# DRMacIver's Notebook

Physical and Topological Limitations to Rational Choice

Physical and Topological Limitations to Rational Choice

Epistemic status: Confident.

Attention conservation notice: So much inside baseball.

Context: This is something I’ve known about for a while but couldn’t find a concise write-up of that I had previously written and still liked, so I thought I’d just rewrite it here.

The Von-Neumann-Morgenstern utility theorem states that if you ask people to choose between finite lotteries over outcomes, any “rational” behaviour looks like picking based on whichever gives the greatest expected utility according to some utility function over the outcomes.

I have a number of objections to this idea, but my main one is this:
Regardless of the axioms you choose for rationality, it’s physically
impossible to implement an agent that can express the sort of total
order over lotteries that VNM rationality is a theorem about, even if
you grant the existence of logically omniscient agents (you can do it if
you grant the existence of *actually* omniscient agents).

Why?

Well, suppose we have some total preorder \(\preceq \subseteq \mathcal{L}^2\), where \(\mathcal{L}\) is the set of lotteries over some finite set of outcomes. Take this total order and define the choice function \(\tau : \mathcal{L}^2 \to \{-1, 0, 1\}\) where \(\tau(u, v) = 0\) if \(u \preceq v\) and \(v \preceq u\), else if \(u \prec v\) then \(\tau(u, v) = -1 = -\tau(v, u)\). i.e. \(\tau\) is a function determining whether we strictly prefer one or are indifferent between the two.

Any choice that a physical agent makes must be based on a finite (but not necessarily bounded up front) set of observations. Each of those observations can only give you information about the world up to some non-zero (but potentially arbitrarily small) tolerance. In particular, if we have lotteries \(u_1 \preceq u_2\), there is some \(\epsilon > 0\) such that if \(d((u_1, u_2), (v_1, v_2) < \epsilon\), we must have \(\tau(v_1, v_2) = \tau(u_1, u_2)\), because we only ever looked at \(u_1, u_2\) up to some finite precision.

In particular this means that \(\tau: \mathcal{L}^2 \to \{-1, 0, 1\}\) is a continuous function!

Unfortunately, \(\mathcal{L}^2\) is a connected topological space and \(\{-1, 0, 1\}\) is totally disconnected. Thus any continuous function must be constant. We know that \(\tau(u, u) = 0\), so we must have \(\tau(u, v) = 0\) for all \(u, v\). i.e. the only physically possible total preorder that we can express is the one that is totally indifferent between lotteries.

If the above argument makes no sense to you, another way to look at it is that you need to know \(u\) and \(v\) to infinite precision at the boundary. If you are on a boundary point where \(\tau(u, v) = 0\), moving slightly in the direction of \(u \prec v\) immediately forces your hand, so you cannot satisfy that continuity property at the boundary.

This problem can be made to go away by removing the indifferent set from the set of lotteries you consider - it’s perfectly physically possible to distinguish the lotteries if you know a priori that you will definitely have a preference for one of them. Unfortunately there are several problems with this:

- Where does that a priori knowledge come from? It’s obviously not true in general - you have to somehow avoid ever being asked to choose between a lottery and itself.
- The VNM axioms rely crucially on the use of indifference in the continuity axiom.
- The implementation of such a choice function is still physically fraught, because as you approach the boundary the amount of precision you require to decide tends to infinity.

It is possible that there is some cunning workaround that lets you rescue VNM choice theory, but I find it very implausible that this is the case. The basic requirement that you construct a discrete function of the world is intrinsically aphysical, and it seems very hard to rescue that.

I have yet to sit down and think through exactly what I would like in its place in any great depth, mostly because nobody except me cares about this problem and I don’t care enough to pursue it solo, but my preferred primitive is to replace the choice function with a continuously varying choice probability, so instead you have a function \(\tau: \mathcal{L}^2 \to [0, 1]\), where \(\tau(u, v)\) is the probability of choosing \(u\) over \(v\), and \(\tau(v, u) = 1 - \tau(u, v)\). This neatly side steps all of the problems with using a discrete choice function, because you don’t need to know the outcome probabilities with infinite precision in order to make your choice, and \([0, 1]\) is a connected topological space so, unlike discrete choices, it’s perfectly possible to construct such functions.