# Definition:Conditional Preference

## Definition

Let $G$ be a lottery.

Let $P$ be a player of $G$.

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

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

Let $\Xi$ be the event space of $G$.

Let $L$ be the set of all plays of $G$.

Let $f$ and $g$ be two lotteries in $G$.

Let $S \subseteq \Xi$ be an event.

A conditional preference is a preference relation $\succsim_S$ such that:

$f \succsim_S g$ if and only if $f$ would be at least as desirable to $P$ as $g$, if $P$ was aware that the true state of the world was $S$.

That is, $f \succsim_S g$ if and only if $P$ prefers $f$ to $g$ and he knows only that $S$ has occurred.

The notation $a \sim_S b$ is defined as:

$a \sim_S b$ if and only if $a \succsim_S b$ and $b \succsim_S a$

The notation $a \succ_S b$ is defined as:

$a \succ_S b$ if and only if $a \succsim_S b$ and $a \not \sim_S a$

When no conditioning event $S$ is mentioned, the notation $a \succsim_\Omega b$, $a \succ_\Omega b$ and $a \sim_\Omega b$ can be used, which mean the same as $a \succsim b$, $a \succ b$ and $a \sim b$.