Editing Shapley value (section)

== Properties ==

The Shapley value has many desirable properties.
Notably, it is the only payment rule satisfying the four properties of Efficiency, Symmetry, Linearity and Null player (or dummy player).<ref name=":2">{{cite book|title=Contributions to the Theory of Games|last=Shapley|first=Lloyd S.|publisher=Princeton University Press|year=1953|isbn=9781400881970|editor-last=Kuhn|editor-first=H. W.|series=Annals of Mathematical Studies|volume=28|pages=307–317|chapter=A Value for n-person Games|doi=10.1515/9781400881970-018|editor2-first=A. W.|editor2-last=Tucker}}</ref>
See<ref name=":1" />{{Rp|147–156}} for more characterizations of the Shapley value.

=== Efficiency ===
The sum of the Shapley values of all agents equals the value of the grand coalition, so that all the gain is distributed among the agents:
:<math>\sum_{i\in N}\varphi_i(v) = v(N)</math>

''Proof'': <math>\sum_{i\in N} \varphi_i(v) = 
\frac{1}{|N|!} \sum_R \sum_{i\in N} v(P_i^R \cup \left \{ i \right \}) - v(P_i^R) 
= \frac{1}{|N|!} \sum_R v(N) = \frac{1}{|N|!}
|N|!\cdot v(N) = v(N)</math>

since <math>\sum_{i\in N} v(P_i^R \cup \left \{ i \right \}) - v(P_i^R)</math> is a [[telescoping sum]] and there are <math>|N|!</math> different orderings <math>R</math>.

=== Symmetry ===
If <math>i</math> and <math>j</math> are two actors who are equivalent in the sense that

:<math>v(S\cup\{i\}) = v(S\cup\{j\})</math>
for every subset <math>S</math> of <math>N</math> which contains neither <math>i</math> nor <math>j</math>, then <math>\varphi_i(v) = \varphi_j(v)</math>.

This property is also called ''equal treatment of equals''.

=== Linearity ===
If two coalition games described by gain functions <math>v</math> and <math>w</math> are combined, then the distributed gains should correspond to the gains derived from <math>v</math> and the gains derived from <math>w</math>:

:<math>\varphi_i(v+w) = \varphi_i(v) + \varphi_i(w)</math>
for every <math>i</math> in&nbsp;<math>N</math>. Also, for any real number <math>a</math>,
:<math>\varphi_i(a v) = a \varphi_i(v)</math>
for every <math>i</math> in&nbsp;<math>N</math>.

=== Null player ===
The Shapley value <math>\varphi_i(v)</math> of a null player <math>i</math> in a game <math>v</math> is zero. A player <math>i</math> is ''null'' in <math>v</math> if <math>v(S\cup \{i\}) = v(S)</math> for all coalitions <math>S</math> that do not contain <math>i</math>.

=== Stand-alone test ===
If <math>v</math> is a [[subadditive set function]], i.e., <math>v(S\sqcup T) \leq v(S) + v(T)</math>, then for each agent <math>i</math>: <math>\varphi_i(v) \leq v(\{i\})</math>.

Similarly, if <math>v</math> is a [[superadditive set function]], i.e., <math>v(S\sqcup T) \geq v(S) + v(T)</math>, then for each agent <math>i</math>:  <math>\varphi_i(v) \geq v(\{i\})</math>.

So, if the cooperation has positive synergy, all agents (weakly) gain, and if it has negative synergy, all agents (weakly) lose.<ref name=":1">{{Cite Moulin 2004}}</ref>{{Rp|147–156}}

=== Anonymity ===
If <math>i</math> and <math>j</math> are two agents, and <math>w</math> is a gain function that is identical to <math>v</math> except that the roles of <math>i</math> and <math>j</math> have been exchanged, then <math>\varphi_i(v) = \varphi_j(w)</math>. This means that the labeling of the agents doesn't play a role in the assignment of their gains.

=== Marginalism ===
The Shapley value can be defined as a function which uses only the marginal contributions of player <math>i</math> as the arguments.