<p>In the context of a Bayesian game, <span class="math-container">$\Gamma = \left&lt; (\Theta_i,\mu_i)_{i \in N}, (A_i)_{i \in N}, (u_i)_{i \in N}\right&gt;$</span> where type of agent <span class="math-container">$i$</span> is drawn from <span class="math-container">$\Theta_i$</span> according to a probability measure <span class="math-container">$\mu_i$</span>, recall that a distributional strategy <span class="math-container">$\sigma_i$</span> is a probability measure on <span class="math-container">$\Theta_i \times A_i$</span> such that the marginal on <span class="math-container">$\Theta_i$</span> is <span class="math-container">$\mu_i$</span>.</p>

<p>Let <span class="math-container">$(\sigma_i)_{i \in N}$</span> be a Bayes Nash equilibrium of <span class="math-container">$\Gamma$</span>.</p>

<p>If <span class="math-container">$\sigma_i^{'} = \sigma_i$</span> almost everywhere, is <span class="math-container">$(\sigma^{'}_{i})_{i \in N}$</span> also a Bayes Nash equilibrium of <span class="math-container">$\Gamma$</span>? </p>

CC BY-SA 4.0
2020-04-09T13:07:16.890Z
Indeed, yes---the players' expected payoffs do not change.
Michael
CC BY-SA 4.0
2020-04-09T23:11:09.283Z
Thank you. The fact that this is not true for behavioural strategies $\beta_i: \Theta_i \rightarrow \Delta(A_i)$ has always bugged me. I consider the question closed.
Walrasian Auctioneer
<game-theory>
Perturbing distributional strategies on a measure zero set
Walrasian Auctioneer
# Perturbing distributional strategies on a measure zero set

(Asked by: username_0 on 2020-04-09)

In the context of a Bayesian game, $\Gamma = \left< (\Theta\_i,\mu\_i)\_{i \in N}, (A\_i)\_{i \in N}, (u\_i)\_{i \in N}\right>$ where type of agent $i$ is drawn from $\Theta\_i$ according to a probability measure $\mu\_i$, recall that a distributional strategy $\sigma\_i$ is a probability measure on $\Theta\_i \times A\_i$ such that the marginal on $\Theta\_i$ is $\mu\_i$.

Let $(\sigma\_i)\_{i \in N}$ be a Bayes Nash equilibrium of $\Gamma$.

If $\sigma\_i^{'} = \sigma\_i$ almost everywhere, is $(\sigma^{'}\_{i})\_{i \in N}$ also a Bayes Nash equilibrium of $\Gamma$?

**Tags:** game-theory

> **Comments:**
> **username_1**: Indeed, yes---the players' expected payoffs do not change.  
> **username_0**: Thank you. The fact that this is not true for behavioural strategies $\beta_i: \Theta_i \rightarrow \Delta(A_i)$ has always bugged me. I consider the question closed.  

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## 0 answers

