“Your vote counts” is an empty slogan or an illusion or a lie. Typically, your vote does not count at all. I am always surprised to find intelligent people who think that an ordinary voter, by using his single vote, has a significant chance of influencing the outcome and consequences of an election. We meet this idea again in Simon Kuper’s Financial Times column (“The Most Powerful Voters Aren’t Who You Think,” October 3, 2022).
Many people seem surprised when an economist or political scientist tells them that, with his single vote, an ordinary and rational voter has no reasonable hope of deciding an election, that is, of changing who is elected (or which proposition is adopted in a referendum) compared to what would have been the case had he voted differently or not at all. Some people either have never reflected on the mathematics of voting, or have never tried to find elections where one vote made a difference, or perhaps they are so engrossed in a simple democratic ideology that they just imagine a reality that matches it.
The basic math are relatively simple. Consider a committee of three persons, including you, who vote between two alternatives. If the probability that each of the other two committee members votes for either alternative is 0.5, the probability that your voice will be decisive is given by the ratio of the two possible tied results to the total number of four possible outcomes, that is, 0.5 also. But if the committee has 4 members besides you, your probability of being decisive decreases to 0.375. If the committee is a group of 1,000 voters plus you, the probability that you will be decisive drops to 0.0189. These probabilities decrease dramatically if the probability of every other voter’s voting one way or the other changes only slightly. For example, if this probability is respectively 0.49 and 0.51 for the two issues, we can calculate that, in an electorate of 1,000 plus you, the probability that you will break a tie goes down to 0.0155; in an electorate of 100,000,000 plus you, it is astronomically lower that the inverse of the number of particles in the observable universe. (See the sidebar “Does Your Vote Really Count?” in my “The Public Choice Revolution,” Regulation, Fall 2004; see also Dennis C. Mueller, Public Choice III [Cambridge University Press, 2003], pp. 304-306.)
Let me answer a standard objection immediately: “But all together, we make a difference.” Of course. If every consumer buys one more tomato, the price of tomatoes and the growers’ incomes will jump. If one voter controlled the votes of 50%+1 of the electors, he would be decisive with certainty. But the probability would drop dramatically if he only controlled 25% of the votes, or 10%; or, ultimately, only his own vote. QED.
Another objection is that rare events do happen. Yes, but not often. Following the November 2017 election in Dictrict 94 four the Virginia House of Delegates, the Republican candidate had a 10-vote lead on all 23,215 ballots. A recount changed the result to a one-vote lead for the Democratic candidate. A three-judge panel then found a vote incorrectly disqualified, which produced a tie. After further litigation and according to Virginia law, a random drawing was held on January 4, which gave the victory to the Republican candidate. At best, every Republican voter can claim to have produced a tie, and that was in a relatively small district.
This simple mathematical approach has been improved, for example by considering that the voter can guess, notably through opinion polls, that a certain number of his fellow voters have dug their heels for a candidate or a party, which decreases the effective size of the decisive set in which he is competing and increases his chances of being decisive. Yet, the probability that a single voter changes the result of a large election remains very low. So low that it very seldom happens.
Considering more real-world complications, namely the districts and the electoral college, Andrew Gelman, Nate Silver, and Aaron Edlin used the 2008 presidential election (Obama-McCain) to evaluate what probability the average American could have reasonable formed that he could elect the president, i.e., that without his vote another president would be elected. It was at most 1 in 10,000,000 depending on locations, and 1 in 60,000,000 million on average over the United States. (See “What Is the Probability Your Vote Will Make a Difference,” Economic Inquiry 50:2 [April 2012], 321-326.)
Gelman et al. also calculated that if a voter in New Mexico could have brought 5,000 of his fellow voters to switch from the other side to his, he would have had a 1.3% chance of flipping the state and a 1 chance in 6,000 of changing the nationally-elected president. No ordinary voter influences that many votes, although a very popular pundit or media personality or sports player or singer may. Not all voters are equal.
To give some perspective, the estimated 1-in-60,000,000 chance of an average American to elect the president he wants is still five times higher than his odds of winning the jackpot in a Powerball drawing (apparently 1 in 292,000,000). We do occasionally see somebody winning the jackpot, but an ordinary voter has never been decisive in a presidential election. The best that Kupel can find is the Bush-Gore 2000 election, when the difference was of “537 possibly miscounted” votes in Florida; Wikipedia would have provided him with better examples, albeit in smaller elections. It is true, of course, that we would need a large number of presidential elections before we can test the 1/60,000,000 probability.
Finally (in this short post), note that the hypothetically decisive voter may end up disappointed because “his” president would break the single promise that motivated the lucky voter.
To be fair to Mr. Kuper, his column was more about the fact that a double citizenship, which both he and his wife hold, allows one to vote in two different countries. Yet, twice a minuscule probability of having an influence in two different countries still means a minuscule probability of having an influence. He should instead have emphasized that the great benefit of double citizenship is not a double vote but the possibility of voting with one’s feet.
I am not denying that there are moral reasons for an individual to vote, at least for a candidate who is very likely to contribute to the maintenance, regeneration, or creation of a free society. I am not denying either that democracy has advantages. I am just reiterating the basic argument that an individual who votes to influence the outcome of an election in anything but a small committee must suffer from cognitive limitation or love gambling. A voter may also just enjoy whispering his opinion to the winds. (See Geoffrey Brennan and Loren Lomasky, Democracy and Decision: The Pure Theory of Electoral preferences [Cambridge University Press, 1993].)
