Uniqueness of the Shapley Value

From Theory

Proof of the Uniqueness

For any $A\subseteq N, a\in R, a\ge 0$ , we define $V A, a (S)$ to be of value $a$ when $A\subseteq S$ , zero otherwise. One can check that this $V A, a (S)$ is valid for a cooperation game. Also, the only possible way (by the axioms) to assign $φ(V)$ for each player i is as following: if $i\in A$ , then $\phi_{i}(V_{A,a})=\frac{a}{|A|}$ ; otherwise zero. Now, we claim that $\{V_{A,1}\}_{A\subseteq N}$ form a basis of the space $\{v \; : \; \mathcal{P}(N) \; \to \Re\}$ . This is because for any $v$ , it can be uniquely written as:

$v(S)=\sum_{A\subseteq N}\prod_{i\in A}S_i\prod_{i\notin A}(1-S_i)v(A)$ , where $S i$ is the i'th coordinate of the indicating vector of the set $S$ .

So $v (S)$ can be written as a polynomial over the variables $S i$ . Now, it is easy to see that $\prod_{i\in A}S_i$ is the same as $V A,1$ , and by the axiom 5 (additivity), $φ(v)$ is unique. Actually, we use a little bit stronger axiom here: $φ i (a v + b w) = a φ i (v) + b φ i (w)$

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Uniqueness of the Shapley Value

From Theory

Proof of the Uniqueness

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