Math — Nash Equilibrium
Best responses and mixed strategies in game theory.
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u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) \quad \forall s_i
B_i(s_{-i}) = \arg\max_{s_i} u_i(s_i, s_{-i})
\sigma_i \in \Delta(S_i)
u_i(\sigma) = \sum_{s} \sigma(s)\,u_i(s)