Advanced Microeconomics/Decision Making Under Uncertainty

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Decision Making Under Uncertainty[edit]

Lotteries[edit]

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[edit]

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[edit]

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}}.