<p>Here's a hint. Consider this game:
<span class="math-container">\begin{array}{|c|c|c|}
\hline
 &amp;L&amp;R\\\hline
U&amp;4,.&amp;0,.\\\hline
M&amp;0,.&amp;4,.\\\hline
D&amp;3,.&amp;3,.\\\hline
\end{array}</span>
What is player 1's best response to <span class="math-container">$L$</span>?
What is player 1's best response to <span class="math-container">$R$</span>?
What is player 1's best response to the mixture <span class="math-container">$\left(\frac12,\frac12\right)$</span>?</p>

CC BY-SA 4.0
2022-04-05T11:05:27.680Z
2022-04-05T11:05:27.680Z
VARulle
<p>Suppose that several pure strategies in a 2-individual game have pure strategy best responses. Can we say that best responses to convex combination of those pure strategies still lie in the convex hull of their best responses?</p>
<p>If not, what are the obvious counter-examples?</p>

<game-theory><nash-equilibrium>
Best response to convex combination of strategies
ask
# Best response to convex combination of strategies

(Asked by: username_0 on 2022-04-05)

Suppose that several pure strategies in a 2-individual game have pure strategy best responses. Can we say that best responses to convex combination of those pure strategies still lie in the convex hull of their best responses?

If not, what are the obvious counter-examples?

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


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## 1 answers

### Answer by username_1 on 2022-04-05. Score: 3 | Accepted answer

Here's a hint. Consider this game:
\begin{array}{|c|c|c|}
\hline
&L&R\\\hline
U&4,.&0,.\\\hline
M&0,.&4,.\\\hline
D&3,.&3,.\\\hline
\end{array}
What is player 1's best response to $L$?
What is player 1's best response to $R$?
What is player 1's best response to the mixture $\left(\frac12,\frac12\right)$?


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