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 $a \sim b$.

This article contains statements that are justified by handwavery. In particular: Myerson is as lax as all the other game theory writers when it comes to defining rigorous concepts. I am going to have to abandon this field of study until I really understand exactly what the underlying mathematical objects are. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding precise reasons why such statements hold. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Handwaving}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1991: Roger B. Myerson: Game Theory ... (previous) ... (next): $1.2$ Basic Concepts of Decision Theory