<p>While it is a bit unusual to describe a <em>strategy profile</em> as being Pareto optimal, especially in the context of Bayesian games, I guess you can still define Pareto optimality in different stages of such games as follows. Recall that in a Bayesian game, a pure strategy is a function <span class="math-container">$s_i:\Theta_i\to A_i$</span>. </p>

<blockquote>
  <p>A strategy profile <span class="math-container">$s(\theta)=(s_1(\theta_1),\dots,s_n(\theta_n))$</span> is
  <strong>ex ante Pareto optimal</strong> if  <span class="math-container">\begin{equation} \mathbb E_\theta [u_i(s(\theta);\theta)]\ge \mathbb
 E_\theta[u_i(s'(\theta);\theta)],\quad\text{$\forall i$ and $\forall
 s'(\theta)$} \end{equation}</span> where the above inequality is strict for
  some <span class="math-container">$i$</span>. [<span class="math-container">$\mathbb E_\theta$</span> means taking expectation over the state
  vector <span class="math-container">$\theta$</span>.]</p>
</blockquote>

<hr>

<blockquote>
  <p>A strategy profile <span class="math-container">$s(\theta)=(s_1(\theta_1),\dots,s_n(\theta_n))$</span> is
  <strong>ex post Pareto optimal</strong> if, given a particular realization of states <span class="math-container">$\bar\theta=(\bar\theta_1,\dots,\bar\theta_n)$</span>, 
  <span class="math-container">\begin{equation} u_i(s(\bar\theta);\bar\theta)\ge
 u_i(s'(\bar\theta);\bar\theta),\quad\text{$\forall i$ and $\forall
 s'(\bar\theta)$} \end{equation}</span> where the above inequality is strict
  for some <span class="math-container">$i$</span>.</p>
</blockquote>

CC BY-SA 4.0
2019-02-14T01:16:21.090Z
2019-02-14T06:51:39.137Z
2019-02-14T06:51:39.137Z
Herr K.
<p>In a normal form game with two players, we call a joint strategy <span class="math-container">$s=(s_1,s_2)$</span> a Pareto optimal outcome if for no joint strategy <span class="math-container">$s'$</span>, for all players <span class="math-container">$i\in\{1,2\}$</span>, we have <span class="math-container">$$u_i(s')\geq u_i(s)\qquad(1)$$</span> and also at least for one player <span class="math-container">$i$</span> <span class="math-container">$$u_i(s')&gt;u_i(s).\qquad(2)$$</span></p>

<p>The social welfare of <span class="math-container">$s$</span> is defined as <span class="math-container">$$\sum_{i=1}^{2}u_i(s).\qquad(3)$$</span> If the social welfare of <span class="math-container">$s$</span> is maximal, then the joint strategy <span class="math-container">$s$</span> is a social optimum.</p>

<p>I want to use these definitions in a Bayesian game where the utility functions are defined as <span class="math-container">$u_i:A\times\Theta\to\mathbb{R}$</span>, where <span class="math-container">$A$</span> is the action set and <span class="math-container">$\Theta$</span> is the type set.</p>

<p>Can we simply replace <span class="math-container">$u_i(s)$</span> in <span class="math-container">$(1),(2),(3)$</span> with <span class="math-container">$u_i(a,\theta)$</span>? Or we have to consider the expected utility of a player?</p>

<p>How should we write the definitions if the game is in the ex-ante stage?</p>

CC BY-SA 4.0
2019-02-12T06:14:43.773Z
See [Bayesian efficiency](https://en.wikipedia.org/wiki/Bayesian_efficiency)
Herr K.
CC BY-SA 4.0
2019-02-12T23:28:31.903Z
@HerrK.This seems to be implementable only to bayesian mechanism design. In my case the bayesian game is finite and I don't consider incentive compatible allocation rules.
johnny09
<game-theory><bayesian-game>
Social Welfare and Pareto Optimality in a Bayesian Game
johnny09
# Social Welfare and Pareto Optimality in a Bayesian Game

(Asked by: username_0 on 2019-02-12)

In a normal form game with two players, we call a joint strategy $s=(s\_1,s\_2)$ a Pareto optimal outcome if for no joint strategy $s'$, for all players $i\in\{1,2\}$, we have $$u\_i(s')\geq u\_i(s)\qquad(1)$$ and also at least for one player $i$ $$u\_i(s')>u\_i(s).\qquad(2)$$

The social welfare of $s$ is defined as $$\sum\_{i=1}^{2}u\_i(s).\qquad(3)$$ If the social welfare of $s$ is maximal, then the joint strategy $s$ is a social optimum.

I want to use these definitions in a Bayesian game where the utility functions are defined as $u\_i:A\times\Theta\to\mathbb{R}$, where $A$ is the action set and $\Theta$ is the type set.

Can we simply replace $u\_i(s)$ in $(1),(2),(3)$ with $u\_i(a,\theta)$? Or we have to consider the expected utility of a player?

How should we write the definitions if the game is in the ex-ante stage?

**Tags:** game-theory, bayesian-game
> **Comments:**
> **username_1**: See [Bayesian efficiency](https://en.wikipedia.org/wiki/Bayesian_efficiency)  
> **username_0**: @HerrK.This seems to be implementable only to bayesian mechanism design. In my case the bayesian game is finite and I don't consider incentive compatible allocation rules.  

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## 1 answers

### Answer by username_1 on 2019-02-14. Score: 5 | Accepted answer

While it is a bit unusual to describe a *strategy profile* as being Pareto optimal, especially in the context of Bayesian games, I guess you can still define Pareto optimality in different stages of such games as follows. Recall that in a Bayesian game, a pure strategy is a function $s\_i:\Theta\_i\to A\_i$.

> A strategy profile $s(\theta)=(s\_1(\theta\_1),\dots,s\_n(\theta\_n))$ is
> **ex ante Pareto optimal** if \begin{equation} \mathbb E\_\theta [u\_i(s(\theta);\theta)]\ge \mathbb
> E\_\theta[u\_i(s'(\theta);\theta)],\quad\text{$\forall i$ and $\forall
> s'(\theta)$} \end{equation} where the above inequality is strict for
> some $i$. [$\mathbb E\_\theta$ means taking expectation over the state
> vector $\theta$.]

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> A strategy profile $s(\theta)=(s\_1(\theta\_1),\dots,s\_n(\theta\_n))$ is
> **ex post Pareto optimal** if, given a particular realization of states $\bar\theta=(\bar\theta\_1,\dots,\bar\theta\_n)$,
> \begin{equation} u\_i(s(\bar\theta);\bar\theta)\ge
> u\_i(s'(\bar\theta);\bar\theta),\quad\text{$\forall i$ and $\forall
> s'(\bar\theta)$} \end{equation} where the above inequality is strict
> for some $i$.


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