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A rule by which each arbitration game (cf. [[Cooperative game|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 [[#References|[1]]] 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 [[#References|[2]]]. 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 [[#References|[3]]].

The arbitration schemes of Nash and Shapley were generalized by J.C. Harsanyi [[#References|[4]]]. 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Nash,   "The bargaining problem"  ''Econometrica'' , '''18''' :  2  (1950)  pp. 155–162</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Shapley,   "A value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305041.png" />-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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.C. Harsanyi,   "A bargaining model for the cooperative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305042.png" />-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</TD></TR></table>

====Comments====
Arbitration schemes are also called bargaining schemes and a Nash solution is also called a bargaining solution. The Shapley solution vector  $  \phi $
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 [[#References|[a1]]], [[#References|[a2]]], respectively [[#References|[a6]]]. For further developments concerning Harsanyi solutions, cf. [[#References|[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, [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Kalai,   M. Smorodinsky,   "Other solutions to Nash's bargaining problems"  ''Econometrica'' , '''43'''  (1975)  pp. 513–518</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.E. Roth,   "Axiomatic models of bargaining" , ''Lect. notes econom. and math. systems'' , '''170''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.C. Harsanyi,   "Papers in game theory" , Reidel  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.J. Aumann,   L.S. Shapley,   "Values of non-atomic games" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Rapoport,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305044.png" />-person game theory: Concepts and applications" , Univ. Michigan Press  (1970)  pp. 168</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Szép,   F. Forgó,   "Introduction to the theory of games" , Reidel  (1985)  pp. 171; 199</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  N.N. Vorob'ev,   "Game theory. Lectures for economists and system scientists" , Springer  (1977)  (Translated from Russian)</TD></TR></table>