<p>Suppose that we have a Bayesian game, where</p>
<ol>
<li>the number of players is <span class="math-container">$I$</span> and we refer to the generic player with <span class="math-container">$i$</span></li>
<li><span class="math-container">$\Theta$</span> stands for the state of the world, where <span class="math-container">$\theta\in\Theta$</span> is the typical element of the state space. Also, <span class="math-container">$\psi$</span> is the common prior beliefs about the state of the world, such that <span class="math-container">$\psi\in\Delta_{++}(\Theta)$</span> (full support)</li>
<li><span class="math-container">$S_1\times S_2\times\cdots\times S_I$</span> is the space of private signals, and <span class="math-container">$s\in S$</span> is the profile of state dependent signals that is drawn according to <span class="math-container">$\pi:\Theta\to\Delta(S)$</span> and every player <span class="math-container">$i$</span> will learn her own private signal <span class="math-container">$s_i$</span>.</li>
<li><span class="math-container">$A=\Pi_{i\in I}A_i$</span> is the compact space of pure strategies where <span class="math-container">$A_i$</span> is the strategy space of player <span class="math-container">$i$</span> and we further assume that <span class="math-container">$|A_1|=\cdots=|A_i|=|A_I|$</span>, namely the strategy spaces are all of the same cardinality. Then the mixed strategy profile is <span class="math-container">$\sigma:S\times\Theta\to\Delta(A)$</span></li>
<li>The utility function is the continuous function <span class="math-container">$u_i:\Theta\times A\to\mathbb{R}$</span> for every player <span class="math-container">$i$</span> or <span class="math-container">$u=(u_i)_{i\in I}:\Theta\times A\to\mathbb{R}^I$</span> as an <span class="math-container">$I$</span> dimensional vector.</li>
</ol>
<p>Suppose that the of profile mixed strategies that constitutes a Bayesian Nash equilibrium is an <span class="math-container">$I$</span>-tuple <span class="math-container">$\sigma^*=(\sigma_1^*,\sigma_2^*,\cdots,\sigma_I^*)$</span> where every player <span class="math-container">$i$</span> choose the strategy <span class="math-container">$\sigma_i^*$</span>, which is her best response strategy with respect to the best response strategies of the other players. Is the mixed strategy <span class="math-container">$\sigma_i^*$</span> chosen by player <span class="math-container">$i$</span> linear? Does this hold for every mixed strategy <span class="math-container">$\sigma_i\in\Sigma_i$</span>?</p>
<p>For example take the case where we have <span class="math-container">$I=4$</span> and <span class="math-container">$|A_1|=\cdots=|A_i|=|A_I|=3$</span>, namely every player's set of action contains three pure strategies.</p>

CC BY-SA 4.0
2022-04-01T08:14:49.897Z
You did not specify any vector space structure on $S_i$, but even if you would, there is no reason why strategies should be linear.
Michael Greinecker
CC BY-SA 4.0
2022-04-01T08:19:57.093Z
@MichaelGreinecker so there must be some extra assumption that I need to make so as the strategies should be linear? If I am not mistaken the mixed-strategy space and the image of the best response correspondence are convex, aren't they? Could this be related with the linearity of the strategies somehow?
studen21
CC BY-SA 4.0
2022-04-01T09:26:51.860Z
Yes, there is a natural linear structure on (a space containing the) space of mixed actions. Assumptions that guarantee linearity will be extremely strong. What do you need linearity for in the end?
Michael Greinecker
CC BY-SA 4.0
2022-04-01T09:30:13.013Z
@MichaelGreinecker Maybe I do not need it. I just wanted to make a connection with an older post https://economics.stackexchange.com/questions/50935/i-present-a-communication-game-could-you-please-make-comments-on-my-assumption

but check this one there. I think I will write some thoughts as an answer...take a look in about 5 minutes...i will delete this post...
studen21
<microeconomics><mathematical-economics><game-theory><bayesian-game>
Is the mixed strategy $\sigma_i^*:S_i\times\Theta\to\Delta(A_i)$ chosen by player $i$ linear?
studen21
# Is the mixed strategy $\sigma_i^*:S_i\times\Theta\to\Delta(A_i)$ chosen by player $i$ linear?

(Asked by: username_0 on 2022-04-01)

Suppose that we have a Bayesian game, where

1. the number of players is $I$ and we refer to the generic player with $i$
2. $\Theta$ stands for the state of the world, where $\theta\in\Theta$ is the typical element of the state space. Also, $\psi$ is the common prior beliefs about the state of the world, such that $\psi\in\Delta\_{++}(\Theta)$ (full support)
3. $S\_1\times S\_2\times\cdots\times S\_I$ is the space of private signals, and $s\in S$ is the profile of state dependent signals that is drawn according to $\pi:\Theta\to\Delta(S)$ and every player $i$ will learn her own private signal $s\_i$.
4. $A=\Pi\_{i\in I}A\_i$ is the compact space of pure strategies where $A\_i$ is the strategy space of player $i$ and we further assume that $|A\_1|=\cdots=|A\_i|=|A\_I|$, namely the strategy spaces are all of the same cardinality. Then the mixed strategy profile is $\sigma:S\times\Theta\to\Delta(A)$
5. The utility function is the continuous function $u\_i:\Theta\times A\to\mathbb{R}$ for every player $i$ or $u=(u\_i)\_{i\in I}:\Theta\times A\to\mathbb{R}^I$ as an $I$ dimensional vector.

Suppose that the of profile mixed strategies that constitutes a Bayesian Nash equilibrium is an $I$-tuple $\sigma^\*=(\sigma\_1^\*,\sigma\_2^\*,\cdots,\sigma\_I^\*)$ where every player $i$ choose the strategy $\sigma\_i^\*$, which is her best response strategy with respect to the best response strategies of the other players. Is the mixed strategy $\sigma\_i^\*$ chosen by player $i$ linear? Does this hold for every mixed strategy $\sigma\_i\in\Sigma\_i$?

For example take the case where we have $I=4$ and $|A\_1|=\cdots=|A\_i|=|A\_I|=3$, namely every player's set of action contains three pure strategies.

**Tags:** microeconomics, mathematical-economics, game-theory, bayesian-game

> **Comments:**
> **username_1**: You did not specify any vector space structure on $S_i$, but even if you would, there is no reason why strategies should be linear.  
> **username_0**: @username_1 so there must be some extra assumption that I need to make so as the strategies should be linear? If I am not mistaken the mixed-strategy space and the image of the best response correspondence are convex, aren't they? Could this be related with the linearity of the strategies somehow?  
> **username_1**: Yes, there is a natural linear structure on (a space containing the) space of mixed actions. Assumptions that guarantee linearity will be extremely strong. What do you need linearity for in the end?  
> **username_0**: @username_1 Maybe I do not need it. I just wanted to make a connection with an older post https://economics.stackexchange.com/questions/50935/i-present-a-communication-game-could-you-please-make-comments-on-my-assumption  but check this one there. I think I will write some thoughts as an answer...take a look in about 5 minutes...i will delete this post...  

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## 0 answers

