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Why Not Eat an Eclair?David Runciman
Vol. 30 No. 19 · 9 October 2008

Why Not Eat an Eclair?

David Runciman

6039 words
Free Riding 
by Richard Tuck.
Harvard, 223 pp., £22.95, June 2008, 978 0 674 02834 0
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Why would anyone vote for Barack Obama? Not why would anyone want to see Obama elected president rather than John McCain (or Hillary Clinton for that matter), but why would anyone who desired that outcome think that his or her individual vote could make the slightest difference in helping to bring it about? General elections are never decided by a single vote, so no one’s vote is ever going to be missed. If you want Obama to win, and plan to vote for him, but you forget, or find yourself otherwise detained, don’t worry – the final result will be unaffected by your failure to show up, even if you happen to live in a swing state like Ohio or Florida. If Obama is winning the state, he will do perfectly well without you; if he is losing, there is nothing you can do to help him get over the line, because the winning line will always be further away than your paltry individual vote. Either way, you are not needed, so why bother to vote at all?

This is a question that has haunted the study of politics for the past fifty years or more. Because political science has been dominated by economics, the problem is often put in terms of costs and benefits. Going to vote is not a costless exercise, since it takes time and effort that could be spent doing other things. You might not consider this much of an outlay, but if you reflect that the benefit you can expect to derive is precisely zero, since your contribution is literally worthless, then it starts to look like a serious waste of your precious resources. One possible way round this problem is to argue that although no election has ever been decided by a single vote, that doesn’t mean it couldn’t happen. The effort of dragging yourself to the polling station is a small price to pay for insuring yourself against the monumental regret you would feel if you stayed at home the one day your vote was really needed. The prospect of waking up to find yourself the person who cost Obama the presidency is so ghastly it’s worth hedging against that possibility, however remote.

It’s a sign of the tangle that political science has got itself into over the problem of voting that this solution was ever seriously canvassed. Not only is it psychologically implausible – is this how you think, even subliminally, when you leave the house on election day? – but it also flies against the evidence of what happens when elections are really close. The assumption is that the tighter an election, the nearer one gets to the holy grail of a contest that could be decided by your personal contribution; in fact, the tighter the election, the less likely it is that any one person’s vote is going to settle it. Just think about what happens during recounts in a parliamentary constituency at a British election: the count doesn’t go on until the correct result is achieved, but until one side gives up in exhaustion or despair. The figures produced by a recount almost never tally with those given first time round, so if the margin were a single vote that would in fact be a reason to count again in order to produce a bigger margin one way or the other. The narrowest margin in modern times is two votes, achieved by Mark Oaten for the Liberal Democrats at Winchester in 1997; his Tory opponent managed to get the result declared void by the courts on the grounds that some of the ballot papers had not been properly stamped. Oaten won the rematch with a majority of more than 20,000. Or think about what happened in Florida in 2000. Does anyone know what the true count was in that election? Perhaps there really was a single vote in it. But the enduring image of squint-eyed counters peering forlornly at hanging chads is enough to confirm that if there was a single vote in it, no one was ever going to find it. When an election is as tight as the presidential contest in 2000, the individual votes that might decide it disappear in a miasma of political confrontation and confusion.

In the end the 2000 presidential election was settled by a single-vote majority, the five-four decision of the Supreme Court to confirm Bush’s victory in Florida. The votes of small bodies often do come down to the choices of individuals. But no one would mistake the Supreme Court’s judgment for a democratic decision; it was a collective judgment of a very different kind. Moreover, because the justices voted along party lines, it is hard to say which one of the five actually swung it (though any of them could have swung it by voting the other way). To know that your vote was the crucial one, what you really need to do is switch sides at the last minute. For example, the second reading of what became the 1832 Reform Act was passed in the House of Commons on 23 March 1831 by a single vote. As Boyd Hilton writes in A Mad, Bad and Dangerous People: ‘The decisive vote was that of John Calcraft, who . . . finding his own seat threatened with disfranchisement, spoke violently against the Reform Bill, but changed his mind at the last moment, and voted for the second reading. He killed himself six months later, correctly imagining himself to be hated by both sides equally.’ Most people, I guess, would be willing to expend considerable effort to ensure they didn’t find themselves in that position.

Perhaps the problem here is that economic political science is too used to thinking about voting in terms of what’s in it for the individual – the assumption being that people put something into the voting system in the hope of getting something out. What if we imagine voters as selfless beings who do not begrudge the time and effort of casting a ballot and simply want to do their bit for democracy? Don’t these people have a reason to participate? Unfortunately, the answer still looks like ‘no’, because the selflessness of one’s motives has no impact on the usefulness of one’s contribution to group activities on this scale. This was a point made by Mancur Olson in 1965 in The Logic of Collective Action, which is often seen as initiating the widespread acceptance of the problem of ‘free riding’ for large group activities, the free rider being the person who, seeing that his individual contribution doesn’t make any real difference to a collective endeavour, decides to withhold it and simply surf off the wave created by the other group members (the problem being that if everyone did that, there would be no group). Olson wrote that the futility of the ‘imperceptible’ contribution holds true ‘whether behaviour is selfish or unselfish’, something he illustrated as follows:

A man who tried to hold back a flood with a pail would probably be considered more of a crank than a saint, even by those he was trying to help. It is no doubt possible infinitesimally to lower the level of a river in flood with a pail . . . but . . . the effect is imperceptible, and those who sacrifice themselves in the interest of imperceptible improvements may not even receive the praise normally due selfless behaviour.

The real-world example Olson gave was farmers in the American grain market. If a selfless farmer, worried about the suffering of his colleagues because of depressed prices, decided to lower his own production levels in order to raise prices, it would be a pointless gesture, because overall price levels would be unaffected by a single decision of this kind. He would simply lose money, and look like a fool. The selfless farmer would be much better off exercising his philanthropy in a way that actually produced a perceptible effect on someone, by some direct act of charity. Olson went on to hint that in a first-past-the-post electoral system similar problems are bound to affect the motivations of individual voters, who will recognise that ‘if their party is going to win, it will as likely win without them.’ He did not push this point, and appears to have felt that voting might be a special case that needed further consideration. However, as Richard Tuck points out in his fascinating new book about the strange hold that the free rider problem has had on political science ever since Olson, such misgivings have not stopped many of his followers from treating voting as the paradigmatic case of the problem of the worthlessness of individual contributions to the actions of large groups.

But if we go back to Olson’s example of the man with the pail, it’s immediately clear that there is something different about voting. A single person trying to hold back a flood with a bucket is going to look and feel extremely odd and ultimately foolish. But if on election day you find yourself the only person at the polling station, though you might feel odd and somewhat uncomfortable, you will not conclude that what you are doing is pointless. Indeed, if you really are the only person who turns up to vote, then your vote will decide the election. Moreover, while seeing someone trying to bail out a river with a bucket gives no one else an incentive to join in, seeing one lonely voter at the polling station does give other people an incentive to take part, if only to prevent that individual from deciding for everyone else. That’s why (unless there has been a boycott) you don’t see elections at which the number of voters drops away to nothing. The closer it gets to zero, the more reason there is to vote, which means it will never get close to zero at all. But there is still a puzzle here: if it is worth voting in order to prevent the number of participants falling so low that a single vote can decide an election, then the worth of an individual vote is measured by how much it ultimately diminishes the worth of an individual vote. Your vote counts only in so far as it guarantees that your vote doesn’t count. So you might still find yourself asking, why bother?

Tuck has an answer to this question that entirely recasts the terms of the free rider problem, by treating voting as a paradigmatic instance of a collective activity in which individual contributions do count. For Tuck, what is distinctive about voting is that success is in the end defined in numerical terms: you always, and only, need to get more votes than the other side to win, so there is a cut-off point at which victory is certain. Tuck calls this a ‘threshold’, and he points out that at the threshold, one vote settles it. This is how elections worked in ancient Rome, where a roll-call of voters would be taken in sequence, until one candidate had enough votes to be guaranteed victory. In this process, the last person to vote for a candidate causes that candidate to be elected. So the crucial fact about voting is that one vote can indeed make all the difference.

But this hardly seems to resolve the problem of the worthlessness of individual contributions, since it only addresses the ultimate worth of the vote of the final person to be called. What about everyone else, both all those who voted before the lucky threshold-crosser and those who would have voted after, but ended up not being needed? What is distinctive about Tuck’s account is that he is able to extrapolate from the value of the vote that tips the balance a value for the other votes as well. It is not just the last vote that counts, because the last vote would not count without all the individual votes that preceded it. So every vote that contributes to a threshold being reached plays a full and equal part in getting a candidate elected. As Tuck says, this understanding fits with common sense and with common law. If it takes six people to lift a lifeboat and only five are on the beach, you don’t say that the sixth person to show up is responsible for lifting the boat on his own. Equally, if it takes six people to lift a boat in order to steal it, the law says that all six are fully responsible for the crime, as though each had committed it on his own. If it takes ten million people to elect a candidate, then all ten million are responsible for the election, even though it only takes one of them to tip the balance.

However, if ten million people are needed to elect a candidate, and fifteen million turn out to vote for that candidate, what about the other five million? Do their votes count for anything? One way of reading the example of Roman voting by roll-call is to imagine that the extra votes are there to guarantee victory. If you need a certain number of votes to win, it would be crazy to send exactly that number of voters to the election in case one of them has an accident along the way (or, worse still, does a Calcraft and switches sides at the last moment). But the fact that other voters are there to step into the breach in case the threshold-crosser goes missing raises a new problem because it suggests that the threshold-crosser was not needed after all – someone else would have tipped the balance instead. Is it possible to believe that you are the cause of a particular result if the same result would have occurred without you? Many philosophers are wedded to a counterfactual conception of causation, which insists that for any given cause there must be an effect that no other cause could have produced, but Tuck shows that there is no reason to think like this. Again, a legal example shows why. If two police officers have their guns trained on an armed man during a bank robbery, and one of them shoots the man dead, we do not think that this officer was not the cause of the robber’s death because if his shot had missed, the second officer would have fired and killed him. You can cause something to happen without being the only person who could bring it about. So your vote can be the cause of a candidate’s victory even if the candidate would have won without you.

Tuck’s argument does imply that once a threshold has been reached, the extra votes don’t actually count for anything. In the Roman case, there might have been some resentment among people who had waited around all day in the hot sun only to be told their contribution was now redundant. This is one obvious advantage of the modern electoral system, which counts votes anonymously and more or less simultaneously. No one can say for sure whether his or her vote was in the necessary set that crossed the threshold or in the redundant set that was simply there as insurance. Voters have an incentive to vote precisely so that they can think of their votes as having been part of the group that tipped the balance. Though Tuck does not mention it, this might explain the growing popularity of postal voting, which is normally justified in terms of convenience (something that seems odd, as so often with voting, since the margins are so small); in fact, there is an obvious attraction in voting early if you want to think of your contribution as laying the foundations for a big enough pile.

What Tuck does say is that seeing voting in these terms helps explain a phenomenon that has otherwise baffled political scientists: the ‘bandwagon effect’, where the more popular candidates pick up great chunks of extra votes as soon as they look like they are going to win. If voting was about trying to make the crucial difference with your individual vote, then the more popular candidates became, the less reason there would be to vote for them. But if voting is about trying to belong to the group that makes the overall difference, then your chances of being in that group increase as the candidate looks more and more certain to win. This fits with the empirical evidence: when a contest is no contest at all, because one side is likely to get almost all the votes, then turn-out falls, because the vast majority of voters fall into the redundant group whose votes weren’t really needed. But turn-out doesn’t necessarily rise the tighter a contest gets. A close contest increases the chances of your vote being decisive, but it also increases the chances that your candidate might lose (at which point your vote will have lost all its causal efficacy). A more one-sided contest increases the chances that your vote will be redundant, but also increases the chances that it will have played a part in actually getting someone elected. Why vote for Obama? Well, one possible answer is that there is a pretty good chance that a vote for Obama might actually cause him to be elected, and that you can wake up the next day with a warm glow. It is part of the genius (or cynicism, depending on your perspective) of Obama’s rhetoric to have tapped into this – you are the change you’ve been waiting for, and voting for me will prove it.

But why vote for Obama in a place like Utah, where you know your vote isn’t going to make any difference because the Republicans have a lock on the state? Tuck hasn’t got much to say about voting for obviously losing candidates, though he accepts that the motivations here maybe complicated. You might simply want to send a signal that nothing and no one in politics should be taken for granted – that there is at least one Communist in Kensington. But Tuck is rightly suspicious of a general trend that seeks to reduce all democratic politics to gestures of this sort, and to understand voting as an ‘expressive’ act in which what matters is not who wins but that you have had an opportunity to feel good about yourself by making your mark. If democracy is not about who wins then it’s not about anything. One reason to vote for a candidate who is going to lose is to send a practical signal that there is something to build on for the future. If you go down to the beach and find only one other person there ready to lift the lifeboat, you’ve still got a reason to stick around. When word gets out that two people showed up, it makes it more likely that others will show up next time, and one day you might get to six. One reason to vote for Obama in Utah is to make it clear that the Democrats could win the state one day if enough people did the same.

Tuck’s threshold argument is compelling but it skirts around a significant fact about real-world elections that I highlighted earlier: although in theory it only requires one vote to take someone over the top, in practice, the closer you get to that threshold the harder it is to find it, as the mist of political enmity descends. One way to address this is provided by Tuck’s account of other cases where the threshold seems to disappear the closer you get to it. This happens when the borderline between two states of affairs is unavoidably vague, even though the process of change is cumulative. Baldness is a classic example. I go bald by losing my hair one strand at a time, but the loss of no one strand is enough in itself to move me from the category of non-bald to bald. So if I consider the loss of my hair on a strand by strand basis, I can’t go bald, not even if I lose it all. The same kind of reasoning can also apply the other way, say to fatness. No single éclair is ever going to make me fat, so I might as well eat this one. But if no single éclair will ever make me fat then, having eaten one yesterday, I might as well eat another one today, and so on, until I become the thing that one éclair at a time isn’t supposed to make me: fat. These are known as ‘sorites’ paradoxes (the ‘sorites’ being a ‘heap’ of the kind that ought never to arise if you add to it one grain of wheat at a time). It is not easy to say how they should be resolved. But Tuck shows that the best way to think about these puzzles is to consider them as not that different from the problem of voting.

With voting, we know that there is a threshold which marks off the winner from the loser. In the case of baldness, we will never find that threshold, because the closer we look, the fuzzier things get; in that sense, baldness is something that can be appreciated only from a distance. But that doesn’t mean we shouldn’t treat baldness as though it were marked out by a threshold, while accepting that we can never discover it for sure. This, Tuck shows, is consistent both with the logic of the problem and with our instincts. The logic is complicated, but our instincts are pretty clear. If a ship’s captain wants to know a safe distance at which to sail past some rocks, he needs to assume that there is some point at which it becomes unsafe. He can also draw some fixed lines around what is and is not safe – two miles out is clearly safe, two yards out is not. What he cannot do is say where the cut-off point is, which means that each time he sails past the rocks he may make a slightly different judgment about how close to go (just as we can make slightly different judgments about whether or not someone is bald, depending on our mood). The captain will want to err on the safe side, but his judgment of where the safe side is will vary. The fact that he can’t pinpoint the threshold doesn’t make it rational to act as though there were no threshold – if he does that, the boat will hit the rocks. And if this is true for baldness and sailors then it is true for collective action as well. Even if you can’t find the threshold – because it’s a question of the size of the crowd, or the weight of public opinion – it still makes sense to act as though it was there somewhere. Why join the march or sign the petition? Because your contribution is crucial to making sure the threshold is passed, even if the judgment of whether it has been passed will vary from day to day.

Real-world elections are more like sailing past dangerous rocks than Tuck allows. There is a threshold there, but the judgment of whether or not it has been passed can vary from election to election, even if the actual number of votes cast is the same. It might, for example, depend on who is sitting on the Supreme Court. Tuck notes that the phrase ‘the straw that broke the camel’s back’ is often used as though it were an illustration of the sorites paradox, when in fact it is a clear case of a threshold event. If we had a sensitive enough weighing machine, and a docile enough camel, we could find the straw that finally tipped the balance. Likewise, if we had enough time and patience (and a relatively foolproof voting method) we could count every vote cast in an election and discover the true result. But we don’t have those things, and in any election of any significance we never will, because there is too much at stake. With the bald man, the camel and the election, we have to act as though there were a threshold even if we can never be sure exactly where it is. But the crucial point is that it makes sense to treat them all as threshold events, and therefore as events in which individual contributions really do count. Tuck thinks we can assume the rationality of collective action given the existence of thresholds, and work from there. In making this claim, he is reversing the central thrust of political science since Olson, which started from the irrationality of the man with the bucket in the flood.

The difference between these two perspectives is enormous. For Tuck, voting makes sense in terms of the contributions of individual voters, and so is something that it makes sense for individuals to do. In a way, we ought to know this already, just as we ought to know that if we eat too many éclairs we will get fat. Of course we can tell ourselves the story about no one éclair making the difference in order to give ourselves an excuse for indulgence, but it would be hard to find anyone who thinks that this sort of self-justification is rational behaviour. Something similar applies to procrastination. You can always tell yourself that it doesn’t matter if you put off a piece of work until tomorrow, but do you really think that this is a rational way to carry on, given the obvious risks, and given the crucial importance of erring on the side of caution? Yet a generation of political thinkers have committed themselves to the view that this is the rational way to think about participation in group activities, and that therefore co-operation is inherently irrational. For Tuck, the question that follows is why anyone would think like this.

In the second half of Free Riding, he provides a compelling answer by locating the free rider problem in its historical context. He shows that the assumption that it really is a problem is much more recent than we might think. Contemporary economists and political scientists often take it for granted that anyone who views human beings as essentially self-interested creatures will inevitably conclude that groups are vulnerable to the defection of individual members who see that they can free ride off the contributions of others. But Tuck shows that before the 20th century philosophers who saw human behaviour in self-interested terms did not conclude this at all. Instead, they took it for granted that individuals will have good reason to co-operate in most circumstances, because it is obvious that the benefits of the group for the individual depend on the contribution of the individual to the group. This was the view, for instance, of David Hume, Adam Smith, Jeremy Bentham and all the utilitarians who followed them. Of course, they did not think that the rationality of collective behaviour meant that this was the way human beings were bound to behave, because they knew that people were often deeply irrational, preferring short-term interests to long-term ones, and immediate gratifications to sensible courses of action (hence the tendency of many human beings to get fat). Equally, they did not assume that the rationality of co-operative behaviour meant that such behaviour was always a good thing for society as a whole. It very much depended on what the group was for: just because it makes sense to collaborate in an act of organised crime does not mean that the victims of the crime should welcome this. But the classical utilitarians would have thought it very odd to imagine that co-operative behaviour was irrational simply because its consequences were not always desirable, or because individuals could tell themselves self-serving stories in an attempt to talk themselves out of it. It is rational to co-operate because your individual contribution causes the benefit you hope to get from the collective endeavour. Tuck shows that you don’t need to overlay this basic truth with additional moral strictures about the need to do your duty by other people – large-scale group activity does not depend on the Kantian imperative to do as you would be done by. If it did, it would be much more precarious than it in fact is.

That co-operative activity is not especially precarious was also something that struck the late 19th and early 20th-century founders of modern economics. Alfred Marshall, Francis Edgeworth and Vilfredo Pareto all recognised that ‘combinations’ of both capitalists and workers – ranging from cartels on the one side to trade unions on the other – were the likeliest outcome of the struggle for competitive advantage. Large-scale groups could regulate the market in ways that individuals could not, which was a reason for individuals to belong to large groups. It was only from the 1930s onwards that the opposite conclusion was drawn from the same set of facts: if individual contributions were negligible in relation to large-scale collective endeavours, it made sense for individuals to free ride off the collective, since their contribution would not be missed. This became the modern doctrine of perfect competition, which states that because small-scale producers can make no impression on the market price by raising or lowering their own level of production, no ‘combination’ of such producers to fix the price will hold together in the long run. Individual producers will always be able to free ride off the group (i.e. by producing more than an agreed limit in order to maximise profits), and when enough individuals do that, the group will collapse. Other things being equal, cartels and trade unions should disintegrate under the pressure of the rational expectations of their individual members. The classic example of perfect competition was the same one that Olson had in mind: farmers in the American grain market.

Tuck draws two broad conclusions from this shift in the understanding of the logic of collective action. The first is that as the idea of free riding escaped from economics and started to colonise political science after the Second World War, it initiated a steady erosion of confidence in the efficacy of democratic participation, which is now reflected in ever declining levels of voter turn-out. This is not entirely plausible. Voter turn-out has declined most strikingly in the past couple of decades, which doesn’t really coincide with a general permeation of free rider arguments in the public consciousness (even if you believe that academic ideas take a while to filter through). It’s more likely that the decline in turn-out has coincided with an increasing scepticism on the part of the public about what their governments are for, and what they are capable of achieving, in the face of international markets. This would fit with Tuck’s overall argument: we instinctively know that our individual votes are worth something, but we are starting to have some doubts about what general elections can achieve, given the saminess of the candidates. In these circumstances, it might make sense to stay at home, notwithstanding the fact that your vote is needed. Obama v. McCain might do something to reverse this trend, but I doubt it will do much.

Tuck’s other claim is more persuasive. He argues that the move to the idea of perfect competition, and hence to the free rider problem, was a product of its time. In the 1930s global politics was dominated by defection from international agreements and general double-dealing. At the same time, Western capitalists, whose entire system was tottering, were hungry for some intellectual ballast to set against the claims of the planned socialist economies of the East.

The doctrine of perfect competition provided what looked like a good reason to think that capitalism was more flexible than its critics had claimed, and as the planned economies eventually revealed their own internal contradictions, the doctrine stuck. But this conception of perfect competition also required a certain historical forgetfulness. At the turn of the 20th century, the US government attempted to confront the monopolies and cartels that dominated American economic life. It was widely assumed that anti-trust legislation was needed precisely because without strong government action rational behaviour would inevitably tend towards large-scale attempts to fix the market. This was certainly the view of late 19th-century economists like Edgeworth, who saw, as Tuck puts it, that ‘by itself, competition would simply degenerate into a war of combinations, and political intervention of some kind would be required to secure a utilitarian outcome from market behaviour.’ But the success of trust-busting in the early 20th century made it easy to forget that lesson, and to assume that government was simply ironing out some of the irrationalities of the market and restoring it to its natural function. If it is assumed that monopolistic or oligopolistic behaviour is inherently irrational, then government intervention becomes little more than a steadying paternal hand on the shoulder. But if it is assumed that co-operative action is rational, and defection is the anomaly, then modern competitive economies will need to operate against a permanent background of threatened state intervention, through which collective actions of certain kinds will be prevented or punished. This is Tuck’s view, and he concludes that, seen in proper historical perspective, the modern idea of perfect competition is a ‘fantasy’.

Free Riding is not an easy book, and it is written in a close academic style that may put off some readers. But it is emphatically not just an academic treatment of politics. If anything, it is the reverse: a defence of politics against the prevalent academic view that reduces it to a faintly absurd, non-natural, marketised activity. Tuck’s account of politics is probably closest to that of Rousseau, who saw political life as a quintessentially collective endeavour, in which the claims of the state as a vehicle of human co-operation had to be asserted against the claims of other more partial groups, which would otherwise distort our co-operative impulses to their own ends. Rousseau would also have thought that the idea that it was possible to rely on the free riding of individuals to confront the power of corporate interests was a fantasy. There is, after all, something incoherent about a view that sees free riders as both the product of group activity and the means of taking on malign groups. If individual contributions really are negligible, then it will need more than individual contributions to bring down the monopolists: it will take the state, which will have to rely on those contributions to achieve its purposes.

Tuck doesn’t try to suggest that politics is an easy business, any more than Rousseau does. It’s a battle against the irrationality and short-termism of individuals as well as against the entrenched power of corporations. Nor does he think it is immune from free rider difficulties round the edges: the threshold argument gives people good reasons to get co-operative projects off the ground, but when they are up and running there will always be some who think the threshold is secure, and bail out. Universal compliance in the achievement of collective goods is as much a fantasy as perfect competition. Nevertheless, Tuck puts this problem where it belongs, at the fringes of political activity, and not where it has been for a generation or more, at the centre. The central questions of contemporary politics are, as they have always been, when, where and how to use the power of the state to achieve the best outcomes for the individuals who give states their power. If international markets are preventing states from achieving these outcomes, that is not a reason to give in to the market. It is a reason to find some new ways of doing politics. If you want to know why it’s worth bothering to vote, this book has an answer: it’s because your vote is what counts, and in the end politics is also what counts.

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Vol. 30 No. 21 · 6 November 2008

David Runciman states that one problem facing the individual deciding whether or not to vote is the fact that the number of votes at which the threshold of victory is reached is impossible to determine, so as a voter I never know whether I made a difference or not, just as in sorites paradoxes I appear never to be able to make a heap of sand by adding a single grain at a time, even though if I carry on adding, I will eventually have a heap (LRB, 9 October). He solves it by saying that, as there is a threshold, it makes sense to act as if my particular vote will be the one that passes it, as the threshold can be passed if everyone behaves in that way. He supports this argument with a number of analogies: the loss of one hair at a time will eventually make me bald; eating single éclairs will make me fat. The problem with using analogies to make arguments is that while they must be similar to the case being made, they must also be different. Here, the achieving of negative outcomes is being used to illustrate collective efforts to achieve positive outcomes, and repeated single actions (eating éclairs) are treated as analogous to the single actions of large groups.

I believe I vote not because ‘the closer you get to that threshold the harder it is to find it,’ but because the threshold is known with complete accuracy, but only after the event. At the time, I can only make an educated guess as to how many voters will turn out for the other party and how many for mine, and part of that guess is based on analysing my own willingness or otherwise to go out to vote and on the assumption that ‘people like me’ are doing the same as I am.

Alex Fox

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