## Tuesday, October 28, 2014

### Blackwell and Girshick: Theory of Games and Statistical Decisions (1954), Ch. 4

There's an interesting representation theorem in Blackwell and Girshick's textbook in statistics: It provides a set of sufficient conditions for a preference ordering over lotteries to be expressible as a prior probability distribution (Th. 4.3.1, p. 118).

I assume the theorem comes from either Wald, Savage, or de Finetti, but no reference is given.

### Well-Behaved Preferences

A lottery can here be defined as a function from the sample space to the real numbers. The conditions in the theorem are then the following:
• The ordering of two lotteries $f$ and $g$ cannot depend on the availability of other lotteries.
• If a lottery $f$ provides a higher payoff than a lottery $g$ at all points in the sample space $\Omega$, then $f$ must be preferred to $g$.
• If $f$ is preferred to $g$, then $f+h$ must be preferred to $g+h$.
An inspection of the proof also shows that they should have included a continuity condition:
• If $f_1, f_2, \ldots$ is a series of lotteries converging to a limit $f$, and if $f_i$ is preferred to $g$ for all $i$, then $f$ must also be preferred to $g$.
When these conditions are met, the preference ordering over the lotteries can be expressed as a distribution over the sample space, unless it categorizes all lotteries as equally good.

### The Large and the Good

The proof of the theorem uses the fact that two convex, disjoint, open sets can be separated by a hyperplane. Here's a sketch:

If we let $e$ be the lottery that pays zero in all situations $\omega \in \Omega,$ then we can define the following sets:
\begin{eqnarray}
F[>] &=& \{f\ |\ \forall \omega \in \Omega: f(\omega) > 0\},
\\
F[\gtrsim] &=& \{f\ |\ f\gtrsim e\},
\end{eqnarray}
and we can further define, in the usual way, $F[>] + F[\gtrsim]$ to be the sums of lotteries from those two sets.

Now, $F[>]$ is an open set, and hence $F[>] + F[\gtrsim]$ is, too. Further, $F[>]$ and $F[\gtrsim]$ are both closed under addition (due to the third assumption), and hence their sum is, too. They are also both closed under multiplication with a scalar, and again, so is their sum — but this latter argument requires a bit more spelling out.

### Rational and Real Convexity

Suppose a lottery $f$ is preferred to the zero lottery, that is, $f \in F[\gtrsim]$. The third assumption of the theorem then tells us that
$$e \;\lesssim\; f \;\lesssim\; f + f \;\lesssim\; f + f + f \;\lesssim\; \ldots \;\lesssim\; nf.$$
By further adding multiple copies of the zero lottery to both sides of this preference ineqality, we can see that
$$me \;\lesssim\; (m-1)e + nf \;=\; nf,$$
where we have selectively used the fact that $e$ is the zero lottery. Putting these facts together, we then have the preference inequality
$$e \;\lesssim\; \left(\frac{n}{m}\right)f.$$
By using a positive sequence of rational approximations $(n/m) \rightarrow \lambda$, we can use this fact along with the continuity assumption to conclude that $F[\gtrsim]$ is closed under multiplication with any positive, real scalar $\lambda$.

I don't think there's a way around this last technicality. It is, incidentally, the same proof technique used to prove that the logarithmic functions are the only continuous functions that turn products into sums.

### Cutting the Cake

At any rate, $F[>] + F[\gtrsim]$ is an open and convex set separated from the singleton set $\{e\}$. We can therefore conclude that there is a lottery (or vector) $p$ which defines the hyperplane $\{f\ |\ f\cdot p=0\}$ separating $F[>] + F[\gtrsim]$ from $\{e\}$. The set $F[>] + F[\gtrsim]$ is thus a subset of the half-space $\{f\ |\ f\cdot p \geq 0\}$.

This vector $p$ must have nonnegative coordinates, since the set $F[>]$ is unbounded in all positive directions. If $p$ had a negative coordinate, $p(\omega) \leq 0$, we could choose a lottery for which the corresponding coordinate, $f(\omega)$, was so large that $f\cdot p < 0$. This would violate the definition of $p$, and $p$ hence has to be a nonnegative vector which can be interpreted as a probability distribution.

It would also have the property that $f\cdot p \geq g\cdot p$ if and only if $f \gtrsim g$. This follows from the fact that $f\cdot p \geq g\cdot p$ if and only if $(f - g) \in F[>] + F[\gtrsim]$, which holds if and only if $f$ can be expressed as the sum of $g$ and some lottery preferable to the zero lottery.