Threshold Strategy in Approval and Range Voting

How to turn polling insight into an optimal ballot — and why anything else is wasted.

“approve of”? What does that mean anyway?

I have written previously about how approval and range voting methods are intrinsically tactical. This doesn’t mean that they are more tactical than other election systems (nearly all of which are shown to sometimes be tactical by Gibbard’s Theorem when there are three or more options). Rather, it means that tactical voting is unavoidable. Voting in such a system requires answering the question of where to set your approval threshold or how to map your preferences to a ranged voting scale. These questions don’t have more or less “honest” answers. They are always tactical choices.

But I haven’t dug deeper into what these tactics look like. Here, I’ll do the mathematical analysis to show what effective voting looks like in these systems, and make some surprising observations along the way.

Mathematical formalism for approval voting

We’ll start by assuming an approval election, so the question is where to put your threshold. At what level of approval do you switch from voting not to approve a candidate to approving them?

We’ll keep the notation minimal:

• As is standard in probability, I’ll write ℙ[X] for the probability of an event X, and 𝔼[X] for the expected value of a (numerical) random variable X.
• I will use B to refer to a random collection (multiset) of ballots, drawn from some probability distribution reflecting what we know from polling and other information sources on other voters. B will usually not include the approval vote that you’re considering casting, and to include that approval, we’ll write B ∪ {c}, where c is the candidate you contemplate approving.
• I’ll write W(·) to indicate the winner of an election with a given set of ballots. This is the candidate with the most approvals. We’ll assume some tiebreaker is in place that’s independent of individual voting decisions; for instance, candidates could be shuffled into a random order before votes are cast, in in the event of a tie for number of approvals, we’ll pick the candidate who comes first in that shuffled order.
• U(·) will be your utility function, so U(c) is the utility (i.e., happiness, satisfaction, or perceived social welfare) that you personally will get from candidate c winning the election. This doesn’t mean you have to be selfish, per se, as accomplishing some altruistic goal is still a form of utility, but we evaluate that utility from your point of view even though other voters may disagree.

With this notation established, we can clearly state, almost tautologically, when you should approve of a candidate c. You should approve of c whenever:

𝔼[U(W(B ∪ {c}))] > 𝔼[U(W(B))]

That’s just saying you should approve of c if your expected utility from the election with your approval of c is more than your utility without it.

The role of pivotal votes and exact strategy

This inequality can be made more useful by isolating the circumstances in which your vote makes a difference in the outcome. That is, W(B ∪ {c}) ≠ W(B). Non-pivotal votes contribute zero to the net expectation, and can be ignored.

In approval voting, approving a candidate can only change the outcome by making that candidate the winner. This means a pivotal vote is equivalent to both of:

• W(B ∪ {c}) = c
• W(B) ≠ c

It’s useful to have notation for this, so we’ll define V(B, c) to mean that W(B ∪ {c}) ≠ W(B), or equivalently, that W(B ∪ {c}) = c and W(B) ≠ c. To remember this notation, recall that V is the pivotal letter in the word “pivot”, and also visually resembles a pivot.

With this in mind, the expected gain in utility from approving c is:

• 𝔼[U(W(B ∪ {c}))] - 𝔼[U(W(B))]. But since the utility gain is zero except for pivotal votes, this is the same as
• ℙ[V(B, c)] · (𝔼[U(W(B ∪ {c})) | V(B, c)] - 𝔼[U(W(B)) | V(B, c)]). But since V(B, c) implies that W(B ∪ {c}) = c, so this simplifies to
• ℙ[V(B, c)] · (U(c) - 𝔼[U(W(B)) | V(B, c)])

Therefore, you ought to approve of a candidate c whenever

U(c) > 𝔼[U(W(B)) | V(B, c)]

This is much easier to interpret. You should approve of a candidate c precisely when the utility you obtain from c winning is greater than the expected utility in cases where c is right on the verge of winning (but someone else wins instead).

There are a few observations worth making about this:

• The expectation clarifies why the threshold setting part of approval voting is intrinsically tactical. It involves evaluating how likely each other candidate is to win, and using that information to compute an expectation. That means advice to vote only based on internal feelings like whether you consider a candidate acceptable is always wrong. An effective vote takes into account external information about how others are likely to vote, including polling and understanding of public opinion and mood.
• The conditional expectation, assuming V(B, c), tells us that the optimal strategy for whether to approve of some candidate c depends on the very specific situation where c is right on the verge of winning the election. If c is a frontrunner in the election, this scenario isn’t likely to be too different from the general case, and the conditional probability doesn’t change much. However, if c is a long-shot candidate from some minor party, but somehow nearly ties for a win, we’re in a strange situation indeed: perhaps a major last-minute scandal, a drastic polling error, or a fundamental misunderstanding of the public mood. Here, the conditonal expected utility of an alternate winner might be quite different from your unconditional expectation. If, say, voters prove to have an unexpected appetite for extremism, this can affect the runner-ups, as well.
• Counter-intuitively, an optimal strategy might even involve approving some candidates that you like less than some that you don’t approve! This can happen because different candidates are evaluated against different thresholds. Therefore, a single voter’s best approval ballot isn’t necessarily monotonic in their utility rankings. This adds a level of strategic complexity I hadn’t anticipated in my earlier writings on strategy in approval voting.

Approximate strategy

The strategy described above is rigorously optimal, but not at all easy to apply. Imagining the bizarre scenarios in which each candidate, no matter how minor, might tie for a win, is challenging to do well. We’re fortunate, then, that there’s a good approximation. Remember that the utility gain from approving a candidate was equal to

ℙ[V(B, c)] · (U(c) - 𝔼[U(W(B)) | V(B, c)])

In precisely the cases where V(B, c) is a bizarre assumption that’s difficult to imagine, we’re also multiplying by ℙ[V(B, c)], which is vanishingly small, so this vote is very unlikely to make a difference in the outcome. For front-runners, who are relatively much more likely to be in a tie for the win, the conditional probability changes a lot less: scenarios that end in a near-tie are not too different from the baseline expectation.

This happens because ℙ[V(B, c)] falls off quite quickly indeed as the popularity of c decreases, especially for large numbers of voters. For a national scale election (say, about 10 million voters), if c expects around 45% of approvals, then ℙ[V(B, c)] is around one in a million. That’s a small number, telling us that very large elections aren’t likely to be decided by a one-vote margin anyway. But it’s gargantuan compared to the number if c expects only 5% of approvals. Then ℙ[V(B, c)] is around one in 10^70. That’s about one in a quadrillion-vigintillion, if you want to know, and near the scale of possibly picking one atom at random from the entire universe! The probability of casting a pivotal vote drops off exponentially, and by this point it’s effectively zero.

With that in mind, we can drop the condition on the probability in the second term, giving us a new rule: Approve of a candidate c any time that:

U(c) > 𝔼[U(W(B))]

That is, approve of any candidate whose win you would like better than you expect to like the outcome of the election. In other words, imagine you have no other information on election night, and hear that this candidate has won. If this would be good news, approve of the candidate on your ballot. If it would be bad news, don’t.

• This rule is still tactical. To determine how much you expect to like the outcome of the election, you need to have beliefs about who else is likely to win, which still requires an understanding of polling and public opinion and mood.
• However, there is one threshold, derived from real polling data in realistic scenarios, and you can cast your approval ballot monotonically based on that single threshold.

This is no longer a true optimal strategy, but with enough voters, the exponential falloff in ℙ[V(B, c)] as c becomes less popular is a pretty good assurance that the incorrect votes you might cast by using this strategy instead of the optimal ones are extremely unlikely to matter. In practice, this is probably the best rule to communicate to voters in an approval election with moderate to large numbers of voters.

We can get closer with the following hypothetical: Imagine that on election night, you have no information on the results except for a headline that proclaims: Election Too Close To Call. With that as your prior, you ask of each candidate, is it good or bad news to hear now that this candidate has won. If it would be good news, then you approve of them. This still leaves one threshold, but we’re no longer making the leap that the pivotal condition for front-runners is unnecessary; we’re imagining a world in which at least some candidates, almost surely the front-runners, are tied. If this changes your decision (which it likely would only in very marginal cases), you can use this more accurate approximation.

Reducing range to approval voting

I promised to look at strategy for range voting, as well. Armed with an appreciation of approval strategy, it’s easy to extend this to an optimal range strategy, as well, for large-scale elections.

The key is to recognize that a range voting election with options 0, 1, 2, …, n is mathematically equivalent to an approval election where everyone is just allowed to vote n times. The number you mark on the range ballot can be interpreted as saying how many of your approval ballots you want to mark as approving that candidate.

Looking at it this way presents the obvious question: why would you vote differently on some ballots than others? In what situation could that possibly be the right choice?

• For small elections, say if you’re voting on places to go out and eat with your friends or coworkers, it’s possible that adding in a handful of approvals materially changes the election so that the optimal vote is different. Then it may well be optimal to cast a range ballot using some intermediate number.
• For large elections, though, you’re presented with pretty much exactly the same question each time, and you may as well give the same answer. Therefore, in large-scale elections, the optimal way to vote with a range ballot is always to rate everyone either the minimum or maximum possible score. This reduces a range election exactly to an approval election. The additional expressiveness of a range ballot is a siren call: by using it, you always vote less effectively than you would have by ignoring it and using only the two extreme choices.

Since we’re discussing political elections, which have relatively large numbers of voters, this answers the question for range elections, as well: Rate a candidate the maximum score if you like them better than you expect to like the outcome of the election. Otherwise, rate them the minimum score.

Summing it up

What we’ve learned, then, is that optimal voting in approval or range systems boils down to two nested rules.

• Exact rule (for the mathematically fearless): approve c iff U(c) > 𝔼[ U(W(B)) | your extra vote for c is pivotal ]. This Bayesian test weighs each candidate against the expected utility in the razor-thin worlds where they tie for first.
• Large-electorate shortcut (for everyone else): because those pivotal worlds become astronomically rare as the field grows, the condition shrinks to a single cutoff: approve (or give a maximum score) to every candidate whose victory you expect to enjoy more than you expected to like the result. (If you can, imagine only cases where you know the election is close.)

We’ve seen why the first rule is the gold standard; but the second captures virtually all of its benefit when millions are voting. Either way, strategy is inseparable from sincerity: you must translate beliefs about polling into a utility threshold, and then measure every candidate against it. We’ve also seen by a clear mathematical equivalence why range ballots add no real leverage in large-scale elections, instead only offering false choices that are always wrong.

The entire playbook fits on a sticky note: compute the threshold, vote all-or-nothing, and let the math do the rest.