The paradox of the paradox of voting

Ehud Lamm
16 min readOct 15, 2021


Remember a former U.S. president whose actions were so bizzare people suggested he was playing a complicated game of 3D political chess? It turned out that it was not great 3D chess, just plain old political chess done poorly. Game theorists sometimes make a similar mistake and assume that they know the games people are playing poorly rather than conclude that they may have misunderstood the games altogether. This mistake underlies discussions of one of the best puzzles game theory uncovered: the paradox of voting.

Photo by Element5 Digital on Unsplash

Perhaps the best thing about game theory is that it enables us to clearly see when our behaviors conflict with our self-congratulatory introspective justifications. It tuns out that in many cases we are just not as rational as we would like to think. A classic example is the ultimatum game. Suppose you are paired with a random individual to anonymously play the following game. You are given $100 to split between the two of you. It is up to you to choose the amount of money to give the other player. They get to choose whether to accept, in which case you both get the specified amounts, or to reject the offer, in which case you both go away empty handed. How much out of the $100 should you offer?

Anyone who had even a slight encounter with game theory inspired ideas knows that the correct answer is to offer the smallest amount possible (one dollar or even one cent if possible). The reasoning goes as follows: the second player has only two options: to accept the small amount offered or to get nothing at all. They clearly prefer something over nothing, so should accept the one dollar. And you end up keeping $99. This is clearly better than offering more to the other player if your motivation is to get the most money.

It turns out that essentially everyone except economics students offer more than $1 and reject offers they think are too low, even if it means getting nothing. People justify their behavior, when they are confronted with this, in various ways. They may insist that they offer more because “that is good behavior” or that they are being “nice.” They may claim that this is what people (in general, everywhere) do. If they are more sophisticated they may claim that the payoff that interests them is not just the money, but includes the happiness or the well-being of others. Modern theory can accommodate motivations of this sort, but without more elaboration cannot tells us what is the source of these motivations.

The surprising behavior of people in the ultimatum game and similar puzzles led to a huge literature, indeed entire research areas that are devoted to explaining its sources. From now on we will put to the side people’s introspective justifications and focus on scientific explanations. There are two complementary approaches to the puzzle. The first is to assume that the analysis just provided is correct and attempt to explain why people are not living up to the rational standard of behavior. People are said to have biases (that’s what behavioral economics is about); that their rationality is bounded by time and computational constraints (a clever idea proposed by Herbert Simon, one of the fathers of AI); that their behavior reflects social norms that they internalized as children. The second approach tries to resolve the puzzle by proposing that the game people play (or assume that they are playing) is not the one the theorists analyzed. For example, people may think the game is going to be repeated several times, with the other player being the proposer in the other rounds. In such cases, though not in the ultimatum game itself, it might make more sense to be generous in this round. Or maybe people are used to games that are not anonymous, so low offers may lead to future grudges, maybe even payback. Or maybe they expect that members of their community will know the amount they offered and will think badly of them if they offer too little.

The two escape routes out of the puzzle are often combined. For example, expecting your community to scoff at low offers may be the result of a social norm of generosity that you have internalized. However, the impetus behind the two approaches is different. The first approach happily accepts that people are not as rational as they tend to think they are. This continues a long history of what Freud described as attacks by science on human arrogance. We are not at the center of the universe (Copernicus), we are not the pinnacle of the great chain of being (Darwin), and most of our mental life is not subject to our conscious control (that was Freud’s own discovery, or so he thought). Indeed so, the thinking goes, and if it turns out that we also have cognitive biases that makes us offer too much in the ultimatum game, that is just further proof that we are not as perfect as we think we are. The second approach tries to save a modicum of human dignity by showing that it was the analysis that was faulty, since the actual game is not the one on which the analysis was based. When the two approaches are combined it is typically the first one that wins out: people’s behavior is better understood if the underlying game is revised, but people are said to misunderstand the game they are instructed to play, for example by assuming repeated rather than one-shot interactions. Again, we are at fault, not the theory. Since I already admitted that these approaches can be combined trying to distinguish between them may seem like quibbling. So let’s consider the paradox of voting to see why it matters.

The paradox goes as follows. A single voter, we are told, very rarely casts the deciding vote in an election, maybe never. So the costs we are willing to spend to vote should be minimal. Expending time and energy beyond this minimal amount is simply not acting rationally. Skipped a shift at work to vote? Not rational. Took the subway? Waste of money. Payed for parking? Silly. Caught covid while waiting in line? Borderline insane. Presumably the 19th century German philosopher Georg Hegel already realized this problem with voting, which I suppose tells you something about the value of reading arcane German philosophers. But back to the issue at hand: why do people vote if the costs outweigh the benefits?

Like the ultimatum game, the paradox of voting tells us something really interesting: it is far from self-evident why people vote. In fact, the logic of the situation goes against voting. But can the paradox tell us something interesting, beyond just being a clever, meta-way of suppressing the vote? The problem, as in the ultimatum game, is the assumption that we know what the game is. That is, that we know the costs of voting (and of not voting) and the purpose of voting (and that they are the same for everyone).

The first, most obvious clue that something is amiss is that the argument works best when what you have in mind is single-member first-past-the-post voting, common in the United States. This voting system is winner takes all: the candidate or measure that gets the most votes wins, making the vote share of the others all but irrelevant. The classic case is a Senate race. Two candidates, one a Democrat and the other a Republican, run for election. The one who gets the more votes, even if by a single vote, wins, and represents the State in the United States Senate for six years. In this system, even a single vote can make a huge difference. On the other hand, since an election is usually decided by more than one vote, it really does not matter how large the margin of victory is. If your candidate is going to lose, losing by a single vote or by a million makes little difference. For example, in the 2018 Senate elections in New York, Kirsten Gillibrand received 4,056,931 votes while the Republican Chele Chiavacci Farley (who?!) received 1,998,220. According to the rules, this huge margin makes no difference, and the practical result is the same as in 1974, when Republican Louis Wyman won a Senate race in New Hampshire by precisely 2 votes.

Clearly, in a proportional voting system the two cases are radically different. In the first, you have a huge margin in parliament, a super-majority in fact. In the second, the parliament is virtually tied. That’s quite a difference. Perhaps unsurprisingly people used to first-past-the-post voting systems are more troubled by the paradox of voting than people from countries with proportional voting. But we can even put this biographical issue aside. Here’s what wikipedia tells us about what really happened in the 1974 New Hampshire race:

On election day, Republican Louis Wyman won with a margin of just 355 votes out of more than 220,000. His opponent John A. Durkin then won the recount by 10 votes. After a second recount, Wyman won by just 2 votes. The Democratic-controlled Senate at first agreed to seat Wyman, who served the last 3 days of Noris Cotton’s term, but began to deliberate again when the new Senate took office. When the Senate deadlocked for months, Durkin agreed to Wyman’s proposal for a new election. The Senate declared the seat vacant and the governor appointed Cotton to hold the seat for six weeks until a special election on September 16. Durkin won the special by 27,000 votes.

It seems that the margin did play a role. Even in a first-past-the-post system what actually happens can depend on the margin. This puts strain on the reasoning underlying the paradox of voting. The game matrix we imagined is simply not a good description of what happens. Notice that this conclusion is not about why people vote given the nature of voting. It is not a bias, and neither am I claiming that people take into consideration cases such as the 1974 case when they decide whether to vote. The claim is much simpler, if a bit subtle: the voting payoffs that lead to the paradox of voting are unrealistic. What people think happens when they vote, whether their behavior exhibts biases, and why, must be secondary questions addressed only after the theorists accurately understand what the game is. The paradox is still very cool; but mostly as a formal conundrum. Deducing from it that people’s willingness to spend time or treasure to vote requires special explanation is far from straightforward since the paradox does not capture what happens in voting.

There are several ways of responding to this worry, some more convincing than others. I want to mention one particularly unsatisfying move. This involves arguing that people do not make the required considerations, and in fact assume that the voting is as prescribed in the paradox of voting. Hence, the argument goes, the fact that they do vote requires explanation. This argument goes against the grain: in setting up the paradox we were not concerned with how people decided whether to vote; this is rarely what this type of analysis consists of. The value of the exercise stems precisely from the fact the behavior is somehow logically suspicious, and people’s introspective explanations are probably rationalizations. Indeed, if this was not the goal, why do we stop with the game theoretic analysis and not ask people why they vote? Because, and here the game theorists are entirely correct, this will not address the same issue: what we are after is whether voting behavior is rational, given the costs and benefits. So this get out of jail free card fails, and is particularly disingenuous when used to support the formal paradox. The problem for the theorist is harder: if it does not matter how people justify the behavior but rather what the real game is, it is important to get the game right.

We saw that first-past-the-post makes the paradox of voting harder. In other words, the paradox of voting helps us realize that first-past-the-post voting makes voting less appealing. The more time or money to spare, the more likely you are to be willing to waste, so the rich and powerful will vote more than the less fortunate. The theorists may inadvertently even exacerbate the problem by highlighting the paradox of voting, in case anyone missed the problem. But the point should not be to interpret the voting system, but to change it. A first step might be to ask who came up with the voting system, in what context, and for what purpose. How was it justified? And, perhaps more crucially, what forces maintain it? These questions are easier to study than the ultimate motivations of the human psyche that supposedly underlie the paradox of voting because they can be addressed with evidence rather than speculation. They are studied by historians.

Looking at what historians say, it is reasonable to conclude that electoral systems changed over time, often intentionally, in order to further reduce the power of weaker parties. For example, the political scientist Amel Ahmed argued that the choice between proportional and plurality voting in Europe was dictated by the overall strategies of various right wing parties to hold back rival socialist parties. Whether this is precisely the case is immaterial for our purposes. It is enough to note two things. The first is that voting is not a well defined notion but rather a vague description of systems that may be radically different and require different strategies. The question “why people vote” may have different answers in different cases, so reflecting on the simple, abstract voting scenario may be less informative than we hoped. The second take home message is that voting plays a role in how voting systems change. You suppress the vote when the opposition succeeds in getting their supporters to vote.

Photo by Pierre Herman on Unsplash

This general lesson reminds us of something that could have saved us from the need to take the history detour. The right to vote (suffrage) is new. In fact, for some groups it is really new. Women got the right to vote in Swiss federal elections in 1971. Even 1928, when women got equal suffrage in the UK, is not that long ago. In the U.S. it took the Nineteenth Amendment in 1920. The history of black suffrage in U.S. is long and convoluted, with a running thread of ingenious attempts to limit and curtail it.

Human culture is a memory system. While we may have genetic adaptations that were shaped over time and reflect past environments, we also acquire behaviors, norms, attitudes, concepts, and so on, that likewise were shaped in the past, and that affect us without us necessarily noticing. Growing up listening to stories about how your grandfather was beaten because he tried to vote may affect your attitude to voting. This goes for education more generally, of course. But, we can hear the game theorists grumbling, this is just an explanation for the irrational behavior described by the paradox of voting; it does not justify it. True enough (though it highlights the critical assumption that only the outcome of the election affects the payoffs of the game). Two things, though. First, our issue was not with the paradox of voting, it was with conclusions that were drawn from it concerning human nature. Universal, possibly genetic, human nature, presumably. Our discussion led us to the realm of cultural specific explanations so it casts doubt on these. Second, let us consider whether it was rational for the proverbial grandmother to fight for voting rights. Are we justifying current irrational behavior by past irrational behavior?

There are two possibilities to consider. If the behavior of the grandmother was rational then we are done. It would mean that voting was rational at that point, and the over-generalized conclusions about the irrationality of voting based on the paradox of voting fail. Assuming, to the contrary, that the grandmother’s behavior was not rational is more problematic for defending the rationality of voting. But can the claim really be that fighting to extend the franchise was irrational? Can we dismiss the French Revolution (and with it most of the father figures of modern rationality, ironically) simply by concluding that it was irrational? And, if one follows this logic, so was the U.S. revolution.

This would clearly be going too far. The French Revolution was an attempt to change a complex, entrenched social order. It involved theorizing about the legitimacy of monarchy, about human rights, about the best way to organize society, and about human nature. These ideas and debates were known to many (decisions were not one-off, and were common knowledge). The Revolution happened when it did because of particular social and economic conditions that affected multiple people in similar ways (which means that people were coordinated from “outside” the game, and knew about others). People did not act just as individuals, there were multiple groups, with different ideologies and goals, and internal power struggles and coalitions. And, of course, as in the 1974 Senate election, the ultimate payoffs were not known in advance. Within this web of relations between individuals and groups, each with its own obligations and expected payoffs, people decided whether and how to overthrow the Ancien Régime. Voting is not different.

Photo by LSE Library on Unsplash

At this point, the theorist may raise the following objection. Even accepting everything that was just noted, the main issue remains, they would argue. For these coordinated actions to happen, human beings must do things for motivations that go beyond the payoffs in the paradox of voting payoff matrix. And this was the conclusion drawn from the paradox! Alas this supposed trump card does not work. There are ways to argue that humans have internal pro-social motivations that underlie our behavior in groups, altruistic behavior, our capacity for coordinated action, our adherence to norms, and so on. If we accept this, we can suggest that they dissolve the paradox of voting, either by changing the payoffs (for example, by making not voting costly for emotional reasons) or by explaining why people stray from the expected behavior. But the paradox of voting itself does not prove that this is the case, given all the alternative explanations we noted. The paradox of voting is, at most, an illustration, and a weak one at that, not a proof or an argument in support for any pro-social let alone a Kantian categorical imperative explanation according to which people behave as they would have everyone behave, something that is sometimes suggested. People surely vote without wishing everyone, including the voters of the rival parties, vote.

What can we conclude from all of this? The most important is that the paradox of voting is not a good argument in favor of humans having innate pro-social, collective action motivations. In fact, voting is one of cases in which these motivations play a small role compared to other motivations. This does not mean we do not have such motivations, only that the paradox of voting does not help making this determination. Of more practical importance is to take note of the assumptions that the paradox of voting makes about voting. Taking the paradox as a starting point makes these assumptions seem more benign than they really are, and the discussion more universal and less parochial than it is. We noted many of these assumptions, but here are the most important ones: that the cost of not voting is assumed to be zero, that voting is meaningfully costly, and that margins are irrelevant to the result of the election. Each of these is a choice, as we saw, and the implied suggestion that they do not matter for who turns out to vote is problematic.

A lot can be learned for idealized games such as the ultimatum game and the voting game. However, we should be aware of the idealizations we are making, that is know what assumptions we are making and why it is ok and useful to make them. The problem is not that the paradox of voting is not an accurate description of voting. The problem is using it, and similar arguments, to explain the role of culture, the evolution of human capacity for norms, and so on — while it is far from clear that the idealized situation captures the original situation, with culture, norms, and so on coming later.

If social life and culture came before or evolved together with human rationality then we need to study other kinds of games. They will be repeated and involve coalitions and memory. The types of games we analyze reflects what we take to be basic or fundamental. Getting the game right is critical (the American electoral system is not the proverbial “state of nature”).

Here then is the prediction: the closer the situation is to these assumptions, that is the closer it is to the situation described by the paradox of voting, the lower the participation rate in elections is going to be. Regardless of any Kantian or non-Kantian tendencies people may have.

The answer to why some people decide to vote tells us relatively little about universal characteristics of human behavior. It tells us about culture specific attitudes to voting and the mechanics of the electoral system. It also tells whether people feel they have other ways of trying to affect politics, that are less costly or more likely to succeed. Once we put aside the conclusions about human nature, which may or may not be correct, but in either case are not supported by the paradox of voting, we can focus more energy on real life voting. As often presented, the paradox of voting is really a question about why Americans vote, given that U.S. elections are first-past-the-post elections, especially why they would vote in cases when an election is not tied to other elections (e.g., in special elections), and in elections with wide margins and turnout that is not in question. Well, compared to many other democracies they indeed do not bother. Changing these factors by making voting less costly and margins more meaningful for the results, will make more people participate, while the best way of making non-voting costly is by celebrating democratic culture, so that people who decide not to vote would be ashamed (as is the case in some other democracies).

Photo by stefan moertl on Unsplash

And this is perhaps the fundamental problem with the paradox of voting. Voting is not simply about the result. It is not even an act of signaling the degree of support or disdain for candidates or measures (which make the margin of the result important), or of expressing our feelings, in particular political anger. It is all of these things. But primarily it is an act of ownership. A democracy without voters is a democracy in name only. Simply put: use it, or lose it. And while this may seem far fetched, the fact that voting is part of a long struggle, encoded in culture and history, for self-governance is not far from people’s minds. They just need to turn on the evening news.



Ehud Lamm

Philosopher and Historian of Biology at Tel Aviv University. Follow me on twitter @ehud