<p>The center allocation <span class="math-container">$z_{i}:=(v(N)/|N|)$</span> for all <span class="math-container">$i \in N$</span> is in the core of the symmetric game <span class="math-container">$v$</span>, whenever <span class="math-container">$z(S) = \sum_{i \in S} z_{i} \ge v(S) $</span> holds for all <span class="math-container">$S \subseteq N$</span>. Now, if <span class="math-container">$v(N)/|N| \ge v(S)/|S|$</span> for all <span class="math-container">$S \subseteq N$</span>, then it holds <span class="math-container">$z(S) = \sum_{i \in S} z_{i} = |S|\cdot v(N)/|N| \ge v(S)$</span>. Thus, <span class="math-container">$\mathbf{z} \in C(v)$</span>.  </p>

<p>We observe that for a symmetric game <span class="math-container">$v$</span>, the core is non-empty if <span class="math-container">$v(N)/|N| \ge v(S)/S$</span> holds for all <span class="math-container">$S \subseteq N$</span>. Otherwise, the core is empty.</p>

CC BY-SA 4.0
2019-11-28T10:44:22.533Z
2019-11-28T10:44:22.533Z
Holger I. Meinhardt
<p>Consider a TU game <span class="math-container">$\langle N,v \rangle$</span> with <span class="math-container">$N = \{1,2,\ldots,n\}$</span> being the set of players and <span class="math-container">$v : 2^N \to \mathbb R$</span>, <span class="math-container">$v(\emptyset) = 0$</span> the characteristic function. The Core of <span class="math-container">$v$</span> is defined by undominated allocations <span class="math-container">$x \in \mathbb R^n$</span>
<span class="math-container">\begin{align}
\mathcal C(v) = \left\{x \in \mathbb R^n ~ \bigg| ~ \sum_{i \in S}{x_i} \geq v(S) ~ \forall S \subseteq N\right\}.
\end{align}</span>
Let <span class="math-container">$m_i(S) = v(S \cup \{i\}) - v(S)$</span> denote the marginal contribution of <span class="math-container">$i$</span> to coalition <span class="math-container">$S$</span>.
We say that the players are symmetric if <span class="math-container">$m_i(S) = m_j(S)$</span> for all <span class="math-container">$i,j \in N$</span> and <span class="math-container">$S \subset N$</span>. 
The marginal contribution of any agent to any coalition is thus identical.</p>

<p>I am wondering whether the following Proposition is true and how one proves it formally.</p>

<blockquote>
  <p><strong>Proposition:</strong> If the players are symmetric and the core is nonempty  <span class="math-container">$\mathcal C(v) \neq \emptyset$</span>, then the equal division of the worth of the grand coalition is an element of the core <span class="math-container">$(v(N)/n,\ldots,v(N)/n) \in \mathcal C(v)$</span>.</p>
</blockquote>

<game-theory><cooperative-game-theory>
If the players are symmetric and the core is nonempty, then $x_i = v(N)/n$ for all $i$ is a Core element
clueless
# If the players are symmetric and the core is nonempty, then $x_i = v(N)/n$ for all $i$ is a Core element

(Asked by: username_0 on 2019-11-28)

Consider a TU game $\langle N,v \rangle$ with $N = \{1,2,\ldots,n\}$ being the set of players and $v : 2^N \to \mathbb R$, $v(\emptyset) = 0$ the characteristic function. The Core of $v$ is defined by undominated allocations $x \in \mathbb R^n$
\begin{align}
\mathcal C(v) = \left\{x \in \mathbb R^n ~ \bigg| ~ \sum\_{i \in S}{x\_i} \geq v(S) ~ \forall S \subseteq N\right\}.
\end{align}
Let $m\_i(S) = v(S \cup \{i\}) - v(S)$ denote the marginal contribution of $i$ to coalition $S$.
We say that the players are symmetric if $m\_i(S) = m\_j(S)$ for all $i,j \in N$ and $S \subset N$.
The marginal contribution of any agent to any coalition is thus identical.

I am wondering whether the following Proposition is true and how one proves it formally.

> **Proposition:** If the players are symmetric and the core is nonempty $\mathcal C(v) \neq \emptyset$, then the equal division of the worth of the grand coalition is an element of the core $(v(N)/n,\ldots,v(N)/n) \in \mathcal C(v)$.

**Tags:** game-theory, cooperative-game-theory

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## 1 answers

### Answer by username_1 on 2019-11-28. Score: 3 | Accepted answer

The center allocation $z\_{i}:=(v(N)/|N|)$ for all $i \in N$ is in the core of the symmetric game $v$, whenever $z(S) = \sum\_{i \in S} z\_{i} \ge v(S) $ holds for all $S \subseteq N$. Now, if $v(N)/|N| \ge v(S)/|S|$ for all $S \subseteq N$, then it holds $z(S) = \sum\_{i \in S} z\_{i} = |S|\cdot v(N)/|N| \ge v(S)$. Thus, $\mathbf{z} \in C(v)$.

We observe that for a symmetric game $v$, the core is non-empty if $v(N)/|N| \ge v(S)/S$ holds for all $S \subseteq N$. Otherwise, the core is empty.


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