Vote Exchanges: The Only Winning Move Is To Realize You Can Play
First, a question: how can we usefully think about voting in the framework of game theory? Naively, it doesn’t seem like there’s much that can be done. We could draw a 2 by 2 by … by 2 payoff matrix, one dimension for every voter in the country, and analyze that object. But what help is this? Each voter has a dominant strategy: vote for the candidate they want to win the most! This is why Americans can vote every four years without spending thousands of hours computing optimal strategies or Nash equilibria: instead of one large, complicated game, voting is just a composition of a large number of “single-player games”, so to speak. More on the game theory later.
This year’s election is unprecedented in modern American history: both candidates have abnormally high unfavorable ratings, the number of undecided voters has refused to go down, and third-party candidates will likely get much more votes than usual. After having a clear lead for months, Hillary Clinton’s has dropped in the polls to a mere one or two points above Donald Trump. At the time of writing, FiveThirtyEight gives an 8% chance to one candidate winning the Electoral College and the other winning the popular vote. This is moderately higher than during a “typical” two-party election, where it would sit at around 7%.
Many Republicans have been hesitant to back Trump, but are also loathe to vote for Clinton, which forces them to look at third-party candidates. These voters are wrestling with two incompatible goals: to vote against (in their minds) the greater of two evils, while not personally voting for someone whom they find disagreeable. Unfortunately, the American voting system is poorly set up for managing both goals. Third-party candidates disproportionately pull votes from the mainstream candidates they are most similar to, so people voting for a third party candidate over a Democrat or a Republican are hurting their own cause. This isn’t particularly important for voters in a safely Democratic state like New York, but a Floridian could very well be faced with a deeply uncomfortable choice: to compromise their principles, and lose the symbolic value of their third-party vote, by grudgingly voting for a major-party candidate. While no individual state is likely to give its electoral votes to a third-party candidate this year, high popular support of a third-party candidate could help catalyze major electoral change.
Voting is more complicated than the picture I painted above: in addition to short-term goals (“make sure candidate X wins the election”), voters have deeper, more long-term ones (“show whether or not I am satisfied with the current state of affairs”; “maintain personal integrity”). Voting is the only way to effect either kind of goal, but has no nuance, which can make it difficult to decide how to vote when the two are in conflict. Mere months ago, more than a few UK voters decided to take the “safe” trade-off of short-term goals for long-term ones, voting for Brexit in the hope that they would merely narrow the margin by which it lost; it was a less safe risk than they realized. These voters demonstrated the tragedy of the commons / prisoner’s dilemma situation at hand: if everyone wants their preferred candidate to win by a narrow margin and too many of them try to “help”, they all wind up unhappy. That example, and the US election of 2000, are well worth keeping in mind in the next few months.
This brings us to vote trading, as discussed in the linked New York Times piece. The idea is simple: if you live in a competitive state and want to vote for a third party, trade your vote with a friend in a safe state who’s voting for the major party candidate you prefer. A vote gets sent in for your third-party candidate, which makes you happy, since you cared a fair bit about increasing their popular vote total. At the same time, your very important Florida or Nevada vote gets used to back the candidate you might have felt obligated to vote for anyway. You actualize your short-term goals without throwing away your long-term ones.
Let’s think back to the payoff matrix from the start, and just focus on the submatrix involving two people’s choices. Say Cari is a solid Clinton fan in New York, and Grant is a Floridian with his eyes on Gary Johnson. In the naive approach, the payoff matrix might look something like this:
where H is a vote for Hillary and J is a vote for Johnson. In this view, Cari gets a util for every Hillary vote, and Grant feels the same way about Johnson; it truly is just two one-player games. But that’s not the full story.
The key insight of vote trading is that there really is an interesting game afoot, and once that’s clear, you and your friends can collude away. Grant would really like Hillary to win, but thinks his principled stand for Johnson is worth something, too, and Cari acknowledges that Grant’s vote matters more than hers does. For sake of illustration, suppose Grant gets 3 utils from each vote for Gary Johnson irrespective of where they come from, and 2 utils if he himself votes for Hillary (since he’s particulary important to her cause, as a swing state voter). Meanwhile, Cari gets 2 utils if she herself votes for Hillary, and 4 if Grant does (Florida could decide it all, of course). Then their payoff matrix is
This makes the true prisoner’s dilemma more clear. Cari’s still quite happy if they both vote for Hillary, as is Grant if they both vote for Johnson, and without cooperation they would both certainly vote for their own candidate, and this is indeed the Nash equilibrium—they’re both just following their dominant strategy, after all. But they’d both be happier if they could cooperate and swap their votes. Grant would get what he cares about (an additional popular vote for his man), and Cari and Grant would both feel better knowing that Hillary’s gained a vote on Trump in Florida. As long as they’re trading votes one-for-one, everyone’s happier here.
As always, the question is: how can we enforce cooperation, especially with a secret ballot? The social contract of vote trading, the participants’ belief in the honor system, and their concern for their friendship all play a part in slapping the wrists of defectors. If the ballot selfie triumphs in the ongoing legal disputes around it, it could provide another way to make sure no one’s cheating. Let’s say some combination of those forces ends up taking away 10 utils from defectors, in some combination of guilt, social losses, and (perhaps) some lead pipe-based social contract enforcement. Then the adjusted payoff matrix is
and now the vote trade situation is the sole Nash equilibrium.