<p>Consider a standard game <span class="math-container">$\Gamma$</span> with incomplete information. There are <span class="math-container">$n$</span> players indexed by <span class="math-container">$i=1,...,n$</span>.
<span class="math-container">$S_i\equiv \{s_{i1},...,s_{iJ}\}$</span> is the set of actions of player <span class="math-container">$i$</span>. <span class="math-container">$S\equiv \times_{i=1}^n S_i$</span>. Each player <span class="math-container">$i$</span> is endowed with a type <span class="math-container">$\epsilon_i\equiv (\epsilon_{i1},..., \epsilon_{iJ})$</span> with distribution <span class="math-container">$f_i\in \Delta(E_i)$</span>. <span class="math-container">$\{\epsilon_{i}\}_{i=1}^n$</span> are mutually independent. <span class="math-container">$\epsilon_i$</span> is player <span class="math-container">$i$</span>'s private information. The utility of player <span class="math-container">$i$</span> from choosing <span class="math-container">$s_{ij}$</span> is <span class="math-container">$u_i(s_{ij}, s_{-i})+\epsilon_{ij}$</span>. <span class="math-container">$u_i$</span> and <span class="math-container">$f_i$</span> are common knowledge for each <span class="math-container">$i$</span>.</p>
<p>I want to define a Bayesian Nash Equilibrium (BNE) of <span class="math-container">$\Gamma$</span>. I propose two definitions.</p>
<p><strong>Def 1:</strong> Let <span class="math-container">$\sigma\equiv (\sigma_1,..., \sigma_n)$</span> where <span class="math-container">$\sigma_i:E_i\rightarrow \Delta(S_i)$</span> for each <span class="math-container">$i=1,...,n$</span>. In particular, let <span class="math-container">$\sigma^{s_{ij}}_i(\epsilon_i)$</span> denote the probability that player <span class="math-container">$i$</span> plays action <span class="math-container">$s_{ij}$</span> when he is of type <span class="math-container">$\epsilon_i$</span>, under <span class="math-container">$\sigma$</span>. Then, <span class="math-container">$\sigma$</span> is a BNE if and only if
<span class="math-container">$$
s_{ij}\in argmax_{s_{ik}\in S_i} \sum_{s_{-i}\in S_{-i}} \int_{E_{-i}} [u_i(s_{ik}, s_{-i})+\epsilon_{ik}]\sigma^{s_{-i}}_{-i}(\epsilon_{-i})f_{-i}(\epsilon_{-i}) \text{ }d\epsilon_{-i}
$$</span>
for each <span class="math-container">$j=1,..., J$</span> such that <span class="math-container">$\sigma^{s_{ij}}_{i}(\epsilon_{i})&gt;0$</span>, for each <span class="math-container">$\epsilon_i\in E_i$</span>, and for each <span class="math-container">$i=1,...,n$</span>.</p>
<p><strong>Def 2:</strong> Let <span class="math-container">$\pi\equiv (\pi_1,..., \pi_n)\in \Delta(S)$</span>. Let
<span class="math-container">$$
R_{ij}(\pi)\equiv \{\epsilon_i\in E_i: s_{ij}\in argmax_{s_{ik}\in S_i} \sum_{s_{-i}\in S_i}[u_i(s_{ik}, s_{-i})+\epsilon_{ik}]\pi_{-i}(s_{-i})\}
$$</span>
<span class="math-container">$\pi$</span> is a BNE if and only if
<span class="math-container">$$
\pi_i(s_{ij})=\int_{R_{ij}(\pi)} f_i(\epsilon_i) d\epsilon_i
$$</span>
for each <span class="math-container">$j=1,...,J$</span> and <span class="math-container">$i=1,...,I$</span>.</p>
<p><strong>Clarification</strong>: Def 2 comes from the fact that I've been taught at school that BNE is a fixed point. Hence, I'm trying to replicate a fixed point equation.</p>
<p><strong>Question 1</strong>: In the case the &quot;argmax&quot; is always unique, are the two definitions equivalent? If yes/no, how do we show it?</p>
<p><strong>Question 2</strong>: In the case the &quot;argmax&quot; may not be unique, are the two definitions equivalent? If yes/no, how do we show it?</p>

CC BY-SA 4.0
2022-01-19T19:29:00.223Z
If there are multiple argmax, I believe the definition does not work and the second definition would give you something different as well. In fact, 1 tries to define a BNE and 2 is trying to define an equilibrium outcome.
Walrasian Auctioneer
CC BY-SA 4.0
2022-01-20T07:14:21.970Z
Thanks. Multiple argmax: I have replaced $=$ with $\in$. At school, I have been taught that BNE is a fixed point. This is what I'm trying to replicate in 2, a fixed point equation. If 2 is wrong, could you tell me where do we get this fixed point equation?
Star
CC BY-SA 4.0
2022-01-21T02:14:32.523Z
Say $s_{ij},s_{ik}\in\arg\max$. It's not necessary that both $s_{ij}$ and $s_{ik}$ be played with positive probability in a BNE. But your Def 2 seems to suggest that they must.
Herr K.
CC BY-SA 4.0
2022-01-21T10:54:47.987Z
Thanks. Can we distinguish the answer in two cases (armax unique versus argmax not unique)? See edited question.
Star
CC BY-SA 4.0
2022-01-27T10:50:46.790Z
I don't understand what exactly $R_{ij}(\pi)$ is supposed to be in your second definition. If you want a definition via fixed points, you could define something like $BR(\sigma)=\{\sigma'\mid \forall i,\epsilon_i:\: \sigma_i'(\epsilon_i) \text{ is a best response against } \sigma_{-i}\}$. A BNE is then a fixed point of the correspondence $BR$.
axelniemeyer
<game-theory><nash-equilibrium><bayesian-game>
Are these two definitions of Bayesian Nash Equilibrium equivalent?
Star
# Are these two definitions of Bayesian Nash Equilibrium equivalent?

(Asked by: username_0 on 2022-01-19)

Consider a standard game $\Gamma$ with incomplete information. There are $n$ players indexed by $i=1,...,n$.
$S\_i\equiv \{s\_{i1},...,s\_{iJ}\}$ is the set of actions of player $i$. $S\equiv \times\_{i=1}^n S\_i$. Each player $i$ is endowed with a type $\epsilon\_i\equiv (\epsilon\_{i1},..., \epsilon\_{iJ})$ with distribution $f\_i\in \Delta(E\_i)$. $\{\epsilon\_{i}\}\_{i=1}^n$ are mutually independent. $\epsilon\_i$ is player $i$'s private information. The utility of player $i$ from choosing $s\_{ij}$ is $u\_i(s\_{ij}, s\_{-i})+\epsilon\_{ij}$. $u\_i$ and $f\_i$ are common knowledge for each $i$.

I want to define a Bayesian Nash Equilibrium (BNE) of $\Gamma$. I propose two definitions.

**Def 1:** Let $\sigma\equiv (\sigma\_1,..., \sigma\_n)$ where $\sigma\_i:E\_i\rightarrow \Delta(S\_i)$ for each $i=1,...,n$. In particular, let $\sigma^{s\_{ij}}\_i(\epsilon\_i)$ denote the probability that player $i$ plays action $s\_{ij}$ when he is of type $\epsilon\_i$, under $\sigma$. Then, $\sigma$ is a BNE if and only if
$$
s\_{ij}\in argmax\_{s\_{ik}\in S\_i} \sum\_{s\_{-i}\in S\_{-i}} \int\_{E\_{-i}} [u\_i(s\_{ik}, s\_{-i})+\epsilon\_{ik}]\sigma^{s\_{-i}}\_{-i}(\epsilon\_{-i})f\_{-i}(\epsilon\_{-i}) \text{ }d\epsilon\_{-i}
$$
for each $j=1,..., J$ such that $\sigma^{s\_{ij}}\_{i}(\epsilon\_{i})>0$, for each $\epsilon\_i\in E\_i$, and for each $i=1,...,n$.

**Def 2:** Let $\pi\equiv (\pi\_1,..., \pi\_n)\in \Delta(S)$. Let
$$
R\_{ij}(\pi)\equiv \{\epsilon\_i\in E\_i: s\_{ij}\in argmax\_{s\_{ik}\in S\_i} \sum\_{s\_{-i}\in S\_i}[u\_i(s\_{ik}, s\_{-i})+\epsilon\_{ik}]\pi\_{-i}(s\_{-i})\}
$$
$\pi$ is a BNE if and only if
$$
\pi\_i(s\_{ij})=\int\_{R\_{ij}(\pi)} f\_i(\epsilon\_i) d\epsilon\_i
$$
for each $j=1,...,J$ and $i=1,...,I$.

**Clarification**: Def 2 comes from the fact that I've been taught at school that BNE is a fixed point. Hence, I'm trying to replicate a fixed point equation.

**Question 1**: In the case the "argmax" is always unique, are the two definitions equivalent? If yes/no, how do we show it?

**Question 2**: In the case the "argmax" may not be unique, are the two definitions equivalent? If yes/no, how do we show it?

**Tags:** game-theory, nash-equilibrium, bayesian-game

> **Comments:**
> **username_1**: If there are multiple argmax, I believe the definition does not work and the second definition would give you something different as well. In fact, 1 tries to define a BNE and 2 is trying to define an equilibrium outcome.  
> **username_0**: Thanks. Multiple argmax: I have replaced $=$ with $\in$. At school, I have been taught that BNE is a fixed point. This is what I'm trying to replicate in 2, a fixed point equation. If 2 is wrong, could you tell me where do we get this fixed point equation?  
> **username_2**: Say $s_{ij},s_{ik}\in\arg\max$. It's not necessary that both $s_{ij}$ and $s_{ik}$ be played with positive probability in a BNE. But your Def 2 seems to suggest that they must.  
> **username_0**: Thanks. Can we distinguish the answer in two cases (armax unique versus argmax not unique)? See edited question.  
> **username_3**: I don't understand what exactly $R_{ij}(\pi)$ is supposed to be in your second definition. If you want a definition via fixed points, you could define something like $BR(\sigma)=\{\sigma'\mid \forall i,\epsilon_i:\: \sigma_i'(\epsilon_i) \text{ is a best response against } \sigma_{-i}\}$. A BNE is then a fixed point of the correspondence $BR$.  

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## 0 answers

