Definition

A compound lottery is a lottery of lotteries. A compound lottery is therefore given by

Lˆ=[q1(L1),q2(L1),...,qJ(LJ)]\hat L = [q_1(L_1), q_2(L_1), ..., q_J(L_J)]

where q1,...,qJq_1,...,q_J are nonnegative numbers summing to 11, and L1,...,LJL_1,...,L_J are lotteries in ℒ\mathcal{L}.

Therefore for each 1≤j≤J1 \leq j \leq J there are nonnegative numbers (pkj)k=1K(p_k^j)_{k=1}^K summing to 11 such that

Lj=[p1j(A1),p2j(A1),...,pKj(AK)]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 LL with probability 11,

Lˆ=[1(L)]\hat L = [1(L)]

References

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