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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130501.png" /> be the set of outcomes, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130502.png" /> be the status quo point, i.e. the point corresponding to the situation in which no cooperative outcome is realized, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130503.png" /> be the corresponding arbitration game and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130504.png" /> be an arbitration solution of it. An outcome <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130505.png" /> is called a Nash solution if
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130506.png" /></td> </tr></table>
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Only a Nash solution satisfies the following axioms: 1) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130507.png" /> is a linear non-decreasing mapping then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130508.png" /> is an arbitration solution of the game <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a0130509.png" /> (invariance with respect to utility transformations); 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305011.png" /> and there is no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305012.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305013.png" /> (Pareto optimality); 3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305017.png" /> (independence of irrelevant alternatives); and 4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305019.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305020.png" /> is symmetric, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305022.png" /> (symmetry).
+
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
  
Another arbitration scheme for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305023.png" />-person game with characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305024.png" /> and player set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305025.png" /> was given by L.S. Shapley [[#References|[2]]]. The Shapley solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305026.png" />, where
+
$$
 +
\prod _ { i } ( u _ {i}  ^ {*} - d _ {i} )  = \max _
 +
{u \in R }  \prod _ { i } ( u _ {i} - d _ {i} ) .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305027.png" /></td> </tr></table>
+
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).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305029.png" /> is the number of elements of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305030.png" />, also satisfies the axiom of symmetry, but, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305031.png" />, and for any two games <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305033.png" /> the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305034.png" /> holds. Arbitration schemes with interpersonal utility comparisons have also been considered [[#References|[3]]].
+
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
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305036.png" /> are two solutions, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305037.png" />, defined by
+
$$
 +
\phi _ {i} (v)  =  \sum _ {S \subset  N } \gamma _ {n} (s)
 +
[ v (S) -v ( S \setminus  \{ i \} ) ] ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305038.png" /></td> </tr></table>
+
$  \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]]].
  
is also a solution if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305039.png" /> belongs to the boundary of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305040.png" />.
+
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.
 
Under suitable conditions an arbitration scheme depends continuously on the parameters of the game.
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====References====
 
====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>
 
<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====
 
====Comments====
Arbitration schemes are also called bargaining schemes and a Nash solution is also called a bargaining solution. The Shapley solution vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013050/a01305043.png" /> 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]]].
+
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====
 
====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>
+
<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,  "$N$-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>

Latest revision as of 07:23, 26 March 2023


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 [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 [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 [3].

The arbitration schemes of Nash and Shapley were generalized by J.C. Harsanyi [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

[1] J. Nash, "The bargaining problem" Econometrica , 18 : 2 (1950) pp. 155–162
[2] 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
[3] 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
[4] 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

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 [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].

References

[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, "$N$-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)
How to Cite This Entry:
Arbitration scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arbitration_scheme&oldid=14856
This article was adapted from an original article by E.I. Vilkas (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article