<p>I am trying to understand the Kohlberg-Mertens theorem. Here is the context and the theorem. We fix <span class="math-container">$N$</span> a set of players and <span class="math-container">$S_i$</span> the finite set of actions of each players with <span class="math-container">$\lvert S_i\rvert =m_i$</span>. We define <span class="math-container">$\Sigma_i$</span> to be the set of probability measure on <span class="math-container">$S_i$</span> i.e. the simplex of <span class="math-container">$\mathbb{R}^{m_i}$</span>. Let <span class="math-container">$S=\Pi_{i\in N} S_i$</span> and <span class="math-container">$\Sigma=\Pi_{i\in N}\Sigma_i$</span> where <span class="math-container">$\lvert S\rvert = m$</span>. We identify a game <span class="math-container">$G$</span> to its payoff functions, here for each player <span class="math-container">$i$</span> a payoff function lies in <span class="math-container">$\mathbb{R}^{\lvert S_{-i}\rvert\times m_i}:=\Gamma_i$</span>, hence a game <span class="math-container">$G$</span> is identified to <span class="math-container">$g=(g_1,…,g_N)\in\mathbb{R}^{Nm}:=\Pi_{i\in N}\Gamma_i$</span>. Let <span class="math-container">$\mathcal{G}$</span> be the set of all such payoff <span class="math-container">$g$</span>. We define the set</p>
<p><span class="math-container">$$
E=\{(g,\sigma)\in\mathcal{G}\times\Sigma :\sigma\:\text{Nash equilibrium of}\: g\}
$$</span>
The theorem is the following</p>
<blockquote>
<p>The projection <span class="math-container">$\pi : E\to\mathcal{G}$</span> is homotopic to an homeomorphism from <span class="math-container">$E$</span> to <span class="math-container">$\mathcal{G}$</span>.</p>
</blockquote>
<p>The intuition is that we can pass continuously from the graph of the Nash correspondence to the set of games. Now my question is on the proof of this result.</p>
<p>They start by saying this</p>
<blockquote>
<p>Let <span class="math-container">$T_{i}=\Pi_{h\neq I}S_{h}$</span>, reparametrize <span class="math-container">$\Gamma_i$</span> the set of all <span class="math-container">$\lvert S_i\rvert\times\lvert T_i\rvert$</span> payoff matrices <span class="math-container">$G_{s,t}^{i}$</span> by <span class="math-container">$G_{s,t}=\tilde{G}_{s,t}^{i}$</span> with <span class="math-container">$\sum_{t\in T_i}\tilde{G}_{s,t}^i=0$</span>, letting <span class="math-container">$z_{s}^i = \sigma_{s}^i +\sum_{t\in T^i}G_{s,t}^i\Pi_{j\neq I}\sigma_{t_j}^j$</span>, <span class="math-container">$(\tilde{G}_{s,t}^{i},z_{s}^i)$</span> can be viewed as belonging to <span class="math-container">$\Gamma_i$</span>. </p>
</blockquote>
<p>However, it is not clear for me that that the sum in the definition of <span class="math-container">$z_{s}^{i}$</span> is well defined. Indeed, if I understand well (this is not defined explicitly in the paper) <span class="math-container">$\sigma_{s}^{i}$</span> is the probability that player <span class="math-container">$i$</span> choses the action <span class="math-container">$s_i$</span> (where <span class="math-container">$s_i$</span> is such that <span class="math-container">$s=(s_1,…,s_i,…,s_N)\in S$</span>) under <span class="math-container">$\sigma^i$</span> , that is a scalar, while <span class="math-container">$\tilde{G}_{s,t}^i$</span> is a matrix.</p>
<p>I think that I am misunderstanding something. If you have any knowledge on this please do not hesitate.</p>
<p>Thank you a lot !</p>

CC BY-SA 4.0
2024-06-02T15:32:50.540Z
That "quote" is not a quote.
Michael Greinecker
CC BY-SA 4.0
2024-06-02T16:39:15.110Z
@MichaelGreinecker Thank you for the remark, I edited the post
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<mathematical-economics><game-theory><decision-theory><theory>
Kohlberg-Mertens theorem
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# Kohlberg-Mertens theorem

(Asked by: username_0 on 2024-06-02)

I am trying to understand the Kohlberg-Mertens theorem. Here is the context and the theorem. We fix $N$ a set of players and $S\_i$ the finite set of actions of each players with $\lvert S\_i\rvert =m\_i$. We define $\Sigma\_i$ to be the set of probability measure on $S\_i$ i.e. the simplex of $\mathbb{R}^{m\_i}$. Let $S=\Pi\_{i\in N} S\_i$ and $\Sigma=\Pi\_{i\in N}\Sigma\_i$ where $\lvert S\rvert = m$. We identify a game $G$ to its payoff functions, here for each player $i$ a payoff function lies in $\mathbb{R}^{\lvert S\_{-i}\rvert\times m\_i}:=\Gamma\_i$, hence a game $G$ is identified to $g=(g\_1,…,g\_N)\in\mathbb{R}^{Nm}:=\Pi\_{i\in N}\Gamma\_i$. Let $\mathcal{G}$ be the set of all such payoff $g$. We define the set

$$
E=\{(g,\sigma)\in\mathcal{G}\times\Sigma :\sigma\:\text{Nash equilibrium of}\: g\}
$$
The theorem is the following

> The projection $\pi : E\to\mathcal{G}$ is homotopic to an homeomorphism from $E$ to $\mathcal{G}$.

The intuition is that we can pass continuously from the graph of the Nash correspondence to the set of games. Now my question is on the proof of this result.

They start by saying this

> Let $T\_{i}=\Pi\_{h\neq I}S\_{h}$, reparametrize $\Gamma\_i$ the set of all $\lvert S\_i\rvert\times\lvert T\_i\rvert$ payoff matrices $G\_{s,t}^{i}$ by $G\_{s,t}=\tilde{G}\_{s,t}^{i}$ with $\sum\_{t\in T\_i}\tilde{G}\_{s,t}^i=0$, letting $z\_{s}^i = \sigma\_{s}^i +\sum\_{t\in T^i}G\_{s,t}^i\Pi\_{j\neq I}\sigma\_{t\_j}^j$, $(\tilde{G}\_{s,t}^{i},z\_{s}^i)$ can be viewed as belonging to $\Gamma\_i$.

However, it is not clear for me that that the sum in the definition of $z\_{s}^{i}$ is well defined. Indeed, if I understand well (this is not defined explicitly in the paper) $\sigma\_{s}^{i}$ is the probability that player $i$ choses the action $s\_i$ (where $s\_i$ is such that $s=(s\_1,…,s\_i,…,s\_N)\in S$) under $\sigma^i$ , that is a scalar, while $\tilde{G}\_{s,t}^i$ is a matrix.

I think that I am misunderstanding something. If you have any knowledge on this please do not hesitate.

Thank you a lot !

**Tags:** mathematical-economics, game-theory, decision-theory, theory

> **Comments:**
> **username_1**: That "quote" is not a quote.  
> **username_0**: @MichaelGreinecker Thank you for the remark, I edited the post  

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## 0 answers

