<p>It is possible for a mixed strategy not to be a best reply to any mixed strategy, even though all pure strategies in its support are best replies to some mixed strategies. Here is an example from the mentioned textbook: Both <strong>T</strong> and <strong>M</strong> are best replies against some mixed strategy of the column player (this follows from being undominated), but playing both with probability <span class="math-container">$1/2$</span> gives a certain payoff of <span class="math-container">$-\frac{1}{2}$</span>, so it would be better to play <strong>B</strong> instead.</p>
<div class="s-table-container">
<table class="s-table">
<thead>
<tr>
<th></th>
<th>L</th>
<th>R</th>
</tr>
</thead>
<tbody>
<tr>
<td><strong>T</strong></td>
<td>1,3</td>
<td>-2,0</td>
</tr>
<tr>
<td><strong>M</strong></td>
<td>-2,0</td>
<td>1,3</td>
</tr>
<tr>
<td><strong>B</strong></td>
<td>0,1</td>
<td>0,1</td>
</tr>
</tbody>
</table>
</div>
<p>Iterated deletion of mixed non-best replies, therefore, leads to non-convex sets. Now, the argument of Fudenberg and Tirole goes, if the column player is uncertain whether the row player plays <strong>T</strong> or <strong>M</strong>, and assigns probability <span class="math-container">$1/2$</span> to both possibilities, then it is as if the row player plays against the dominated mixed strategy above. Such mixture can be represented by the convex hull.</p>
<p>However, if there are more than two players, then the same argument can be used to argue that if a player is uncertain over which reasonable strategy profile of the other players is played, then they should also allow for convex combinations. But this will make the beliefs over the actions of the other players not stochastically independent anymore and they do not correspond to mixed strategy profiles anymore. If one allows for such correlation, one obtains the solution concept of <em>correlated rationalizability</em> instead, for many people (me included) a more sensible concept.</p>

CC BY-SA 4.0
2022-01-05T16:44:58.027Z
2022-01-06T19:56:26.440Z
2022-01-06T19:56:26.440Z
Michael Greinecker
<p>In <em>Game Theory</em> by Fudenberg and Tirole, they define the set of rationalizable strategies in the following way.</p>
<blockquote>
<p><strong>Definition 2.3 in FT</strong>: Set <span class="math-container">$\tilde{\Sigma}_{i}^{0}=\Sigma_{i}$</span> where <span class="math-container">$\Sigma_{i}$</span> is the space of mixed strategies for player <span class="math-container">$i$</span>. For each <span class="math-container">$i$</span>, define recursively <span class="math-container">$$\tilde{\Sigma}_{i}^{n}=\{\sigma_{i}\in \tilde{\Sigma}_{i}^{n-1}|\exists \sigma_{-i} \in \times_{j\not=i} \text{convex hull} \,\left(\tilde{\Sigma}_{j}^{n-1}\right) s.t \,\,u_{i}(\sigma_{i},\sigma_{-i})\geq u_{i}(\sigma_{i}',\sigma_{-i}) \,\,\forall \sigma_{i}'\in \tilde{\Sigma}_{i}^{n-1}\}$$</span>
The rationalizable strategies for player <span class="math-container">$i$</span> are <span class="math-container">$R_{i}=\cap_{n=0}^{\infty}\tilde{\Sigma}_{i}^{n}$</span></p>
</blockquote>
<p><strong>The convex hull</strong> of a set <span class="math-container">$X$</span> is the smallest convex set that contains <span class="math-container">$X$</span></p>
<p>My question is that, why do we need the convex hull operator in the definition?</p>
<p>I think <span class="math-container">$\tilde{\Sigma}_{i}^{n}$</span> is the set of strategies that survive the <span class="math-container">$n-1$</span> rounds of deletion of never-best-response strategies, I don't know why we need convexity here.</p>

<game-theory>
Convex hull operator in the definition of rationalizable strategies
John
# Convex hull operator in the definition of rationalizable strategies

(Asked by: username_0 on 2022-01-05)

In *Game Theory* by Fudenberg and Tirole, they define the set of rationalizable strategies in the following way.

> **Definition 2.3 in FT**: Set $\tilde{\Sigma}\_{i}^{0}=\Sigma\_{i}$ where $\Sigma\_{i}$ is the space of mixed strategies for player $i$. For each $i$, define recursively $$\tilde{\Sigma}\_{i}^{n}=\{\sigma\_{i}\in \tilde{\Sigma}\_{i}^{n-1}|\exists \sigma\_{-i} \in \times\_{j\not=i} \text{convex hull} \,\left(\tilde{\Sigma}\_{j}^{n-1}\right) s.t \,\,u\_{i}(\sigma\_{i},\sigma\_{-i})\geq u\_{i}(\sigma\_{i}',\sigma\_{-i}) \,\,\forall \sigma\_{i}'\in \tilde{\Sigma}\_{i}^{n-1}\}$$
> The rationalizable strategies for player $i$ are $R\_{i}=\cap\_{n=0}^{\infty}\tilde{\Sigma}\_{i}^{n}$

**The convex hull** of a set $X$ is the smallest convex set that contains $X$

My question is that, why do we need the convex hull operator in the definition?

I think $\tilde{\Sigma}\_{i}^{n}$ is the set of strategies that survive the $n-1$ rounds of deletion of never-best-response strategies, I don't know why we need convexity here.

**Tags:** game-theory

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## 1 answers

### Answer by username_1 on 2022-01-05. Score: 8 | Accepted answer

It is possible for a mixed strategy not to be a best reply to any mixed strategy, even though all pure strategies in its support are best replies to some mixed strategies. Here is an example from the mentioned textbook: Both **T** and **M** are best replies against some mixed strategy of the column player (this follows from being undominated), but playing both with probability $1/2$ gives a certain payoff of $-\frac{1}{2}$, so it would be better to play **B** instead.

|  | L | R |
| --- | --- | --- |
| **T** | 1,3 | -2,0 |
| **M** | -2,0 | 1,3 |
| **B** | 0,1 | 0,1 |

Iterated deletion of mixed non-best replies, therefore, leads to non-convex sets. Now, the argument of Fudenberg and Tirole goes, if the column player is uncertain whether the row player plays **T** or **M**, and assigns probability $1/2$ to both possibilities, then it is as if the row player plays against the dominated mixed strategy above. Such mixture can be represented by the convex hull.

However, if there are more than two players, then the same argument can be used to argue that if a player is uncertain over which reasonable strategy profile of the other players is played, then they should also allow for convex combinations. But this will make the beliefs over the actions of the other players not stochastically independent anymore and they do not correspond to mixed strategy profiles anymore. If one allows for such correlation, one obtains the solution concept of *correlated rationalizability* instead, for many people (me included) a more sensible concept.


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