Advanced Microeconomics/Decision Making Under Uncertainty

Lotteries
A simple lottery is a tuple $$(p_1, \dots, p_N)$$ assigning probabilities to N outcomes such that $$\sum_{n=1}^{N}p_k = 1$$.

A compound lottery assigns probabilities $$(\alpha_1,\dots,\alpha_K)$$ to one or more simple lotteries $$ L_1, \dots, L_K$$

A reduced lottery may be calculated for any compound lottery, yielding a simple lottery which is outcome equivalent (produces the same probability distribution over outcomes) to the original compound lottery.

Consider a compound lottery over the lotteries $$L_1,\dots,L_K$$ each of which assigns probabilities $$p_1,\dots,p_N$$ to N outcomes. The compound lottery implies a probability distribution over the N outcomes which, for any outcome n, may be calculated as $$\sum_{k=1}^K\alpha_k\cdot p_n^k$$ In words, the probability of event n implied by a compound lottery is the probability of event n assigned by each lottery, weighted by the probability of a given lottery being chosen.

Example
Consider an outcome space $$\{1,2,3,4,5,6,7,8,9,10\}$$. A (fair) six sided dice replicates the simple lottery $$\left( \frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},0,0,0,0 \right)$$ and a (fair) ten sided dice replicates the simple lottery $$ \left(\frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}\right) $$

Now imagine a person randomly draws a dice from an urn known to contain nine six sided dice and one ten sided dice. This draw represents a compound lottery defined over the outcome space. The probability of any outcome $$\in [1,6] = \frac{9}{10}\cdot\frac{1}{6}+\frac{1}{10}\cdot\frac{1}{10} = \frac{16}{100}$$ and the probability of an outcome $$\in [7,10] = \frac{9}{10}\cdot 0 + \frac{1}{10}\cdot \frac{1}{10} = \frac{1}{100}$$. Producing a reduced lottery, $$\left(\frac{4}{25},\frac{4}{25},\frac{4}{25},\frac{4}{25},\frac{4}{25},\frac{4}{25}, \frac{1}{100},\frac{1}{100},\frac{1}{100},\frac{1}{100}\right)$$

Preferences and Uncertain Outcomes
Let $$\mathbf{\mathcal{Z}}$$ represent a set of possible outcomes (consumption bundles, monetary payments, et cetera) with a space of compound lotteries $$ \Delta \mathbf{\mathcal{Z}}$$.