If Voting Third Party Is Throwing Away Your Vote, so Is Voting Major Party
(Under certain plausible assumptions)
Writing up this post really made me wish Substack supported LaTex. Doing math on this platform is a pain in the ass.
Preliminaries
Suppose, for the sake of computational simplicity, we have a set Can of candidates and an (American format) election with three candidates A, B, and C, corresponding to two major parties that are almost guaranteed, between them, to win, and a single third party candidate with a snowball’s chance in hell. We will limit our considerations to a single voter Alice and her preferences regarding the outcomes. Suppose, further, we have a function U mapping Can to R1 that assigns to each candidate a real number U(X) signifying, broadly, the “utility” Alice would receive in the outcome where X wins. I want to be very clear that I’m remaining agnostic about what sort of utility we mean here; it could be as simple as measuring how happy Alice would be with the outcome. It’s worth thinking about at some point, but not important to the calculations that are coming.
Now suppose, further, that Alice’s first choice is the third party candidate, C.
We will define a preference function that maps ordered pairs in Can x Can to a real number in the following way:
(1) Pr(X, Y)=U(X)-U(Y)
The larger Pr(X, Y), the greater Alice’s preference for X over Y, and the more that is to be gained by an X win averting a Y win. The smaller Pr(X, Y), the more indifferent Alice is about which candidate wins.
Finally, let P(X) denote the probability that a given candidate X will win if Alice doesn’t vote.
Now, we can calculate the expected value E of the election for Alice if she doesn’t vote like so:
(2) E=P(A)U(A)+P(B)U(B)+P(C)U(C)
If we let x, y, and z denote, respectively, the odds that Alice’s individual vote (for C, her third party preference) will sway the outcome of the election away from A and toward C, away from B and toward C, and away from either A or B toward C, then our new expected value, E’, of the election given that Alice votes looks like this, ignoring, for now, the costs associated with voting, which I will bring in later:
(3) E’=(P(A)-x)U(A)+(P(B)-y)U(B)+(P(C)+z)U(C)
But this isn’t quite what we’re interested in, either. We want to know what utility was generated for Alice by the act of her voting. In other words, (3) takes into account a ton of background stuff that would have been there whether or not Alice voted, namely the stuff from (2). So what we’re really interested in is the difference in expected values:
(4) E’-E=zU(C)-yU(B)-xU(A)=Pr(C, B)x’+Pr(C, A)y’
Here we are defining x’=P(A)-x, y’=P(B)-y; these new probabilities represent the odds of a win for A and B respectively given that Alice votes for C2. I'll leave showing these two equalities as an exercise for the reader.
The way you want to think of this is as follows: the utility we get from voting for C comes from the probability that our vote will sway the election times however much utility is to be gained by swaying the election.
So we come to the question: what does it mean to throw away your vote? You hear people say a vote for a third party is thrown away all the time. I don’t think it can be as simple as “your vote is thrown away if your candidate doesn’t win,” because then half of the voting population throws away their votes every election. That doesn’t seem right. I will take Alice has thrown away her vote to mean that E’-E<e, where e represents the various costs associated with voting (opportunity costs, energy costs, etc.); in other words, her vote cost more than the expected value it gave her.
I claim that, under certain assumptions, the following propositions are equivalent:
Any vote by Alice would be thrown away
A vote for a third party by Alice would be thrown away
Clearly 1—>2. We’d like to show that, under the right assumptions, 2—>1.
The Argument
Suppose the expected value E’-E of a vote for a third party is thrown away, i.e.:
(5) E’-E=Pr(C, B)x’+Pr(C, A)y’<e
Given this assumption, we aim to show that a vote for Alice’s second choice, B, would also be thrown away.
We calculate the expected value E’’ of her vote (this is important; I’m skipping a step here, but we’re using what we established in the preliminaries to jump straight to talking about the EV of her vote, rather than of the election) as follows:
(6)E’’=Pr(B, A)x’’+Pr(B, C)z’’
Where x’’ and z’’ are defined in the obvious way, given how we defined x’ and y’ above; x’’ represents the odds that Alice’s vote will sway the election from an A win to a B win, and z’’ represents the odds that Alice’s vote will sway the election from a C win to a B win.
Notice that Pr(B, C) is negative; hence we have
(7)E’’<Pr(B, A)x’’=x’’(U(B)-U(A))
Finally, if we make the assumption that x’’<(Pr(C, B)x’+Pr(C, A)y’)/(U(B)-U(A))—which I will motivate later, although it will not always hold—then we have the following string of inequalities:
(8)E’’<Pr(B, A)x’’<Pr(C, B)x’+Pr(C, A)y’<e
Which is to say, Alice’s vote for B was thrown away, and we conclude the proof.
The controversial part here, of course, will be the assumption we made:
(9) x’’<(Pr(C, B)x’+Pr(C, A)y’)/(U(B)-U(A))
It helps to think through when this sort of thing will be true. I can’t mathematically constrain the utility functions in a nice way, but we can get a handle on what sorts of circumstances make this inequality true, and which make them false. And in general, we can assume that the probabilities x’’, x’, and y’ will all be pretty small. For a more thorough treatment of probabilities like those and related voting issues, see the excellent paper Why You Should Vote to Change the Outcome (Barnett, 2020).
We can make the relevant fraction very large by either making the numerator large or making the denominator small. The obvious way to make the denominator small is to assume our preference between the two major party candidates is very small; the obvious way to make the numerator large is to have a big preference between C and B or between C and A.
Intuitively, we can understand the conditions as follows: if Alice has little preference between B and C, then there’s no reason Alice should impose upon herself the attendant costs of voting in order to try to sway the election from one to the other. And if Alice had a very large preference for C over B or A to begin with and that vote was still wasted, it seems likely that a vote for her second choice will also be wasted. That, in English, is what inequality (9) is saying. Of course, there will be edge cases where we won’t be sure whether voting is rational because, as I mentioned, there’s not a nice way to constrain the utility function and we don’t know exactly what the probabilities will be.
A Caveat
I mentioned that (9) might not always hold, and indeed, we can easily imagine a scenario where it fails. The prototypical sort of example of a third party Alice for whom (9) will most probably fail is a Green Party voter. Her preference for the greens over the democrats will likely be very modest in comparison to her massive preference for the democrats over the republicans, and in cases like that, a vote for a third party might be wasted and a vote for a major party might not.
The real numbers.
I am taking the small liberty of assuming that whether any given eligible voter votes is independent of who else votes, i.e., that by Alice voting for C, she doesn’t magically make a million people vote for A or some nonsense.