Lottery
For a finite set of outcomes $O$, a lottery is a probability distribution on $O$. The degenerate lottery at $o\in O$ assigns probability one to $o$.
Given lotteries $L_1,L_2$ and a scalar $\alpha\in[0,1]$, the compound lottery $$ [\alpha L_1,(1-\alpha)L_2] $$ is the lottery assigning each outcome the probability $$ \alpha L_1(o)+(1-\alpha)L_2(o). $$
Implementation note
Lottery 𝕜 O is a domain-flavored abbrev for stdSimplex 𝕜 O. The
constructors Lottery.pure and Lottery.mix are definitional aliases for
stdSimplex.pure and stdSimplex.mix from Core.Simplex; the canonical
algebra (Lottery.expectedValue_pure, Lottery.expectedValue_mix,
Lottery.expectedValue_mono, Lottery.expectedValue_const) is one-line
wrappers over wsum_pure_apply, wsum_mix, wsum_le_wsum, wsum_const.
See [[math.simplex.mix]] for the underlying convex-combination vocabulary.
References
- [MSZ, Chapter 2, Definitions 2.9-2.11] Maschler, Solan, and Zamir, Game Theory. Lotteries, degenerate lotteries, and compound lotteries.