Article published in Liberty, September 1998, pp. 43 ff.
If you participate in three-member committee, and if each of the two other members votes among two alternatives with a probability of 50-50, the probability that you will influence the outcome of the vote is 0.50, for such is the probability that there would be a tie without your vote. This is the same as the probability of getting one head and one tail when flipping two coins. If there are four committee members besides you, the probability of your vote deciding the result drops to 0.375, i.e., the probability of getting two heads or two tails when flipping four coins.
These probabilities are quite easy to calculate with probability theory and combinatorial analysis. With a small number of voters, one can even figure them out by listing all the possible outcomes and taking the proportion of favorable ones. Obviously, a single voter’s influence diminishes as the total number of voters increases. What is the impact of an individual vote in the context of state elections or referenda with large numbers of voters and more complicated institutional setups?
The impact of an individual vote, or influence of an individual voter, is defined as its agency in producing a result significantly different than what would obtain if the person casting this vote did not vote. Increasing by one vote the score of an alternative that already gets millions of votes and wins by a wide margin is not significant. Compare with the consumer who buys a dozen of tomatoes. The significant result of this action is that he will now have 12 more tomatoes to eat. His purchase does exert an infinitesimal impact on tomato prices, but it is so insignificant that he will not take it into account. Altruists don’t buy less tomatoes thinking that this reduces the diet cost of the poor.
Voting is even more problematic since an individual voter will not get anything unless his vote switches the majority one way or another. He will not get one millionth of his expected subsidy if he increases by such proportion the loosing party’s score. Nor will he get a larger subsidy if his preferred party would have won anyway without his vote. Voting is like buying tomatoes and not getting any — unless 50% plus one also buy tomatoes. Therefore, the influence of an individual voter should really be defined as his agency in substituting one winning alternative for another as compared to what would have happened if his vote had not been cast.
In most cases, then, an individual voter will have no impact; in rare cases, his vote will break up a tie and swing the majority. Consequently, it is natural to consider the voter’s influence in probabilistic terms, i.e., as the probability of his causing a different alternative to be adopted as the result of his vote.
The Meaning of Chance
In probability theory, an event that occurs with certainty has a probability of one; an impossible event, a probability of zero. The higher the probability between 0 and 1, the more likely is the event to happen. To say that a coin has a probability of 0.5 (or 50%) of falling on tail means that if one throws a coin a large number of times, one would expect to get 50% tails.
What exactly is chance? Do random events really exist, or is everything determined in the universe? We will not be able to go very far into this complicated philosophical and epistemological question. For believers in universal determinism, like Pierre-Simon de Laplace (1749-1827), chance is only a label for our ignorance of causes. Given all the physical information about the roulette table, the throwing mechanism and the environment, it would in principle be possible to calculate exactly in which slot the ball will stop. In this perspective, we use probabilities only because, in practice, we cannot measure everything. We study some events as if they were random, probability theory is only a methodological trick to deal with our ignorance.
This approach has been undermined by Eisenberg’s Uncertainty Principle, which puts some unreducible randomness in the very heart of matter. Lately, it has also been contradicted, in a different way, by chaos theory, which operates a theoretical reconciliation between determinism and unpredictability. Moreover, man’s free will introduces non-determinism in the world. Chance may very well be an inescapable feature of the universe.
Another way chance drives a wedge in determinism lies in the meeting of independent causal series (although one may still ask why these series met). A carpenter working on the roof accidentally drops his hammer on the sidewalk. An IRS bureaucrat who has never audited this carpenter just happens to walk in this street for the first time of his life, and gets the hammer on the head.
We now have some keys to interpret what we mean when we model electors as voting randomly. On the one hand, probabilistic modeling can be conceived as a mere methodological device to study deterministic voter’s choice, of which we do not know all the circumstances. On the other hand, perhaps there is some essential randomness in voting: voter’s rational ignorance combined with the complexity of political choices could mean that voters actually vote just as if there were flipping coins or throwing dice. We may also say that a voter’s choice is the product of independent causal series, or is related to his free will. A related interpretation is that voters have subjective preferences and change their minds, and we use probabilities to estimate the likelihood of preference changes.
Quantifying the Gamble
Consider again the simplest case: each voter choosing between two alternatives or political parties with probabilities of 50%. It can be easily calculated (with Mathematica, for example) that he probability of a tie and, consequently, the probability of an additional voter’s influence is only 0.008 with 10,000 voters, and drops to 0.00008 (8 chances out of 100,000) with 100,000,000 voters.
There might be a certain number of committed voters, hard-core "Democrats" or "Republicans," whose probability of voting for their preferred party is close to 1. Let’s then relax our assumption that all voters vote as if flipping a coin, and assume that 90% of the 100,000,000 voters are hard-core partisans. The pool of voters deciding the election is then 10,000,000, and the probability of a tie is 0.0025 — 25 chances out of 10,000, or 1 out of 400.
These calculations assume that the number of voters is always even, for if it is odd, the marginal individual’s vote could only, at best, create a tie if he voted for the otherwise loosing side. With a 50-50 probability that the number of voters will turn out to be even, the probability of an individual impact is only half the values given above. Yet, one chance out of 800 still looks like a good bet — much better than the one chance in a few million of winning the lottery jackpot.
The picture changes dramatically when we drop the assumption of an equal (50-50) probability that a voter will choose one of two alternatives. With asymmetric probabilities, i.e., one party having a head start, an individual voter’s influence is much smaller and decreases more rapidly to infinitesimal values. Take our previous example: 100,000,000 voters, of which 90,000,000 are hard-core voters, half for one side, half for the other. Out of the remaining 10,000,000 voters, the probability that any one will vote for party A is, say, 0.501 — and, of course, 0.409 that he will vote for B. In this case, the probability that one individual vote will change the election result is 5 X 10-13, i.e., 5 chances out of 10 trillion. If the asymmetric probabilities are a bit more skewed, let’s say 0.54 than an uncommitted voter will vote for A (and 0.46 that he will give his vote to B), then the probability that one individual vote will change the election result is 2 X 10-13946, i.e., 2 chances out of 10 followed by 13,945 zeros.
Such probabilities are infinitesimal. Compare them with the total number of elementary particles in the universe, which physicists estimate to be no more than 10100. Proudhon was right in claiming that "the universal franchise is a real lottery".
Counterintuitively and interestingly enough, creating many (smaller) electoral districts or ridings does not improve the weight of an individual vote. Intuition might suggest that creating two ridings where there was none would improve the voter’s influence, but a little (messier) statistical analysis shows this intuition to be misleading. What apparently happens is that the higher probability of a tie within a smaller riding never completely compensates for the lower probability of a tie among more numerous ridings.
Gambling on more than two horses
What happens to the impact of an individual vote with more than two challengers? This problem is much more complex analytically and is related to the so-called "the classical occupancy problem": if a certain number of balls fall randomly into a certain number of cells, what is the probability that two cells will receive a equal number of balls and that no other cell will get more? Apparently no closed-form solutions has been found for this problem, so simulations have to be made or analytical estimations devised.
Simulations of many elections (say, 1,000) with large numbers of voters (say, 10,000,000) require much more computing power than the typical PC can marshal. The more limited runs I made seem to confirm the intuition that a voter’s influence increases with the number of alternatives, as long as probabilities do not differ among parties. To get an intuitive feel for this conclusion, consider the extreme case where each voter except one has formed his own political party: then, one single partyless voter will decide the result with a probability of one. Of course, the more equally popular are any number of political parties, the lower will be the winner’s popular vote.
The most interesting, and analytically difficult, case is when you have many parties and asymmetric probabilities for individual voting. Prof. Fred Huffer, a statistician at Florida State University, has devised an ingenious (and complex) formula for the case where the probabilities favoring the two main contending parties are substantially higher than for the other parties. Whether the presence of multiple parties increases or decreases the probability of a voter’s influence depends on the actual probabilities but, in most realistic cases, this probability remains infinitesimal. For example, if the respective probabilities that any (uncommitted) voter will vote for any one of four political parties are 0.38, 0.37, 0.2 and 0.05, and if we have 10,000,000 uncommitted voters, Huffer’s estimates give 4 X 10-294 for the probability of an individual voter’s influence.
Why Do People Vote?
A rational individual will not vote in order to exert an influence on the result, for the same reason that he will not produce one more tomato to push down tomato prices. Why, then, do 50%, 60% or 70% of the electors take the trouble to go and vote? True, voting does not take much time if one votes blind — i.e., if one does not spend time studying the issues — but then, it does not produce any noticeable effect either.
One type of motivation is that the act of voting itself gives satisfaction ("utility" in economic terms) to the voter; or, what is the same, his abstention brings him disutility. Depending on individual preferences, there are many ways in which voting can be rewarding. Some individuals like to gamble: they buy lottery tickets and vote. Some enjoy the feeling of participation in a crowd, as when they applaud their favorite team at the stadium, even if individual hand clapping does not raise the noise level more than one vote contributes to an electoral victory. Some people like to express their opinions, and elections provide a cheap way to do this. Moreover, given the democratic mystique and state propaganda, you might suffer from disapprobation if he is known not to have performed his "civic duty."
The rationalization according to which one votes because democracy would crumble or other catastrophes would happen if nobody voted, does not stand strictly speaking. The abstention of one individual exerts no perceptible influence on the system, especially if one declines to vote without fanfare or if one falsely claims that he has voted. Of course, many non-voters may have an influence, but the individual decides only for himself, and his decision will have no significant impact on the number of non-voters. It is true that an individual with some influence in public debates can marshal more than one vote, but this has nothing to do with the decision of a quidam to throw one vote that has only a tiny chance of achieving any result. If you can persuade many people to boycott tomatoes, you may end up influencing their prices; but you will have no noticeable influence if you secretly by one less tomato.
One way to see the inconsistency between infinitesimal influence and the argument "If everybody does it..." is to reconsider the impact of the non-voter. Those who do not vote increase the influence of anyone who does: non-voters render a service to voters. Consequently, an altruist might be expected to refrain from voting to help his human brothers increase their political influence. "I don’t vote because if all rich men like me voted, there wouldn’t be any political influence left to the poor." The problem with the argument is obvious: What’s the use of an action that has no impact on the result to be achieved? From the point of view of the individual, any service he renders either by voting or abstaining is infinitesimal. Both the individual voter and the individual none-voter make insignificant gestures. We are back to reasons of personal utility for voting: gambling, the pleasure of the crowd, or the expression of one’s opinion.
The Moral Element
Or are we? Not exactly. A rational individual may choose to make insignificant gestures for moral reasons. We do this all the time — when, for example, we give a quarter to a beggar, or refrain from throwing an empty pack of cigarettes on the highway, or open the door for a woman. Indeed, morality is the only way to construe the argument "If everybody does it..." Infinitesimal gestures are gauged not by their consequences but by their intrinsic moral value.
As far as voting is concerned, I may do it despite its having no impact, simply because I believe that morality requires that everybody does it. Whether one interprets his moral duty in terms of a Kantian universalization principle or in terms of some other ethical theory, the moral sentiment is a constituent part of the human mind and human action. And whether one reinterprets moral motivation in terms of personal interest ("I feel good") does not change the fact that people make inconsequential actions for the sake of doing what’s right to do.
People, then, may choose to vote — and certainly often do — out of a sense of moral, or civic, duty. Whether the underlying moral theory is mistaken or not is another matter. Indeed, it is mistaken when an individual believes that his vote contributes to some dangerous notion of "social welfare." But one must not discard the idea that some moral duty is involved in maintaining a social order based on liberty, whether it means contributing to private charity in an anarcho-capitalist society or voting in a minimal state context. Voting may be a moral duty if it contributes, however infinitesimally, to affirming individual liberty.
Of course, the argument cuts both ways. One’s moral (or "civic") duty may also require one to abstain from voting when there is no morally acceptable alternative on the ballot. Indeed, low voter turnout may partly reflect people’s sense that "something is rotten in the state of Denmark." I, for one, do believe my moral duty is to abstain when state elections are totally rigged against liberty — or when voting requires one to register on a permanent voters’ list. Non-voting is then a way to express a moral refusal of the system, even if it contributes only infinitesimally to changing it.
Leaving all values to be decided by voting was bound to generate, at best a stupid gamble, at worst a perilous game for liberty.
Liberty Magazine, © Copyright 1997, Liberty Foundation