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?
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 $\left(\frac12,\frac12\right)$?
