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A lottery is a game in which the consequence of each move is determined by the realization of a random variable.

Let $X$ denote the set of prizes which a player may receive.

Let $\Omega$ denote the set of possible states.

A lottery is a mapping $f: X \times \Omega \to \R_{\ge 0}$ such that:

$\forall t \in \Omega: \displaystyle \sum_{x \mathop \in X} f \left({x, t}\right) = 1$

Probability Model

In a probability model, a lottery is a probability distribution over a set of prizes.

State-Variable Model

In a state-variable model, a lottery is defined as a mapping from a set of possible states into a set of prizes.