Let $G$ be a lottery.
Let $P$ be a player of $G$.
Let $\Xi$ be the event space 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 $S$.
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 $a \sim b$.