mixed extension

Definition

Let $G = (N, (S_i)_{i \in N},(u_i)_{i \in N})$ be a strategic-form game in which for every player $i \in N$, the set of pure strategies $S_i$ is nonempty and finite. Denote by $S := S_1 \times S_2 \times ... \times S_n$ the set of pure strategy vectors. The mixed extension of $G$ is the game $$\Gamma = (N, (\Sigma_i)_{i \in N}, (U_i)_{i \in N})$$ in which for each $i \in N$, player $i$'s set of strategies is $\Sigma_i = \Delta(S_i)$, and his payoff function is $U_i : \Sigma \to \mathbb{R}$, which associates each strategy vector $\sigma = (\sigma_1,...,\sigma_n) \in \Sigma = \Sigma_1 \times ... \times \Sigma_n$ with the payoff $$U_i(\sigma) = \mathbf{E}_\sigma[u_i(\sigma)] = \sum_{(s_1,...,s_n)\in S} u_i (s_1,...,s_n)\sigma_1(s_1)\sigma_2(s_2)...\sigma_n(s_n)$$

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 147.