Mixed strategies play an important role in many game-theoretic analyses. Let \(\Delta(X)\) denote the set of probability measures over the finite[5]  set \(X\). A mixed strategy for player \(i\), is an element \(m_i\in \Delta(S_i)\). If \(m_i\in\Delta(S_i)\) assigns probability 1 to an element \(s_i\in S_i\), then \(m_i\) is called a pure strategy (in such a case, I write \(s_i\) for \(m_i\)). Mixed strategies are incorporated into a game-theoretic analysis as follows. Suppose that \(G=\langle N, \{S_i, u_i\}_{i\in N}\rangle\) is a finite strategic game. The mixed extension of \(G\) is the strategic game in which the strategies for player \(i\) are the mixed strategies in \(G\) (i.e., \(\Delta(S_i)\)) and the utility for player \(i\) (denoted \(U_i\)) of the joint mixed strategy \(m\in \Pi_{i\in N} \Delta(S_i)\) is calculated in the obvious way (let \(m(s)=m_1(s_1)\cdot m_2(s_2)\cdots m_n(s_n)\) for \(s\in \Pi_{i\in N} S_i\)):