Wolfgang Schwarz

Preference and the Principal Principle

Decision theoretic representation theorems show that one can read off an agent's probability and utility functions from their preferences, provided the latter satisfy certain minimal rationality constraints. More substantive rationality constraints should therefore translate into further constraints on preference. What do these constraints look like?

Here are a few steps towards an answer for one particular constraint: a simple form of the Principal Principle. The Principle states that if cr is a rational credence function and ch=p is the hypothesis that p is the chance function, then for any E in the domain of p,

(1) cr(E/ch=p) = p(E).

Following Jeffrey, we can compute the utility (desirability) of ch=p as follows, assuming that E ranges over some partition of the relevant chance events (individual outcomes, say):

(2) u(ch=p) = \sum_E u(E) cr(E/ch=p).

Combining (1) and (2) we get

(3) u(ch=p) = \sum_E u(E) p(E).

Now let q be some other candidate chance function, and let xp+(1-x)q be the mixture of p and q that assigns probability xp(E)+(1-x)q(E) to each event E. By the same reasoning as above, the desirability of the hypothesis that xp+(1-x)q is the actual chance function is

(4) u(ch = xp+(1-x)q) = \sum_E u(E) xp(E)+(1-x)q(E).

Rearranging the right-hand side in (4) gives us

(5) u(ch = xp+(1-x)q) = x u(ch=p) + (1-x) u(ch=q).

(5) is von Neumann and Morgenstern's linearity condition on utility. Their representation theorem shows that the condition is satisfied whenever

1. preference is a weak order, and
2. if you prefer ch=p to ch=q, then for any chance function r, you prefer ch = xp+(1-x)r to ch = xq+(1-x)r.

I.e. a mixture of a better chance with r is better than a mixture of a worse chance with r. (There's also an Archimedean third condition, but I don't think it is required for linearity.)

So the Principal Principle shows up as the constraint (b) in von Neumann and Morgenstern's axiomatization. Unfortunately, (b) is a little too weak to revocer the full Principle, since we can't go back from (4) to (1).