Is it possible to have one pure strategy and infinitely many mixed strategies?

Question

I'm wondering if it is possible to construct a game in which there is one pure strategy and infinitely many mixed strategies?

I don't believe this is true, since mixed strategies mix over pure strategies, but I'm not sure.

Answer 1

No, say you only have one pure strategy $s_1$ so $S_1=\{s_1\}$, then there is only one mixed strategy $\sigma$, i.e. probability distribution over $S_1$, s.t. $\sum_{s_i\in S_1}\sigma(s_i)=\sigma(s_1)=1$.