Assumed background: Acyclic preferences, Dutch Book theorems

There are fairly elementary arguments that, in the absence of uncertainty, any preferences not described by a utility function are problematic - this is the circular preferences argument. There are also fairly elementary arguments that, if we handle uncertainty by taking weighted sums of utilities of different outcomes, then the weights should follow the usual rules of probability - these are the Dutch Book arguments. But in the middle there’s a jump: we need to assume that taking weighted sums of utilities makes sense for some reason. There are some high-powered theorems which make that jump (specifically the complete class theorem), but they’re not very mathematically accessible.

(If any of that sounds new, you should read Yudkowsky’s excellent intro to this stuff before reading this post.)

It turns out that there is a relatively simple theorem which bridges the gap between deterministic utility and Dutch Book arguments. But rather than hanging out in decision theory textbooks, it’s been living it up in finance. It’s called the Fundamental Theorem of Asset Pricing (FTAP).

Here’s the setup. Just like the Dutch Book arguments, we have a bunch of tradable assets - i.e. betting contracts, like stock options or horse race bets. We have a bunch of possible outcomes - i.e. possible prices of an underlying stock at expiry, or possible winners of the horse race. Each asset's final value will depend on the outcome. Then the FTAP states that either:

There exists some portfolio of assets which costs $0 to buy (can include short sales) and is guaranteed a positive payout (i.e. arbitrage), or
There exists a probability distribution such that the price of each asset is the expected value of its payout (i.e. price is a weighted sum of possible outcome values).

Note that this is exactly what we need to round out the Dutch Book arguments: either there exists an arbitrage opportunity, or we compare assets using a weighted sum of possible outcome values.

Let’s prove it. First, we’ll name some variables:

$V_{i j}$ : a big matrix which contains the value of each asset i under each possible outcome j.
$S_{i}$ : current price of asset i (we need P for probability, so S represents price).
$p_{j}$ : probability distribution over outcomes j (which may or may not exist)
$q_{i}$ : arbitrage portfolio (which may or may not exist)

FTAP says that either:

Arbitrage portfolio exists: profit $\sum_{i} q_{i} V_{i j} > 0$ for all outcomes j, and the portfolio currently costs $\sum_{i} q_{i} S_{i} = 0$ .
Probability distribution exists: $S_{i} = \sum_{j} V_{i j} p_{j}$

I’ll state the proof informally - if you know a little linear algebra, it’s easy but tedious to formalize and see that it works. The key question is: how many assets, and how many possible outcomes? With N assets and M outcomes, our arbitrage condition has N variables (the q’s) and M+1 equations (one for each outcome plus the current cost constraint). Conversely, our probability distribution condition has M variables (the p’s) and N equations. We generally expect the system to be solvable when the number of variables is at least as large as the number of equations. So, either:

N > M (more assets than outcomes), and the arbitrage system (typically) has a solution, or
M >= N (at least as many outcomes as assets), and the probability system (typically) has a solution.

I’m brushing some stuff under the rug here - i.e. maybe there are more assets than outcomes, but the prices line up perfectly. That’s where the linear algebra comes in - the above works for full-rank V, but rank-deficient V requires checking the usual corner cases. If you take a math finance class, you’ll probably go through that tedium in its full glory, along with some more interesting extensions of the theorem.

Anyway, what have we shown? We actually haven’t established that the “probability distribution” p_j is a probability distribution - we’ve shown that the prices are described by some weighted sum of outcome values, but the weights could still be negative or not sum to 1. That’s fine - the usual Dutch Book arguments show that the weights are a probability distribution (or else there’s an arbitrage opportunity). We’ve bridged the gap.

All the usual considerations of the Dutch Book theorems still apply. “Arbitrage” means exactly the same thing here that it means in the Dutch Book theorems. As usual, we’re formulating things with “bets” and “contracts” and “arbitrage” and “prices”, but that can model a much wider range of phenomena.

One interesting point: the probability distribution may not be unique. There may be more than one possible distribution which satisfies the conditions. This works fine with the Dutch Book arguments: each possible distribution corresponds to a different prior.