= Mixed strategy
{title2=$\sigma_i\in\Delta(S_i)$}
{wiki}
A <mixed strategy> is a <probability distribution> over a player's <pure strategies>. For a <finite game> with $m_i$ such strategies, the possible <mixed strategies> form the <probability simplex>
$$
\Delta(S_i)=\left\{\sigma_i\in\mathbb R^{m_i}:\sigma_i(a)\geq0,\quad\sum_{a\in S_i}\sigma_i(a)=1\right\}.
$$
A <pure strategy> is included as a <mixed strategy> placing all <probability> on one choice. The <support of a mixed strategy> consists of the choices with positive <probability>. In a sequential game, sampling a complete <pure strategy> once is the definition; it is not automatically equivalent to fresh randomization at every move.
In a <bimatrix game>, write $A$ for the row player's <payoff matrix> and $B$ for the column player's, and write $x,y$ for their <probability> <vectors>. <Independent> randomization gives <expected payoffs>
$$
U_1(x,y)=x^{\mathsf T}Ay,\qquad U_2(x,y)=x^{\mathsf T}By.
$$
For example, in <matching pennies>, choosing heads with <probabilities> $p=3/4$ and $q=1/4$ gives joint <probabilities> $3/16,9/16,1/16,3/16$ for $(H,H),(H,T),(T,H),(T,T)$. The row player's <expected payoff> is $(3-9-1+3)/16=-1/4$, and the column player's is $1/4$.
Randomization can prevent an opponent exploiting a predictable choice. It does not mean every choice is equally good against every opponent: only <best responses> belong in the <support of a mixed strategy> at a <Nash equilibrium>. A shared random signal can instead produce a <lottery over joint action profiles> whose joint <probabilities> do not factor into <independent> <mixed strategies>.
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