u \in R } \prod _ { i } ( u _ {i} - d _ {i} ) .$$

Only a Nash solution satisfies the following axioms: 1) if $f$ is a linear non-decreasing mapping then $f \overline{u}$ is an arbitration solution of the game $[ fR, fd ]$ (invariance with respect to utility transformations); 2) $\overline{u} \geq d$, $\overline{u} \in R$ and there is no $u \in R$ such that $u \geq \overline{u}$ (Pareto optimality); 3) if $R ^ \prime \subset R$, $d ^ \prime = d$, $\overline{u} \in R ^ \prime$, then $\overline{u} ^ \prime = \overline{u}$ (independence of irrelevant alternatives); and 4) if $d _ {i} = d _ {j}$, $i, j = 1 \dots n$, and $R$ is symmetric, then $\overline{u} _ {i} = \overline{u} _ {j}$, $i, j = 1 \dots n$ (symmetry).

Another arbitration scheme for an $n$-person game with characteristic function $v$ and player set $N = \{ 1 \dots n \}$ was given by L.S. Shapley . The Shapley solution $\phi (v) = ( \phi _ {1} (v) \dots \phi _ {n} (v) )$, where

$$\phi _ {i} (v) = \sum _ {S \subset N } \gamma _ {n} (s) [ v (S) -v ( S \setminus \{ i \} ) ] ,$$

$\gamma _ {n} (s) = (s-1) ! (n-s) ! / n !$ and $s$ is the number of elements of the set $S$, also satisfies the axiom of symmetry, but, moreover, $\sum _ {i} \phi _ {i} (v) = v (N)$, and for any two games $u$ and $v$ the equality $\phi (u+v) = \phi (u) + \phi (v)$ 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 $u ^ {*}$ and $u ^ {**}$ are two solutions, then $\overline{u}$, defined by

$$\overline{u} \geq \mathop{\rm min} _ {i \in N } ( u _ {i} ^ {*} , u _ {i} ^ {**} ) ,$$

is also a solution if and only if $\overline{u}$ belongs to the boundary of the set $R$.

Under suitable conditions an arbitration scheme depends continuously on the parameters of the game.

How to Cite This Entry:
Arbitration scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arbitration_scheme&oldid=52469
This article was adapted from an original article by E.I. Vilkas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article