Occasionally, I discuss strategic voting in some of my game theory classes. In particular, in past classes I have highlighted the trivial point that if people have only two choices (or candidates) to vote for, then it is a (weakly) dominant strategy to vote for your more preferred choice: voting for your more preferred choice can never lead to a worse outcome for you than voting for your less preferred choice. So, everyone will vote for their favorite choice and the vote winner reflects the majority preference. I regarded it as a bit of an embarrassment that such a voting game also has other equilibria, in which some people vote for their less preferred choice. For instance, if every one of the n-1 people other than you (with n ≥ 3) vote for choice A and you prefer choice B, it is immaterial if you vote for A or B because your vote will simply not matter. One can, in theory, even have the paradoxical situation that everyone prefers A over B and yet everyone votes for B. This is an equilibrium! Just not a very plausible one, I would have argued. I have recently learnt not to too quickly discard the weakly dominated equilibria of such a voting game.

Well, it all comes down to the question as to how well your game theoretic model captures the real-life situation you are interested in. And sometimes, even when your game-theoretic model is pretty close to the real-life situation, it can still produce empirically irrelevant predictions. Suppose that the real-life situation is like this: There are n people voting to determine an executive ruler for a given period of time. This executive ruler then implements policies that affects every one of the n individuals. There are many decisions that this ruler must make. Many of these decisions are about representing the whole group of these n individuals to other groups. But, in addition to all these, the ruler also decides who gets how much money for activities that the individuals would like to undertake. In fact, it may not even be about allocating money to individuals. Sometimes the ruler has to decide simply whether or not to allow an activity that the individuals like to undertake and that actually do not incur any costs to the collective.

Now suppose that there are two candidates for this position, both among the n individuals we are here dealing with (but this is not important). Candidates A and B. Suppose candidate A is the incumbent, the current executive ruler, and candidate B is his challenger. The n individuals have preferences over which candidate they prefer to see in office and they vote. Unfortunately, in our case, the rules of voting are not completely clear. The constitution governing how voting should be done is imprecise, vague, incomplete, and subject to interpretation. Suppose that, for some reason that we should come back to explore later, the vote is non-secret: everyone is just asked to raise their hand for the candidate they vote for.

This game now has an equilibrium in which everyone votes for the incumbent, candidate A, regardless of their preference. This equilibrium is in fact strict, meaning that everyone strictly “prefers” to vote for candidate A, given that everyone else votes for candidate A. How come? Well, the eventual vote winner, the incumbent candidate A, simply has to explicitly (or even just implicitly – words are often not needed) threaten any individual voting against him, by punishing them with a lack of funding or general lack of support for whatever activities they would like to undertake. This threat is quite credible, as the incumbent does not himself have any particularly strong preference whether or not these activities should happen. The game seen like this, even though only one of two candidates can be elected, has in fact more than two possible outcomes, as punishments can be tailored to the individuals who have voted against the incumbent. In the presence of such threats, and if you believe that everyone else votes for the incumbent and that, therefore, your vote makes no difference anyway, you’d rather vote for the incumbent and avoid the punishment you would receive by openly voting against him.

This means, however, that even if everyone (except the incumbent himself one would assume) would prefer candidate B, there is a (strict!) equilibrium in which everyone votes for the incumbent candidate A!

Would it help if voting were secret? I am sure it would be better, but it does not necessarily remove the problem entirely. The incumbent could (explicitly or implicitly) threaten to start a witch hunt (not necessarily openly) if there are any who vote against him.

Why do I single out the incumbent? Isn’t there an analogous equilibrium in which everyone votes for the challenger, candidate B? In principle yes. Only I feel that it is typically much easier for the incumbent to make clear what he would do if people do not behave. Also, voting for the incumbent seems a natural focal point, after all he was elected last time. In fact, one could easily imagine – and perhaps imagination is not even necessary – that there was a first election in which candidate A won with honest majority support, and over the years support dwindled and yet elections were won with the aid of more and more explicit or implicit threats.

How could such a situation be overcome? Well, I guess it needs some sort of small-scale revolution. Of course, if a revolution fails (or even just while it is happening) many revolutionists may suffer a bit. Because of that, potential revolutionists would like to know how likely it would be that a revolution would be successful. To know that, one would like to know how many other individuals are also in favor of a revolution. How can you gauge that? Well, the outcome of the election is certainly no help – as it does not reveal how many people are against the incumbent. This is another, or perhaps the prime, reason why the incumbent favors the outcome in which all vote for him. And then, even if it is indeed the case that a majority favors candidate B, this is not sufficient to start a revolution, if people are unsure about this fact or at least unsure if others know this fact, or are unsure whether others are sure that others know this fact. You probably need something closer to common knowledge that this is the case, and this is hard to establish. But this seems to lead me to a different topic, which may necessitate a separate treatment.