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In this paper, "Savages’ Subjective Expected Utility Model" by Edi Karni, he gives a definition of "conditional preferences." See here:

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What situation is this supposed to capture? It seems like it might be more useful to define conditional preferences as $f \succeq_E f'$ if $f(s) \succeq f'(s)$ for all $s \in E$. This definition seems like it might be more ... useful? For one, it seems to resemble conditional expectation that way (or does it?). What idea is the definition as given in the paper supposed to capture (this might be a simple question)?

Also, would it be more general to say, in the definition given in the paper, that $f(s) \sim f'(s)$ instead of $f(s) = f'(s)$?

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1 Answer 1

The statement $f\succcurlyeq f^\prime$ and for every $s$ not in $E$, $f(s)=f'(s)$ says that $f$ and $f^\prime$ have the consequences unless $E$ occurs, and $f$ is preferred to $f'$. Loosely, the sure thing principle says that if two acts differ only in their consequence should $E$ occur, their ordering should depend only on their consequences in $E$ and not on states outside of $E$. What the sure thing principle gets you is that if $f\succcurlyeq_E f^\prime$ then $f$ must be conditionally preferred to $f^\prime$ because of the consequences of $f^\prime$ in $E$. Otherwise this would not be known.

Here's a plain English interpretation of conditional preference. Imagine I put together two portfolios, $A$ and $B$. I start both portfolios off with the same asset, let's just say some money in a mutual fund. In portfolio $A$ I put an additional asset that pays a dollar in all states of the world $E$ where a democrat wins the 2016 election. I give you a choice between $A$ and $B$. In action $f$ you choose $A$ and in action $f'$ you choose $B$. Presumably you prefer $f$ to $f'$. But the payoffs of $f$ and $f'$ differ only should a democrat win the 2016 election. So by the definition Karni has given you $f$ is preferred to $f'$ given $E$. With the sure thing principle, I can separate your preference in $E$ states from your preferences in not $E$ states. So I now know your preference for $f$ must stem from $f$ being in some sense better than $f'$ in $E$ states.

Technically (read: obnoxiously) preferences are defined over the functional space $F$, so the abstruse definitions come from the fact that the notation $f(s)\succcurlyeq f'(s)$ or $f(s)\sim f'(s)$ is not yet defined. Instead Karni has to work with the $f_E$ notation. In this notation you'd write your idea for conditional preference as $f\succcurlyeq_E f'$ iff $f_Eh\succcurlyeq f'_Eh$ (the first part of the sure thing principle). I agree that this is more in line with my notion of what conditional preference should be. Savage mentions both senses in his discussion:

What technical interpretation can be attached to the idea that $f$ would be preferred to $g$, if $B$ were known to obtain? Under any reasonable interpretation, the matter would seem not to depend on the values $f$ and $g$ assume at states outside of $B$. There is, then, no loss of generality in supposing that $f$ and $g$ agree with each other at states outside of $B$, that is, that $f(s)=g(s)$ for all $s \in \sim B$. Under this unrestrictive assumption, $f$ and $g$ are surely to be regarded as equivalent given $\sim B$, that is, they would be considered equivalent, if it were known that $B$ did not obtain. The first part of the sure-thing principle can now be interpreted thus: If, after being modified so as to agree with one another outside of $B$, $f$ is not preferred to $g$; then $f$ would not be preferred to $g$, if $B$ were known. The notion will be expressed formally by saying that $f\leq g$ given $B$.

(Here $\sim$ is set notation instead of preference ordering.)

The sense that Karni involves two acts that agree with each other outside of $B$. However Savage's definition of conditional preference is exactly $f\succcurlyeq_E f'$ iff $f_Eh\succcurlyeq f'_Eh$. If there really is no loss of generality in the rest of the proofs it doesn't matter which definition you pick. However Savage's definition is in line with both yours and mine, and the definition of conditional expectation. I'll look to see if that has any real consequences for Karni.

ADDENDUM Using equality instead of $\sim$ is plenty general, simply because if the two acts only differ in consequences that do not have an effect on the preference ordering, you could just redefine the consequence space such that the equality was still maintained. Which is a cop out, but also true.

I believe I've figure it out. The only place Karni seems to use his conditional preference definition is in P7. Savage uses an analogous P7 but with his own definition of conditional preference. P7 says (Savage's wording):

If $f \leq g(s)$ given $B$ for every $s\in B$, then $f \leq g$ given $B$.

This is a statement of conditional dominance: "a certain consequence $g(s)$ is better than an (uncertain) $f$ for every possible $s$ in $B$, so $g$ should be preferred to $f$ given $B$." Because we have already accepted the sure-thing principle, P7 from Savage and P7 from Karni are equivalent. If we had not accepted the sure-thing principle, Savage's would be more restrictive. So Karni is actually being less restrictive with his definition of conditional preference, and that is presumably the purpose.

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