Post-Election Litigation and the Paradox of Voting

Peter N. Salib1Climenko Fellow and Lecturer in Law, Harvard Law School. Assistant Professor of Law, The University of Houston Law Center (Fall 2021). & Guha Krishnamurthi2Assistant Professor, South Texas College of Law.

I.  Introduction

Economists will tell you that your vote does not matter. Or at least it does not matter if what you care about is who wins a large election. It is therefore something of a mystery why anyone bothers to vote at all. There are other reasons to do so, aside from electing an official. Maybe voting makes you feel good because you think of it as fulfilling a civic duty or complying with a kind of golden-rule-style moral imperative.3See also Andre Blais, To Vote or Not to Vote: The Merits and Limits of Rational Choice Theory (2000) (link). Or maybe failing to vote feels bad because your friends or family think you ought to vote, and they will shame you if you do not. You may even belong to some powerful political interest group whose leader can plausibly change the election’s outcome and who will reward everyone in your group for helping her do so.

But if you vote because you believe your ballot—single and solitary as it is—might make a difference on the election’s outcome, economists say that you probably should not bother. Elections are won by whomever gets the most votes. Garnering just one vote more than one’s opponent is sufficient to win.4This is at least generally true of American elections. It is, however, possible to imagine other systems of voting and governance that might not give rise to the paradox discussed here. And crucially, the candidate who wins by 30,000 votes is not any more the winner than the candidate who wins by thirty.5But see infra note 18 and accompanying text on the mandate theory of rational voting. As a result, all of the votes beyond the single vote that puts one candidate ahead of the other have absolutely zero effect on the outcome. They are wasted. And thus, the only election in which your individual vote matters at all is the election that, without you, would end in a tie. Ties are vanishingly rare, especially for elections in which tens of thousands or millions of people vote. As a result, if voting is even mildly inconvenient—think of the lines, the traffic, or even the missed opportunity for coffee with a friend—you are almost certainly better off abstaining.6See also Jean-Antoine-Nicolas de Caritat Marquis de Condorcet, Condorcet: Foundations of Social Choice and Political Theory 245 (Iain McLean & Fiona Hewitt eds. & trans., 1994) (link) (identifying that the lack of decisional capacity may impact individual interest in voting); G.W.F. Hegel, The Philosophy of Right 202–03 (T.M. Knox trans., 1942) (link) (“As for popular suffrage, it may be further remarked that especially in large states it leads inevitably to electoral indifference, since the casting of a single vote is of no significance where there is a multitude of electors.”).

That, at least, is the traditional story. The most recent American presidential election, between Joe Biden and Donald Trump, however, tells a different one. Nor does the story of 2020 stand on its own. It has a prequel in the presidential election of 2000: Bush versus Gore and, crucially, Bush v. Gore. Taken together, these episodes show that, at least in close, high-stakes American elections, votes do not decide the winner. Litigation does.

The 2000 presidential election famously came down to Florida, and Florida came down to just 537 votes. That is the official figure—George W. Bush won by 537. But that is not because 537 more people voted for Bush than Gore. Had Florida been left to its own devices, its supreme court would have mandated the manual recount of 9,000 “under-votes” exhibiting an infamous array of “chads”7Florida’s voting machines punched holes in ballots to indicate voter choices. Excised paper fragments from the hole punches are called “chads.” And imperfect voting machines can produce chads that, in various ways, fail to detach fully from the ballot.—“pregnant,” “hanging,” and otherwise. It is still unknown whether Bush would have retained his lead after such a recount.

Instead of allowing Florida to proceed with its count, Bush sued. The case proceeded rapidly to the Supreme Court, which stopped the recount and handed Bush his victory. The Court divided 5–4, along neatly partisan lines, with the conservative majority’s key argument generously described as “perplexing to most constitutional law scholars.”

The 2020 presidential election, too, ended with litigation. But its result was not decided there. The major difference between the two elections was their respective vote margins. In 2000, throwing out a few hundred votes in a single state would have changed the result. But Biden won by tens of thousands of votes in several states, each of which was sufficient to guarantee his victory in the electoral college. That, of course, did not stop the Trump team from filing lawsuits in several of those states. But those lawsuits were longshots—moonshots, even. There was no colorable legal theory under which Trump could change enough votes to win.

Here, in the space between the post-election litigation of 2000 and the post-election litigation of 2020, one can see why the traditional rational-choice story of voting is wrong. Individual votes can matter, even in the absence of a tie. When elections are close, as in 2000, they are more susceptible to decision by litigation. Partisans, either at bar or on the bench, can muster a wider and stronger range of legal arguments to invalidate sufficient votes to change the outcome. But as the pre-litigation margin of victory grows, as in 2020, the available arguments become fewer and less credible. Then, even highly motivated reasoners are unlikely to find a way to invalidate enough votes to swing the result.

Moreover, litigation is costly. Thus, as higher vote margins make a court-ordered reversal less likely, rational candidates become less likely to mount a challenge at all. Even the Trump campaign—rational only in fits and starts—did not file post-election litigation in California, although that state’s 55 electoral votes would have secured Trump’s victory. What emerges from these observations is that it is beneficial for a candidate to accrue votes beyond the tipping point of the election, at least past the “margin of litigation”—the number of votes that post-election litigation could plausibly overturn.8Richard L. Hasen, The Voting Wars 131 (2012) (link).

In the remainder of this Essay, we systematize the above observations about vote margins and post-election litigation. We relate those observations to the formal economic arguments on the voting paradox. Our primary claims are twofold. First, the mechanics of post-election litigation complicate core assumptions of the traditional economic argument that voting to change outcomes is irrational. In particular, we show that it is not just the tipping-point, half-plus-one vote that affects an election’s outcome. Rather, in reasonably close races, many votes have the potential to change the outcome. This changes the math substantially, making it possible that at least some votes should rationally be cast in expectation of having an electoral effect.

We also explore a second possible source of value for individual votes beyond tiebreakers: legitimacy. When voters believe that courts have coopted their power to pick their representatives, the public may view electoral results as illegitimate. Such perceptions of legitimacy may have a variety of negative downstream effects. Most benignly, officials perceived as illegitimate may have a harder time enacting their agendas. More ominously, experiences from abroad show that perceived interference in high-stakes elections can give rise to civil unrest, violence, or even armed conflict. But as electoral margins increase, arguments that courts have stolen the election—by either action or inaction—become less appealing. This may reduce perceptions of illegitimacy and ward off such dangerous outcomes.

II.  The Paradox of Voting

A fundamental question about voting, and indeed any kind of action, is what is its motivation: Why should anyone vote? The obvious answer is that one has reason to vote in order to play a part in (collective) decision making, to help decide our collective course of action. But in a technical sense, one’s own vote is only a decision-maker if it is a tiebreaking vote.9There is also the possibility that one’s vote could be a tie-making vote. Suppose there were 500,000 potential voters in total, and, without Mike’s vote, the total is 250,000 to 249,999. Mike’s vote could result in a tie, which would then presumably trigger some other default mechanisms to decide between ties. For simplicity, we subsume this under the term “tiebreaking vote.” However, when votes are sufficiently numerous, we know that the chances that one’s vote will be a tiebreaker are incredibly low. At the same time, as the economist Anthony Downs observed, “every rational man decides whether to vote just as he makes all other decisions: if the returns outweigh the costs, he votes; if not, he abstains.”10Anthony Downs, An Economic Theory of Democracy 260 (1957).

Put in mathematical terms, in rational choice models, the utility of a vote is equal to pjB – C. The term pj is the probability that one’s vote will break a tie in favor of candidate j. B is the value to the voter of electing candidate j over the alternative. And C is the cost—in terms of things like time, inconvenience, gas, or lost wages—of voting. A given individual ought to vote when utility is positive and abstain when it is negative.

In sufficiently large elections, the utility of voting will almost always be negative. This is because ties are so extraordinarily rare—perhaps occurring in just one in several billion elections—that pj reduces B to nearly zero.11The important question here is the probability that there is a tie (and that the additional vote breaks the tie). In such a case, every vote in the majority is a tiebreaking vote. And on the flip side, suppose in an election between B and T with 105,000 voters, the breakdown is 55,000 for B and 50,000 for T. It is not the case that there is one 50,001st voter who breaks the tie. It’s arbitrary in what order the votes are cast or counted. This is an election in which no individual’s vote was dispositive and thus, the rational choice model tells us, one in which no vote was cost-benefit justified. Thus, based on the fact of the sheer unlikelihood of a tie, the “return” of one’s vote, at least in terms of decisional capacity, is nearly nil. And because voting takes some effort, it would seem rational to abstain from voting and irrational to vote.12Downs, supra note 10, at 260–70.

The paradox arises because this result intuitively seems wrong. First, a massive number of people around the world vote in democratic elections. One could retort that they are acting irrationally, but that is an intellectually costly claim.13Error theories are intellectually costly because they require the proponent to posit that there is systematic error by numerous individuals. That is certainly possible, but it imposes a burden to explain how such widespread error occurs. See Javier González de Prado Salas, Rationality, Appearances, and Apparent Facts, 14 J. Ethics & Soc. Phil. 83, 99 (2018) (“[I]t would have the costs generally associated with error theories; in particular, it would be necessary to offer an explanation of why ordinary [people] are systematically mistaken about their [ ] practices.”). Moreover, we needn’t only point to others; introspecting, we might count ourselves as rational and as making the rational choice to vote. So that imposes an explanatory burden: What makes voting rational, when it seems like the decisional capacity of voting would not?

A few answers look initially plausible. First, we can imagine some instances where the expected value of one candidate winning—B, above—is unusually high. Assuming some reasonable costs, even if the chance that one’s vote will matter is infinitesimal, voting could be rational if the magnitude of the beneficial result from a favorable decision is great enough. Imagine a voter—call him David—who can choose between two hypothetical candidates B and T or abstain. B is a status quo-ist. T vows to nuke the world and kill all earthly life. David decides to vote for B. Even if the likelihood that David’s vote will make a difference is very low, the magnitude of the benefit if it did make a difference is extremely high.14This assumes that a continually existing world is of sufficient benefit to David. This is plausible, of course. David’s life is surely important to him, and so is a world that he can interact with. Now, it may not be the case that the benefits of an existing and operating world inure to David, or indeed any rational agent. David may not interact with much of the world, and so those benefits may have to be excised from David’s rational calculus. Nevertheless, we think it’s plausible that this would still be enough to weigh in favor of David voting. And it plausibly could be high enough to overcome the standard cost of voting.

Of course, such dire votes are rare, and many people apparently find it rational to vote even when it is not so dire. Even in a consequential election, such as in the 2016 election between Hillary Rodham Clinton and Trump, or in the 2020 election between Trump and Biden, it is not obvious that the delta between the choices is enough to compel one to vote on the traditional rational-choice accounts. Nevertheless, voters cast hundreds of millions of ballots in these races.

Another possible way to understand voting as rational is to consider the possibility of universal or widespread voter abstinence. We can posit that that would be a disastrous result, for it may lead to an undemocratic replacement or an erosion of the legitimacy of the system. And thus even if it is very unlikely, voting will ensure that that does not occur, which in turn may outweigh any standard costs of voting.15Downs, supra note 10, at 269 (“Since the consequences of universal failure to vote are both obvious and disastrous, and since the cost of voting is small, at least some men can rationally be motivated to vote even when their personal gains in the short run are outweighed by their personal costs.”). Importantly, here the ultimate focus is not necessarily about a particular desired result in the election but rather upholding the democratic system of decision-making itself. Thus, with this interest in mind, it may be that marginal and aggregated votes matter. For each additional person who votes, it may be the case that democratic norms are proportionately affirmed and thus healthier, or that elections may become more representative of the people as a whole. Even if the upholding of the system is a threshold, rather than a continuum, wherein the system is upheld if a “sufficient” number of people vote, it’s likely that the threshold is vague. If so, there could be added utility in marginal and aggregated votes to ensure that the vague threshold is met.

Another strategy to explain the rationality of voting is to expand the benefits of voting beyond the decisional capacity and institutional impacts. For example, William Riker and Peter Ordeshook have argued that there is a psychic benefit component to voting that may motivate one to vote. This psychic benefit can include “the satisfaction from compliance with the ethic of voting”; “the satisfaction from affirming allegiance to the political system; the satisfaction from affirming a partisan preference; the satisfaction of deciding, going to the polls”; and “the satisfaction of affirming one’s efficacy in the political system.” Such accounts may explain the descriptive fact of people’s voting, but it does little to explain why voting is normatively rational—because, for example, why does the ethic of voting, the allegiance to the system, or the satisfaction of deciding or affirming one’s efficacy matter if there is no decisional or institutional impact? That remains unaddressed.

Taking another tack, other explanations have reexamined the assumption that one’s vote has little chance of being impactful on the decision. As Thomas Palfrey and Howard Rosenthal observe, what matters is the interactions between voters. For example, if David knows that all of the other voters are going to abstain, then David’s vote is decisive, and so the chance that his vote will have decisional capacity is one. But in sufficiently large elections, this approach fails to rationalize voting. This is for two reasons: First, a precondition to having high turnout is that each voter must think their vote can be decisive, which in turn requires an even number of votes for each candidate. But the higher the expected number of voters, the less likely this is.16See also Howard Margolis, Probability of a Tie Election, 31 Pub. Choice 135, 135–38 (1977). Second, even if there is a large number of potential voters evenly distributed between the candidates, there is uncertainty about whether they will vote, so a voter having such information about other voters is not possible, and voter turnout is disincentivized.

There are certainly other theories that plausibly explain how voting might be rational, such as that voters may have tangible interests in demonstrating their voting for a particular candidate, even if the vote is not efficacious; that even large elections can be minimized into small elections, like winning a precinct, which can have tangible benefits for the precinct; and that group leaders may be able to more specifically incentivize individual voters through some means. But there is a dearth of evidence showing that any of these is correct. So there remains a lacuna in explaining the motivational paradox of voting.17Rick Hasen has set forth a putative explanation that relies on the “social norm” of voting. Such an explanation, in our view, gives further depth to understanding the mechanics, but it is in a sense a “buck-passing account,” because our attention simply shifts to why such a “social norm” exists and whether that is rational. 

III.  The Impact of Litigation

This Part examines two ways in which the effects of post-election litigation challenge the standard accounts of Downs’s paradox. The first effect is on election outcomes themselves. We argue that margins of electoral victory—not just the tiebreaking vote—matter to the outcomes of post-election litigation. The bigger a pre-litigation lead one candidate has, the less likely it is that litigation will intervene and change the no-litigation result. And at least at a first approximation, this probabilistic effect is continuous across some range of possible vote margins. If a candidate is losing by enough votes, she has no chance of changing her fate with litigation. Conversely, if she is winning by that same margin, she has no chance of having her victory snatched away by a judicial ruling. Every vote for that candidate between those two margins has some effect—though not necessarily equal across the whole range—of increasing her probability of victory.

The second effect of post-election litigation is a knock-on to the first. When close, high-stakes elections are decided by courts—or could have been—the public may perceive the results as illegitimate. That, in turn, can have two kinds of effects. First, officials seen as illegitimate may face stronger opposition to implementing their agenda. This argument echoes an older one in the rational-voting literature. Some have contended that marginal votes may matter to bolster a candidate’s popular mandate.18Jason Brennan, The Ethics and Rationality of Voting, Stan. Encyclopedia Phil. (2020) (discussing voting to change the mandate); Robert A. Dahl, Myth of the Presidential Mandate, 105 Pol. Science Q. 355, 355–60 (1990) (discussing the mandate theory); Alexander A. Guerrero, The Paradox of Voting and the Ethics of Political Representation, 38 Phil. & Pub. Affairs, 272, 272–80 (2010); Lawrence J. Grossback, David A.M. Peterson & James A. Stimson, Comparing Competing Theories on the Causes of Mandate Perceptions, 49 Am. J. Pol. Sci. 406 (2005). But, as discussed below, the two theories are distinct.19See infra Part III.B. Even if a candidate’s perceived popularity does not, as the empirical literature suggests, affect her policymaking efficacy, her perceived legitimacy might. Second, perceived illegitimacy can have more dire effects. In extreme cases, perceptions that an election has been stolen, and thus that the ruling coalition is imposing its power unlawfully, can lead to civil unrest, violence, or even armed conflict.

As margins of victory become greater, both of these effects are mitigated. When a candidate wins by a sufficient number of votes that no clever litigation strategy could overturn them, even extreme partisans must eventually admit defeat. As Donald Trump’s myriad longshot litigations wound down, figures including Fox News commentators, down-ballot Republicans, and Trump’s own advisors began to acknowledge President Joe Biden. Certainly, not everyone agrees that Biden is the legitimate Forty-Sixth President of the United States. Nor did everyone agree that George Bush was legitimately the Forty-Third. But are likely fewer Biden deniers than there might have been following a tighter race. Biden’s comfortable margin likely reduced tail risks like widespread political violence.

A.  Decisional Impact

As discussed above, in rational choice models, the utility of a vote is equal to pjB – C. And as already discussed,20See supra Part II. pj is the important variable here. For elections of sufficient size, the probability of casting the tiebreaking vote is extraordinarily small, perhaps on the order of one in eight billion. Thus, unless a voter values a win for candidate j at eight billion times her cost of voting, her utility calculation will come out negative.

Note however that pj is infinitesimal only because a vote matters exclusively in the extraordinarily unlikely case of a counterfactual tie. In every other possible circumstance—the other eight billion or so—everyone’s individual vote is worth precisely zero. But if votes affected outcomes not only at the pivot point between a tie and a majority but also on the margin as majorities became larger, the calculus would change substantially.

Imagine, for example, that elections were decided by a dice roll: Each candidate got one six-sided die per vote, they rolled them all, and the candidate with the highest total won. Then pj would not depend on the impossibly unlikely event of a tie. The marginal probabilistic effect of an individual vote would be a function of the expected value of one additional die,21Each fair six-sided die is worth, on average, 3.5: the sum of the sides (1+2+3+4+5+6=21) divided by six. as compared with the expected results of the candidates’ rolls without the extra die. Then, every additional vote would have value up to some margin at which one candidate had so many dice that she literally could not lose.22Consider an election in which one candidate ends up only with a single six-sided die, while the other has seven. Even if the former rolls a six, the latter wins with any possible roll, including seven ones.

This, we think, is the right way to understand voting when post-election litigation is added into the mix. Legal arguments are a bit like dice rolls. Some are stronger and some weaker. But as prudent lawyers will agree, it is generally impossible to say ex ante with absolute certainty which will succeed, which will fail, and what the ultimate effects will be. All of that can only be known once the arguments are tried and, so to speak, the die is cast.

The probabilistic strength or weakness of the legal arguments necessary to reverse an election’s pre-litigation results will generally depend on the margin of victory. So long as the election was conducted with reasonable care and competence,23There is no evidence to the contrary in either the 2000 presidential election or the 2020 presidential election. genuine errors will be small. Then, the strongest legal arguments will change just a handful of votes and can reverse only a very close pre-litigation result. By contrast, as the number of vote reversals or invalidations needed to change the pre-litigation outcome becomes larger, the available arguments become fewer and weaker. This is not to say that no election could ever suffer from clear and pervasive error, such that a straightforwardly compelling argument could change many votes. It is just to say that such situations are comparatively rare.

Consider again the illustrative contrast between the 2000 and 2020 presidential elections. In 2000, Gore trailed Bush by just a few hundred votes in Florida. Gore argued that the state ought to manually examine 9,000 ballots that automated counting machines did not register as containing a vote for president. Both the Florida and the United States Supreme Court agreed, at least in principle. That should be unsurprising; this was a modest argument, both legally and in terms of its likely net effect on votes. Bush then argued that the state’s arbitrary and inconsistent approach to such manual review violated the Due Process and Equal Protection Clauses. That argument, too, was at least convincing enough to sway seven U.S. Supreme Court Justices, including two liberals.

The case’s most “perplexing” and partisan holding was about the appropriate remedy. Relying on doubtful constitutional and pragmatic propositions, the conservative majority overrode the Florida Supreme Court’s interpretation of Florida law and called off the recount entirely. Crucially, however, Bush did not necessarily need to win this long-shot argument to sway the election via litigation. Had the U.S. Supreme Court not called off the recount, it would have been left to Florida to devise a constitutionally adequate recount process. Then, Gore and Bush would have duked it out over the appropriate standard, perhaps validating or invalidating just enough votes to win. The closeness of the pre-litigation result made either outcome plausible.

The 2020 election, by contrast, was nowhere near as close. To change the pre-litigation result, Trump would have had to invalidate tens of thousands of votes across multiple states. There were no remotely plausible legal arguments for doing so. That is not to say that Trump did not try. Trump and his affiliates filed over fifty lawsuits, advancing multifarious legal theories for throwing out different numbers of ballots. They succeeded in invalidating exactly zero votes.

The closest they came was obtaining a preliminary order from Justice Samuel Alito that required Pennsylvania to segregate mail-in votes that arrived after election day. The Pennsylvania Supreme Court held that those votes should be counted, invalidating a state law to the contrary under state constitutional principles. If the U.S. Supreme Court had heard this case on the merits, it could have reversed the state court, citing Chief Justice William Rehnquist’s novel Article II, § 1, cl. 2 argument from Bush v. Gore. That clause delegates to state legislators the power to appoint presidential electors. Rehnquist therefore contended that, if a state supreme court misinterprets state statutes governing presidential elections, it thereby violates the United States Constitution. This is doubtful. At any rate, the Court has now denied certiorari in the Pennsylvania case, in part on mootness grounds. Biden led by over 100,000 votes in Pennsylvania, so throwing out a comparatively small number of late-arriving ballots would have made no difference.

In the end, the Trump camp raised dozens of wild legal theories designed to disenfranchise thousands upon thousands of Americans. None of those theories succeeded because they are crazy. And the least crazy, though still dubious, theory would reverse nowhere near enough votes to make any difference in the election’s outcome.

These differences between 2000 and 2020 illustrate our point nicely. Close elections are more susceptible to reversal via litigation than landslides. The vote margin between those two extremes therefore matters to the outcome. And if that is right, then the standard rational-choice model is wrong. Votes do not matter to the outcome only in the extraordinarily unlikely event of a tie. Instead, they matter whenever the election is close enough that the probability of litigation changing the outcome is above zero. For such elections, the value of pj is therefore the single-vote-induced shift in probability that litigation will result in the installment of one’s preferred candidate.

Skeptics may argue that, even under our model, votes are still almost always valueless because the relationship between vote margin and litigation outcomes is stepwise. That is, because of such stepwise effects, even in the face of post-election litigation, it is still only tiebreaking votes that matter. Consider a simple model in which there are only two legal arguments available in post-election litigation. Argument 1 will invalidate 1,000 votes, and it is 20% likely to succeed. Argument 2 will invalidate 5,000 votes, and it is 1% likely to succeed.

In this model, it does not matter, for example, whether Candidate A’s pre-litigation margin is 2,000 votes or 4,000. Either way, Candidate B must make Argument 2 and will win with only 1% probability. In this model, the only votes that matter are the 1,001st vote and the 5,001st.24As noted above, we simplify slightly in considering only tiebreakers, not tie-makers. See supra Part I. Those votes shift pj, by putting the pre-litigation margin of victory outside the range of reversal by one litigation argument. The 1,001st vote takes Argument 1 off the table and thus shifts Candidate B’s chances from 20% to 1%. The 5,001st vote makes Argument 2 worthless and further shifts Candidate B’s chances from 1% to zero. But the 2,001st vote has no such effect, nor do any of the others. Thus, in this simple model, pj, is a function of the probability that the election will end in either a straight tie, a 1,000-vote margin, or a 5,000-vote margin, multiplied by the change in probability at each of those thresholds that Candidate B flips the election.

Here, then, it is again only tiebreaker votes that matter. However, the number of possible ties in each election is increased substantially from the traditional rational-choice model. Tiebreakers occur not only as one candidate’s pre-litigation vote count pulls ahead of the other’s. Instead, there is a potential tie associated with each potential post-election litigation argument.

Even if this were the end of the story, our model would make voting look less irrational than the standard ones do. Depending on the number of litigation arguments—and thus the number of potential tiebreaker votes—the odds of one’s vote mattering might go up substantially. If our model makes the likelihood of a tie 100 or 1,000 times greater,25That is the case if there are this many potential post-election litigation arguments. As discussed below, we think that there are at least this many, when the full complexity of litigation is factored in. See infra note 29 and accompanying text. then that at least pushes the expected value of voting toward the positive range.26Even if it remains negative, this could matter substantially if one posits additional hedonic benefits to voting beyond those associated with changing the outcome. See supra Part II. 

We do not think that this is the end of the story. In fact, we think that the best model of post-election litigation treats the function from vote margin to pj as more or less continuous. That is, every marginal vote, or close to it, ranging from a tie to a majority beyond “the margin of litigation” matters. To understand this, consider that every potential post-election litigation argument is not really just one argument. Rather, each theory harbors numerous potential variations, and each variation has the potential to invalidate a slightly different number of votes. Even small changes in vote margins might affect which variations on which arguments the candidates ultimately make.

Take again Bush v. Gore as a model. There, a majority of the Supreme Court determined that Florida’s non-uniform recount procedures were constitutionally deficient. What if, instead of terminating the recount, the Court had, per Justice John Paul Stevens’s dissent, remanded for the Florida Supreme Court to promulgate a uniform standard? Then, the parties would have litigated over the proper standard. The possibilities would have been numerous, with each variation resulting in a slightly different number of ballots counted.27As the Bush Court noted, each county in Florida used a different standard, and the candidates could doubtless have come up with many more.

The parties’ advocacy in litigation of one proposed voting standard, as opposed to others, would not be random. They would be strategic. If the lawyers were doing their jobs well, the parties would choose the arguments most likely to help them win. In post-election litigation, this means maximizing two often-competing desiderata. On the one hand, the parties must advocate arguments—or variations of arguments—that, if accepted, will invalidate (or validate) enough votes to ensure electoral victory. On the other hand, they want to advance the arguments most likely to be accepted by the court.

The best way to balance these goals depends directly on pre-litigation vote margins. If Gore needed 800 votes, instead of 537, to beat Bush, he might have advocated a more aggressively Gore-friendly recount process. Perhaps less so if he knew he only needed 600. And if he did only need 600, advocating the slightly less Gore-friendly standard would likely maximize his probability of success. Courts should—and do—cast a skeptical eye on litigation arguments plainly designed to disenfranchise one candidate’s voters but not the other’s. And in an already close election, the most evenhanded approaches to counting votes will, absent some oddity,28Here, “oddity” means a situation in which the votes under special review—like a recount—are unlike the general population of votes in some way that favors one candidate over another. One could imagine situations like this. A fairly conducted manual count of rural undervotes might naturally favor a Republican candidate, and an urban one a Democrat. usually yield a small net advantage for one side or the other. The 600-vote argument will thus generally be more likely to succeed than the 800-vote argument.

Skeptics may here object that even this dynamic does not show that every vote between a tie and the margin of litigation counts. Are there really two different versions of a recount standard that would net Gore 800 and 600 votes, respectively? Even if there are, there surely are not, they would argue, two different versions that would net him 600 and 601 votes, respectively, such that every single vote matters. Suppose that the range between a tie and a victory beyond the margin of litigation spans 50,000 votes. Are there really 50,000 different legal arguments and variations available? Perhaps not. Or at least there may not be that many differences that can be rightly described as legal, per se.

However, once one considers the innumerable complexities of litigation beyond legal theories, the possibility of 50,000 variations seems more plausible. Lawyers make all kinds of subtle choices that balance marginal variations in the effect of winning a given argument against marginal variations in the likelihood of success.29This balance is a key question in litigation strategy and risk analysis. Gore’s lawyers could choose to make both the 600-vote argument and, in the alternative, the 800-vote argument. Or they could make just one. Insofar as the 800-vote argument is weaker, merely making it could make Gore seem less credible generally. That could reduce the probability of victory as to the 600-vote argument. And even if they make both, how much space should each get in the briefing? Which cases should they cite? How aggressive should the tone be? Who should argue the motion? If they lose both arguments, should they file a motion for reconsideration? If so, as to both arguments or just one? How should they allocate space in that brief? What about appeals? The list goes on and on.

Now we think it is possible to see how the function from pre-litigation vote margin to pj might be best modelled as continuous. As margins grow, the trailing candidate needs to be more and more ambitious in litigation to win. But on average, ambition in litigation—at least litigation designed to change vote counts—will be inversely correlated with success. Candidates can deploy ambition in many different kinds of decisions, ranging from raising totally new legal arguments, to varying those arguments, to presenting those variations differently via briefing, motion practice, and oral argument. And the resulting panoply of available ambition levels means that, at least in theory, even very small differences in vote margin could affect behavior. That is, even the smallest difference in margin—a single vote—could make a commensurately small difference in behavior.

We now turn to one final point about decisional impact. In our model, the fact that every vote between a tie and an unassailable victory counts does not mean that every vote counts the same amount. Votes here are likely subject to diminishing value on the margin. A Biden lead of 100,000 votes in Pennsylvania may not be very much more likely to be overturned by litigation than a lead of 101,000. Likely the marginal security of moving from a 1,000-vote lead to 2,000 is greater. The curve may even have regions with increasing marginal value; that is, where each additional vote is worth more than the previous one. Perhaps certain small leads are as likely to be attributable to random noise in counting than anything else, such that even a straightforward statutorily mandated recount might well eliminate them. A 100-vote margin may be almost exactly as likely to be reversed as a ten-vote margin, but maybe a 500-vote margin is appreciably harder to overturn than a margin of 100.

To summarize, our model challenges the traditional rational-choice account of voting to change the outcome. Most previous scholars have contended that such voting is unlikely to be rational because only tiebreaking votes matter. And because ties are so rare, pj is vanishingly small. We think that many more votes matter than tiebreakers. This is because, for sufficiently close races, litigation, not vote counts, may decide the outcome. Moreover, every vote from a tie up to a victory beyond the “margin of litigation” makes the probability of a candidate’s electoral defeat less likely. This changes the math substantially. Rather than a function of the likelihood of a tie, pj is, on our view, a function of the effect of increasing pre-litigation vote margins on probable litigation outcomes. This makes it at least plausible that, for those living in sufficiently competitive jurisdictions,30Our account has little to offer those who live in districts where elections are usually not close, with vote margins regularly beyond the margin of litigation. This, however, comports with a fairly widespread intuition that it makes less sense to vote in districts where the same party invariably wins in a landslide. The puzzle in these districts is why many people who themselves acknowledge the futility of, e.g., voting for a Democrat in a statewide election in Mississippi, nevertheless show up to the polls. it might be rational to vote for the purpose of ensuring a candidate’s victory.

B. Legitimacy Impact

An associated effect of post-election litigation arises from the fact that when elections are decided by courts, or appear as though they could have been decided by courts, the public may question the legitimacy of the election. So why does the perception of legitimacy of elections matter? That is because when people see elections as illegitimate, they also see election winners as illegitimate—and thus not eligible to govern. It is the lawful democratic election that gives the winner the license to govern. The absence of such electoral legitimacy can impact the ability of an elected politician to govern, and worse, it can derail the institutions of government.31Questions about election legitimacy can take at least two forms. As James Gardner has explained, “The normative point of an election is to identify and install in office those whom the people wish to appoint. The substantive electoral legitimacy of an election thus requires that it employ methods and procedures reasonably calculated to permit the accurate identification of those individuals. The procedural electoral legitimacy of an election, in contrast, requires only that it be conducted faithfully pursuant to authoritatively established procedures, whatever they may be.”

With that in mind, suppose we consider whether a governed person perceives the election as legitimate. In this role as election observer, that person’s determination about the legitimacy of the election may include many considerations, including substantive political views. But we can surmise that a primary consideration for the observer is whether the purported winner received more votes than the other candidates, per the electoral laws. This last clause is critical: If there is contested litigation that may put the vote tally in doubt, then the observer may have reason to doubt the legitimacy of the election. Importantly, here, it may be the case that the election is not in fact in doubt, because, say, any contested litigation would actually fail. What is critical though is the perception of the election observer, and that includes any (reasonable) epistemic gaps of the observer. Many individual observers could have such doubts about the election. If there are such widespread doubts, then that can result in serious concerns about the legitimacy of the election.

It is perhaps distressing that we do not have to reach far for such examples. President Trump’s attorneys, and other attorneys purporting to represent him, have filed numerous lawsuits challenging the election results. Some of the theories undergirding these suits are viewed by most reasonable legal minds as far-fetched and bizarre. But there are a number of people who seem to be sufficiently swayed by their promise to have doubts about the legitimacy of the election and Joe Biden’s win. Now we might be skeptical that these lawsuits are brought in good faith—perhaps they are simply an antidemocratic wolf in sheepish legal clothing. But that in fact strengthens our point. The extreme unlikelihood of those lawsuits’ success suggests that accrued votes, past the numerical tipping point of the election, can rebut antidemocratic actions.

Consider instead, then, a less extreme example. In its first session of 2021, the Pennsylvania Senate’s Republican majority forcefully wrested control of the proceedings to block the swearing-in of a newly elected Democrat. The Democrat, Jim Brewster, won his election by just 67 votes. Those votes included ballots missing their handwritten dates, which the Pennsylvania Supreme Court ruled should be counted. Pennsylvania Republicans have nevertheless sued in federal court, effectively seeking the invalidation of an apex state court’s holding about state law. In the meantime, Senate Republicans have refused to treat Brewster as a legitimately elected official, voting to omit him from the list of newly elected senators. Had Brewster’s margin been large enough that he did not need any undated ballots to win, he would presumably have been seated alongside the chamber’s other duly elected Democrats.

As mentioned in Part II, our argument about legitimacy sounds similar to the argument that voting tallies may relate to the strength of the winning politician’s mandate to govern.  That hypothesis, however, is belied by the empirical evidence, which seems to demonstrate that vote margins beyond a majority have little effect on officials’ ability to get things done.32Brennan, supra note 18, (“Political scientists have done extensive empirical work trying to test whether electoral mandates exist, and they now roundly reject the mandate hypothesis. A winning candidate’s ability to get things done is generally not affected by how small or large of a margin she wins by.”); Dahl, supra note 18, at 360–70 (rejecting the mandate theory). Our argument, however, is distinct. That is, because this argument is focused on legitimacy, it impacts the very question as to whether the winner has the authority to govern at all. It is one thing to oppose an official simply because the populace believes them to have won by a slim margin; it is another to oppose them because the populace believes they did not win at all.

The potential impact of such legitimacy problems is wide-ranging. At the lowest end, if a consensus of observers believes that the purported election winner is in doubt, then the consensus may also believe that the election winner does not have genuine license to govern. This may lead to greater opposition to the election winner’s agenda—for example, people might think that such a contested winner should pursue a caretaker agenda rather than anything more ambitious, given the doubts over legitimacy. More seriously, a legitimacy crisis may lead people to ignore the commands and actions of the winning politician, which in turn can straightforwardly lead to governance problems. Indeed, we have seen such disruptions resulting from election litigation, including from Bush v. Gore and the 2008 U.S. Senate election between Al Franken and Norm Coleman.33See Hasen, supra note 8, at 131–58.

In extreme cases, this could even lead to disastrous failures in governance and civil unrest. And things could get even worse. It could lead to widespread doubts about the functionality of our elections, which in turn might lead to upheaval of our democratic institutions. Indeed, on January 6, 2021, the United States witnessed this firsthand: A mob of insurrectionists attacked the Capitol, killing an officer, injuring dozens, and causing chaos. The insurrectionists were seeking to overturn the results of the 2020 presidential election being certified by Congress at that time. And they were spurred on by President Trump’s fraudulent claims that the election was rigged against him.

An individual’s vote can have an impact on the legitimacy calculus in similar ways as it does with the decisional calculus. To see this, first consider the election observer’s legitimacy calculus again. That will principally be determined by the vote tally and potential post-election litigation. In that sense, whether an observer believes an election winner is legitimate will, with good reason, approximate whether one wins an election, including post-election litigation. And this gets aggregated across all election observers, which in turns produces a consensus, or lack thereof, on legitimacy. But it is an approximation, because there is necessarily a gap between any individual observer’s understanding of the merits of post-election litigation and the actual merits of post-election litigation. As a consequence, there may be a discordance, or at least a fuzziness, between how post-election litigation impacts the actual outcome and how it may impact legitimacy.34How election observers understand the merits of potential post-election litigation is a difficult question. There is plausible reason to think, however, that people observing the election may overestimate the potential for success of such post-election litigation. That is, some election observers may think that there was a good chance that post-election litigation would have changed the result of an election, even if, in fact, it did not. If so, then it may mean that people’s legitimacy determinations are more sensitive to post-election litigation than the decisional impact of post-election litigation.

Now consider two dimensions of an election: (a) the breadth of the margin—which may be close or not close; and (b) whether there is outcome-changing litigation. There are then four scenarios:

  • A close election without outcome-changing litigation;
  • A close election with outcome-changing litigation;
  • An election that is not close, without outcome-changing litigation; and
  • An election that is not close, with outcome-changing litigation.

In either close-election scenario, potential problems of legitimacy arise. When an election is close, and there is outcome-changing litigation, (some) people observing the election may decry the judicial intervention. And when there is no outcome-changing litigation, (some) people observing the litigation may believe, or assume, that litigation should have changed the outcome.

Contrast this with when the election is not close. If there is no outcome-changing litigation, most will understand that the potential and actual arguments raised in litigation were weak, and indeed too weak to overturn a clear win. This results in less legitimacy concerns, even if some persist.

That leaves a final scenario: when an election is not close and litigation changes the outcome. This poses a perilous result for legitimacy, but the chances of a legitimacy crisis are, for the reasons described above,35See supra Part II.B. low. Moreover, they are made less likely as the margin of victory grows.36See supra Part II.B. Thus, marginal votes—that is, the accrued votes past the tipping point of the election—can substantially reduce the possibility of this very bad outcome.

Consequently, just as with the decisional impact of voting in light of post-election litigation, voting beyond the tipping point of the election can have an impact on the legitimacy of the election. In assessing voting margins’ decisional impact, we noted that every vote from a tie up to a victory beyond the “margin of litigation” makes the probability of a candidate’s electoral defeat less likely. For legitimacy, similarly, every vote from a tie up to a victory beyond the margin of potentially successful litigation, in the view of election observers, makes a candidate’s electoral illegitimacy less likely. As noted, the margin of potentially successful litigation in the view of election observers may be broader than the actual “margin of litigation” for an electoral decision. So there may be a band where a candidate would likely win any post-election litigation but remains illegitimate in some voters’ eyes. If so, this means there is an added benefit for a voter to vote, even past the margin of litigation.

Indeed, that the “margin of legitimacy” may be broader than the margin of litigation is exemplified by the January 6, 2021 Capitol attack. By all informed accounts, President Trump’s resounding electoral loss was neither rigged nor otherwise significantly impacted by fraud.37See supra Part I. But many of Trump’s followers seem to believe that Biden’s electoral victory was illegitimate. And it seems that if Biden’s electoral victory had been larger, even if some of the Trump acolytes would not budge in their belief of a rigged, fraudulent election under any circumstances, some of those Trump supporters would likely have been convinced that Biden’s electoral victory was legitimate.

At the same time, we can consider the counterfactual where Biden’s electoral victory was much thinner. Under the substantial electoral victory that Biden actually obtained, a frighteningly large number of Republicans questioned the legitimacy of the victory. But a robust majority of Republican senators, where they enjoyed a majority of the chamber, rejected calls to seek to overturn the election. Had the election been significantly closer, we could imagine how much worse it would have been. The official attacks on the election may have been more elaborate and garnered more support, and that in turn could have led to further unrest.

This suggests a relationship of the sort we have identified: As the margin of victory becomes larger, the more people believe that the election is legitimate; as it becomes smaller, the less people believe so. And, in a functioning democracy, there is some margin of victory that ensures a sufficient consensus will believe that the election was legitimate. These observations about legitimacy build upon our prior mathematical understanding that, rather than a function of the likelihood of a tie, pj is a function of the effect of increasing pre-litigation vote margins on probable litigation outcomes. Taken together, these arguments make it at least plausible that people living in competitive jurisdictions rationally ought to vote to ensure that their preferred candidates win and are viewed as legitimate.

C. The Voter’s Total Calculus

We now return to the rational-choice calculation of voting’s utility: pjB – C. The paradox of voting arises because, given the sheer unlikelihood of a tie, it seems that the expected effect of voting would be negligible, and thus the benefits of voting are unable to overcome even the slightest of costs.

We have challenged this notion by focusing on the pj term. What we have shown is that the probability of there being an effect is not so negligible, because the effect does not manifest only in cases of the voter being a tiebreaker. Rather, marginal votes beyond the tiebreaking threshold can have expected effects in terms of both post-election litigation outcomes and knock-on perceptions of electoral legitimacy. Further, we have argued that the effect of these votes is continuous or nearly so, such that each added vote has some probabilistic impact, up until the margin of litigation. Consequently, pj can be expanded into ptiebreaker + ppost-election litigation decision + ppost-election litigation legitimacy. Because the second and third terms are nonnegligible, pj is also nonnegligible. And thus, when pj is appropriately multiplied by B, there is some nonnegligible expected benefit to voting.38There are different ways of characterizing B, and it may have different components. For example, you could separate out of B different benefits—Bfavorable electoral decision, Belectoral legitimacy, Bsystemic legitimacy. Understood this way, in order to compute the total benefit B, you would have to multiply the component benefits with the respective probabilities.

We think that expected benefit could plausibly outweigh the costs of voting. At the very least, it robustly pushes the voters’ calculus in that direction. Some may have doubts. To that end, it behooves us to look at costs—C, in the equation above. Most take it for granted that there are some such costs associated with voting, even if it is just the effort to walk to a polling booth, the gas expended to drive to a polling booth, the time waiting in line, whatever.

One initial line of response is to ask whether these should actually count as marginal costs of voting that would not have been otherwise incurred. If, for example, the polling booth is on the way to work, there is literally no wait, and the person is already informed based on their personal interest in politics, the additional costs of filling out a ballot may be negligible. We could also imagine voting as simply something to do, filling the role of other miscellaneous events in one’s day. For example, and from personal experience, someone might take a break from work to go get coffee, without any particular desire for coffee—they simply want a walk and moment of respite. But on Election Tuesday, if voting can fit snugly in that break, then it is really costless, because the person was going to go on an excursion regardless. The cost-over-replacement of voting was nothing.

Of course, we recognize that voting isn’t so easy for everyone, and their jobs and personal obligations and the conditions of their polling places may not be so accommodating. Consequently, there may be real obdurate costs to voting that are not easily overcome. Here, we are still motivated to question this cost side of the ledger. Such conditions simply point to a prima facie obligation on the government to drive those costs down by, for example, creating more polling places, shortening wait times at polling places, bettering the conditions of voting places, expanding alternative ways of voting like vote by mail, making the act of voting easier with better machines, and promoting voter education.39In Harper v. Virginia State Bd. of Elections, the Supreme Court stated that “a State violates the Equal Protection Clause of the Fourteenth Amendment whenever it makes the affluence of the voter or payment of any fee an electoral standard. Voter qualifications have no relation to wealth nor to paying or not paying this or any other tax . . . The principle that denies the State the right to dilute a citizen’s vote on account of his economic status or other such factors by analogy bars a system which excludes those unable to pay a fee to vote or who fail to pay.” We think that this generalizes into the principle that the government cannot impose effective poll taxes by making voting needlessly and unreasonably expensive. See Crawford v. Marion Cty. Election Bd., 553 U.S. 181, 191 (2008) (“However slight that burden [on voting] may appear, as Harper demonstrates, it must be justified by relevant and legitimate state interests ‘sufficiently weighty to justify the limitation.’”). Many of these solutions are feasible with better technology. Most significantly, we imagine that a lot of the costs may be significantly cut if we can, say, make feasible voting from one’s mobile or personal devices, or perhaps simply embrace vote by mail even further.40Some scholars have suggested that the important question is not whether voters turn out, but whether the sample of those who do turn out is representative of the population.

One further point about Downs’s paradox concerns the criminalization of fraudulent voting. Federal law provides for rather severe penalties, even in the case of a singular, uncoordinated fraudulent vote—stiff fines and up to five years imprisonment. However, under the traditional view of the rationality of voting, these punishments are unjustified. If it is the case that such singular cases can have practically no decisional impact, singular cases have no great rational impact on legitimacy, and that voters are acting irrationally in casting their ballot at all, then punishing such singular cases of fraudulent voting makes no sense. But our intuition is that there is plausible reason to criminalize and punish voter fraud to some extent. All else being equal, voting, including fraudulent voting, may be rationally justified, and marginal votes, including fraudulent votes, can have decisional impact and an impact on legitimacy. Thus, we have reason to stop fraudulent voting. Without condoning the level of punishment, our account of the rationality of voting provides some reason to think that there is potential value in criminalizing and prosecuting fraudulent voting. In particular, we may wish to obtain some specific and general deterrence with respect to fraudulent voting.

IV.  Conclusion

We have argued that the paradox of voting is less puzzling than it has previously seemed. Even when the benefits of voting are limited to political outcomes, and the consumption value of voting is excluded, we posit that the benefits might outweigh the costs. This is because, contrary to the standard story, it is not only tiebreaking votes that affect outcomes. Rather, as modelled in the U.S. presidential elections of 2000 and 2020, litigation can determine the outcomes of close elections. And the winner’s pre-litigation vote margin, in turn, affects litigation. The more one candidate is winning by, the less likely the courts will overturn the pre-litigation result. This has knock-on effects for political legitimacy. When voters believe that litigation did or should have changed an election’s result, they may view the official winner as illegitimate, with all the possible ill effects that entails. But insofar as larger victories make people less likely to believe that courts, not voters, are picking winners, the bad effects of perceived illegitimacy are curtailed.

* * *

Peter N. Salib is a Climenko Fellow and Lecturer on Law at Harvard Law School. Starting in the autumn of 2021, he will be an Assistant Professor of Law at the University of Houston Law Center.

Guha Krishnamurthi is an Assistant Professor at South Texas College of Law.

The authors thank Susannah Barton Tobin, Colin Doyle, Greg Elinson, Christopher Havasy, Lael Weinberger, Saul Levmore, Louis Kaplow, and Rick Hasen for insightful comments and questions and Michael Hornzell for excellent research assistance.

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