Just take the midpoint?

An intuitive response to imprecise probability, and its limitations

Some people react to the idea that we should use imprecise probabilities and that this leads to consequentialist cluelessness by saying: Just take the midpoint (of your interval-valued probabilities or expectations)! No more indeterminacy in the comparisons of expected values.

I think there’s something to this suggestion. It can be motivated by a symmetry intuition that I share to some extent. But I think that there are serious costs to any such view, and it might not be action-guiding anyway, on account of the vagueness of the intervals we’re taking the midpoints of. Let’s look at different ways to just-take-the-midpoint and their problems.

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For agents with imprecise probabilities, the decision rule to beat is the “Maximality” rule. Recall that on the imprecise model, our beliefs are represented by a set of probability distributions called our “representor”, and Maximality says that

• Action a is better than b if it has a higher expected value for every distribution in our representor (call this relation Dominance);

• Action a is permissible iff there’s no action Dominance-better than a.

Sadly, if our beliefs are severely imprecise — as they arguably should be when it comes to, e.g., the long-run consequences of our actions — lots of things are going to be permissible. This is because, for lots of actions a and b, we’ll be able to find a precise distribution in our representor according to which a is better, and a different distribution so that b is better, so neither is better than the other in the sense of Dominance; the comparison is indeterminate.

But you might think that this rule isn’t discerning enough. Suppose we have a prospect that could either give us +1 utility or -1 utility, and are totally clueless as to how likely the respective outcomes are. Our probabilities are not 1/2 and 1/2, but ?? and ??. Still, one might have the intuition that our beliefs about possible utilities are "symmetrical around zero" – (+1, ??) mirrors (-1, ??) – in a way that should lead us to being indifferent between this prospect and a certainty of 0 utility.¹

If we were using imprecise probabilities, our valuation of this prospect would naturally be represented by an interval-valued expectation (-1, 1). The aforementioned symmetry shows up in the symmetry of this interval around zero. It's tempting to say that we ought to just take the midpoint of this interval, on the grounds of symmetry. The question is then how to make this move without using a second-order probability distribution over our representor, which I think we should reject for much the same reasons we’re adopting imprecise probabilities in the first place. (E.g., what is the parameterization of the representor over which you specify this higher-order prior? The choice seems arbitrary.)

Besides using higher-order probabilities, there are a few other versions of “just take the midpoint” that I’ll look at in this post:

• (Undominated) difference-of-midpoints. Takes the action whose expected utility interval has the largest midpoint. The undominated version (which I think is preferable) restricts attention to actions that aren’t Dominance-worse than some other action. Problem: Preferences can flip under arbitrarily small perturbations of one’s expectations.

• Midpoint-of-differences. First takes the difference of expected utilities between a and b for each probability distribution in the representor, and then prefers a to b if the midpoint is larger than zero. Problem: Generates preference cycles.

(Much of this ground is covered in Bradley (2012, chap. 5)’s discussions of EHurwicz_ρ (our “difference-of-midpoints” when ρ=½ ), LHurwicz_ρ (similar to our “undominated difference-of-midpoints”), and pairwise regret (equivalent to midpoint-of-differences). I recommend checking out that chapter if you’re particularly interested in decision rules for imprecise probabilities.)

Difference-of-midpoints

First some more notation. Let

• P be our representor,

• u_p(a) be the expected utility of action a under precise distribution p;

• U(a) be the set of expected utilities associated with a, i.e., U(a) = {u_p(a) : p in P};

• M(U(a)) be the midpoint of U(a) (we’ll assume for simplicity that U(a) is always an interval).

The difference-of-midpoints rule tells us that a is better than b iff U(a) has a larger midpoint than U(b). Unfortunately, difference-of-midpoints is at odds with Dominance, which is a pretty appealing property. Consider a decision problem with these expected utilities:

• u_p(a) = p;

• u_p(b) = p²;

• In our representor, p ranges over (0,1).

Dominance says a is better than b, since it’s better than b for each element of the representor. But U(a) = U(b) = (0,1), so they have the same midpoint!

So we could instead use the undominated difference-of-midpoints rule, which says: Prefer a to b iff M(U(a)) > M(U(b)) or a Dominates b. That seems better, but it still has some weird properties. For example, preferences can flip due to arbitrarily small perturbations to our expectations. Consider the same decision problem as above, with a new action:

• u_p(a) = p;

• u_p(b) = p²;

• u_p(b⁺) = p² + ε, for a fixed ε arbitrarily close to 0;

• In our representor, p ranges over (0,1).

We now have a > b, because of Dominance, but b⁺ > a, since U(a) = (0,1) and U(b⁺) = (ε, 1+ε)!

Seems non-ideal. (Maybe one could try a defense along these lines: the Dominance and Midpoint relations are accounting for two different kinds of decision-theoretic value. While discontinuity is weird if one only changes prospects along one dimension of value, it’s not so strange if things flip when one value relation goes silent and the other becomes active.)

Midpoint-of-differences / pairwise regret

Here’s a different way of cashing out “just take the midpoint”. Now let U(a,b) be the set of differences in the expected utility of a and b, ranging over the representor. I.e., U(a,b) = {u_p(a) - u_p(b) : p in P}. The midpoint-of-differences rule says: Strictly prefer a to b iff U(a,b) > 0.

(As it happens, this is equivalent to the pairwise regret rule, which says to prefer a to b iff max_{p in P} u_p(a) - u_p(b) > max_{p in P} u_p(b) - u_p(a).)

Clearly, midpoint-of-differences will say that a is better than b whenever Dominance does. And it doesn’t have the same discontinuity issue as undominated difference-of-midpoints. But it’s got its own problem: It can lead to cycles.

Consider the following case. There are three states; tuples (x, y, z) will represent utilities at each of the three states, given a particular action. Beliefs about which state obtains are maximally imprecise.

Payoff tuples for three actions are:

• a: (-1, +1, 0)

• b: (-0.1, -0.1, -0.1)

• c: (0.8, -0.2, -1.2)

So we’ve got

• M(U(a,b)) = M((-0.9, 1.1)) = 0.1. So a > b;

• M(U(b,c)) = M((-0.9, 1.1)) = 0.1. So b > c;

• M(U(c,a)) = M((-1.2, 1.8)) = 0.3. So c > a.

Unlike many folks, I’m actually not so bothered by cyclicity per se. But I am bothered by the possibility that rules for resolving cyclic preferences fail to give the answers that made the whole take-the-midpoint thing intuitive to begin with. The comparison between a and b in this example is a central example of a case where we wanted a > b: Our beliefs about a’s payoffs are perfectly “symmetrical around zero”, and b has a precise and negative expected value. And yet:

The most straightforward choice rule for cyclic preferences is the top cycle rule, which says that the set of permissible actions is the smallest set A of actions a such that a is better than everything outside of A. In this example, the top cycle is just {a, b, c}, so a isn’t to be chosen over b. There are many refinements of the top cycle, for all I know it’s possible for such methods to choose b-like actions over a-like ones in some contexts.

(Maybe one could argue for some way of refining the top cycle in a way that will always get the intuitive answer. (Taking the element of the top cycle whose U(a) has the highest midpoint could do it, but this seems ad hoc?) Or one could try to say something like, betterness is fundamentally comparative, and a really is better than b; but choice can come apart from betterness in the context of cyclic betterness. That is, we might have to choose counter to our betterness relation, so that our choice function as a whole is coherent. Neither of these seems very promising to me, but I could be wrong.)

Vagueness

Finally, even supposing we end up liking one of these rules, I’m not sure how much it helps with the indeterminacy issue. In real life, our beliefs should not just be imprecise but vague. I’m usually not going to have U(a,b) = (-0.924, 1.07), I’m going to have U(a,b) = (vague negative number, vague positive number of similar magnitude). And then I’m back to indeterminacy, because some precisifications of the midpoint of such an interval will be less than zero and others will be greater than zero.

The rules could still be action-guiding in cases where there is reason for us to have asymmetric intervals. As in, U(a,b) = (vague small negative number, vague big positive number) would determinately favor a under midpoint-of-differences. It’d be interesting to think about whether there are cases with such asymmetries when it comes to longtermist cause prioritization.

Acknowledgements

Thanks to Anthony DiGiovanni for helpful comments.

References

Bradley, S. 2012. “Scientific Uncertainty and Decision Making.” PhD thesis.

1. This line of thought was inspired by a quadrilemma for agents with imprecise beliefs, presented by Sylvester Kollin in a forthcoming manuscript.