# Definition:Lottery

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## Definition

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.

## Sources

- 1991: Roger B. Myerson:
*Game Theory*... (previous) ... (next): $1.2$ Basic Concepts of Decision Theory - 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $2.1$: Strategic Games