Compound lottery

Definition

A compound lottery is a lottery of lotteries. A compound lottery is therefore given by $$\hat L = [q_1(L_1), q_2(L_1), ..., q_J(L_J)]$$ where $q_1,...,q_J$ are nonnegative numbers summing to $1$, and $L_1,...,L_J$ are lotteries in $\mathcal{L}$.

Therefore for each $1 \leq j \leq J$ there are nonnegative numbers $(p_k^j)_{k=1}^K$ summing to $1$ such that $$L_j = [p_1^j(A_1), p_2^j(A_1), ..., p_K^j(A_K)]$$

Identification with simple lottery

Every simple lottery can be identified with compound lottery that yields the simple lottery $L$ with probability $1$, $$\hat L = [1(L)]$$

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 14-15.