The Beginning of Infinity

by David Deutsch

  In Germany (formerly West Germany) between 1949 and 1998, the Free Democratic Party (FDP) was the third largest.* Though it never received more than 12.8 per cent of the vote, and usually much less, the country’s proportional-representation system gave it power that was insensitive to changes in the voters’ opinions. On several occasions it chose which of the two largest parties would govern, twice changing sides and three times choosing to put the less popular of the two (as measured by votes) into power. The FDP’s leader was usually made a cabinet minister as part of the coalition deal, with the result that for the last twenty-nine years of that period Germany had only two weeks without an FDP foreign minister. In 1998, when the FDP was pushed into fourth place by the Green Party, it was immediately ousted from government, and the Greens assumed the mantle of kingmakers. And they took charge of the Foreign Ministry as well. This disproportionate power that proportional representation gives the third-largest party is an embarrassing feature of a system whose whole raison d’être, and supposed moral justification, is to allocate political influence proportionately.

  Arrow’s theorem applies not only to collective decision-making but also to individuals, as follows. Consider a single, rational person faced with a choice between several options. If the decision requires thought, then each option must be associated with an explanation – at least a tentative one – for why it might be the best. To choose an option is to choose its explanation. So how does one decide which explanation to adopt?

  Common sense says that one ‘weighs’ them – or weighs the evidence that their arguments present. This is an ancient metaphor. Statues of Justice have carried scales since antiquity. More recently, inductivism has cast scientific thinking in the same mould, saying that scientific theories are chosen, justified and believed – and somehow even formed in the first place – according to the ‘weight of evidence’ in their favour.

  Consider that supposed weighing process. Each piece of evidence, including each feeling, prejudice, value, axiom, argument and so on, depending on what ‘weight’ it had in that person’s mind, would contribute that amount to that person’s ‘preferences’ between various explanations. Hence for the purposes of Arrow’s theorem each piece of evidence can be regarded as an ‘individual’ participating in the decision-making process, where the person as a whole would be the ‘group’.

  Now, the process that adjudicates between the different explanations would have to satisfy certain constraints if it were to be rational. For instance, if, having decided that one option was the best, the person received further evidence that gave additional weight to that option, then the person’s overall preference would still have to be for that option – and so on. Arrow’s theorem says that those requirements are inconsistent with each other, and so seems to imply that all decision-making – all thinking – must be irrational. Unless, perhaps, one of the internal agents is a dictator, empowered to override the combined opinions of all the other agents. But this is an infinite regress: how does the ‘dictator’ itself choose between rival explanations about which other agents it would be best to override?

  There is something very wrong with that entire conventional model of decision-making, both within single minds and for groups as assumed in social-choice theory. It conceives of decision-making as a process of selecting from existing options according to a fixed formula (such as an apportionment rule or electoral system). But in fact that is what happens only at the end of decision-making – the phase that does not require creative thought. In terms of Edison’s metaphor, the model refers only to the perspiration phase, without realizing that decision-making is problem-solving, and that without the inspiration phase nothing is ever solved and there is nothing to choose between. At the heart of decision-making is the creation of new options and the abandonment or modification of existing ones.

  To choose an option, rationally, is to choose the associated explanation. Therefore, rational decision-making consists not of weighing evidence but of explaining it, in the course of explaining the world. One judges arguments as explanations, not justifications, and one does this creatively, using conjecture, tempered by every kind of criticism. It is in the nature of good explanations – being hard to vary – that there is only one of them. Having created it, one is no longer tempted by the alternatives. They have been not outweighed, but out-argued, refuted and abandoned. During the course of a creative process, one is not struggling to distinguish between countless different explanations of nearly equal merit; typically, one is struggling to create even one good explanation, and, having succeeded, one is glad to be rid of the rest.

  Another misconception to which the idea of decision-making by weighing sometimes leads is that problems can be solved by weighing – in particular, that disputes between advocates of rival explanations can be resolved by creating a weighted average of their proposals. But the fact is that a good explanation, being hard to vary at all without losing its explanatory power, is hard to mix with a rival explanation: something halfway between them is usually worse than either of them separately. Mixing two explanations to create a better explanation requires an additional act of creativity. That is why good explanations are discrete – separated from each other by bad explanations – and why, when choosing between explanations, we are faced with discrete options.

  In complex decisions, the creative phase is often followed by a mechanical, perspiration phase in which one ties down details of the explanation that are not yet hard to vary but can be made so by non-creative means. For example, an architect whose client asks how tall a skyscraper can be built, given certain constraints, does not just calculate that number from a formula. The decision-making process may end with such a calculation, but it begins creatively, with ideas for how the client’s priorities and constraints might best be met by a new design. And, before that, the clients had to decide – creatively – what those priorities and constraints should be. At the beginning of that process they would not have been aware of all the preferences that they would end up presenting to architects. Similarly, a voter may look through lists of the various parties’ policies, and may even assign each issue a ‘weight’ to represent its importance; but one can do that only after one has thought about one’s political philosophy, and has explained to one’s own satisfaction how important that makes the various issues, what policies the various parties are likely to adopt in regard to those issues, and so on.

  The type of ‘decision’ considered in social-choice theory is choosing from options that are known and fixed, according to preferences that are known, fixed and consistent. The quintessential example is a voter’s choice, in the polling booth, not of which candidate to prefer but of which box to check. As I have explained, this is a grossly inadequate, and inaccurate, model of human decision-making. In reality, the voter is choosing between explanations, not checkboxes, and, while very few voters choose to affect the checkboxes themselves, by running for office, all rational voters create their own explanation for which checkbox they personally should choose.

  So it is not true that decision-making necessarily suffers from those crude irrationalities – not because there is anything wrong with Arrow’s theorem or any of the other no-go theorems, but because social-choice theory is itself based on false assumptions about what thinking and deciding consist of. It is Zeno’s mistake. It is mistaking an abstract process that it has named decision-making for the real-life process of the same name.

  Similarly, what is called a ‘dictator’ in Arrow’s theorem is not necessarily a dictator in the ordinary sense of the word. It is simply any agent to whom the society’s decision-making rules assign the sole right to make a particular decision regardless of the preferences of anyone else. Thus, every law that requires an individual’s consent for something – such as the law against rape, or against involuntary surgery – establishes a ‘dictatorship’ in the technical sense used in Arrow’s theorem. Everyone is a dictator over their own body. The law against theft establishes a dictatorsh
ip over one’s own possessions. A free election is, by definition, one in which every voter is a dictator over their own ballot paper. Arrow’s theorem itself assumes that all the participants are in sole control of their contributions to the decision-making process. More generally, the most important conditions for rational decision-making – such as freedom of thought and of speech, tolerance of dissent, and the self-determination of individuals – all require ‘dictatorships’ in Arrow’s mathematical sense. It is understandable that he chose that term. But it has nothing to do with the kind of dictatorship that has secret police who come for you in the middle of the night if you criticize them.

  Virtually all commentators have responded to these paradoxes and no-go theorems in a mistaken and rather revealing way: they regret them. This illustrates the confusion to which I am referring. They wish that these theorems of pure mathematics were false. If only mathematics permitted it, they complain, we human beings could set up a just society that makes its decisions rationally. But, faced with the impossibility of that, there is nothing left for us to do but to decide which injustices and irrationalities we like best, and to enshrine them in law. As Webster wrote, of the apportionment problem, ‘That which cannot be done perfectly must be done in a manner as near perfection as can be. If exactness cannot, from the nature of things, be attained, then the nearest practicable approach to exactness ought to be made.’

  But what sort of ‘perfection’ is a logical contradiction? A logical contradiction is nonsense. The truth is simpler: if your conception of justice conflicts with the demands of logic or rationality then it is unjust. If your conception of rationality conflicts with a mathematical theorem (or, in this case, with many theorems) then your conception of rationality is irrational. To stick stubbornly to logically impossible values not only guarantees failure in the narrow sense that one can never meet them, it also forces one to reject optimism (‘every evil is due to lack of knowledge’), and so deprives one of the means to make progress. Wishing for something that is logically impossible is a sign that there is something better to wish for. Moreover, if my conjecture in Chapter 8 is true, an impossible wish is ultimately uninteresting as well.

  We need something better to wish for. Something that is not incompatible with logic, reason or progress. We have already encountered it. It is the basic condition for a political system to be capable of making sustained progress: Popper’s criterion that the system facilitate the removal of bad policies and bad governments without violence. That entails abandoning ‘who should rule?’ as a criterion for judging political systems. The entire controversy about apportionment rules and all other issues in social-choice theory has traditionally been framed by all concerned in terms of ‘who should rule?’: what is the right number of seats for each state, or for each political party? What does the group – presumed entitled to rule over its subgroups and individuals – ‘want’, and what institutions will get it what it ‘wants’?

  So let us reconsider collective decision-making in terms of Popper’s criterion instead. Instead of wondering earnestly which of the self-evident yet mutually inconsistent criteria of fairness, representativeness and so on are the most self-evident, so that they can be entrenched, we judge such criteria, along with all other actual or proposed political institutions, according to how well they promote the removal of bad rulers and bad policies. To do this, they must embody traditions of peaceful, critical discussion – of rulers, policies and the political institutions themselves.

  In this view, any interpretation of the democratic process as merely a way of consulting the people to find out who should rule or what policies to implement misses the point of what is happening. An election does not play the same role in a rational society as consulting an oracle or a priest, or obeying orders from the king, did in earlier societies. The essence of democratic decision-making is not the choice made by the system at elections, but the ideas created between elections. And elections are merely one of the many institutions whose function is to allow such ideas to be created, tested, modified and rejected. The voters are not a fount of wisdom from which the right policies can be empirically ‘derived’. They are attempting, fallibly, to explain the world and thereby to improve it. They are, both individually and collectively, seeking the truth – or should be, if they are rational. And there is an objective truth of the matter. Problems are soluble. Society is not a zero-sum game: the civilization of the Enlightenment did not get where it is today by cleverly sharing out the wealth, votes or anything else that was in dispute when it began. It got here by creating ex nihilo. In particular, what voters are doing in elections is not synthesizing a decision of a superhuman being, ‘Society’. They are choosing which experiments are to be attempted next, and (principally) which are to be abandoned because there is no longer a good explanation for why they are best. The politicians, and their policies, are those experiments.

  When one uses no-go theorems such as Arrow’s to model real decision-making, one has to assume – quite unrealistically – that none of the decision-makers in the group is able to persuade the others to modify their preferences, or to create new preferences that are easier to agree on. The realistic case is that neither the preferences nor the options need be the same at the end of a decision-making process as they were at the beginning.

  Why don’t they just . . . fix social-choice theory by including creative processes such as explanation and persuasion in its mathematical model of decision-making? Because it is not known how to model a creative process. Such a model would be a creative process: an AI.

  The conditions of ‘fairness’ as conceived in the various social-choice problems are misconceptions analogous to empiricism: they are all about the input to the decision-making process – who participates, and how their opinions are integrated to form the ‘preference of the group’. A rational analysis must concentrate instead on how the rules and institutions contribute to the removal of bad policies and rulers, and to the creation of new options.

  Sometimes such an analysis does endorse one of the traditional requirements, at least in part. For instance, it is indeed important that no member of the group be privileged or deprived of representation. But this is not so that all members can contribute to the answer. It is because such discrimination entrenches in the system a preference among their potential criticisms. It does not make sense to include everyone’s favoured policies, or parts of them, in the new decision; what is necessary for progress is to exclude ideas that fail to survive criticism, and to prevent their entrenchment, and to promote the creation of new ideas.

  Proportional representation is often defended on the grounds that it leads to coalition governments and compromise policies. But compromises – amalgams of the policies of the contributors – have an undeservedly high reputation. Though they are certainly better than immediate violence, they are generally, as I have explained, bad policies. If a policy is no one’s idea of what will work, then why should it work? But that is not the worst of it. The key defect of compromise policies is that when one of them is implemented and fails, no one learns anything because no one ever agreed with it. Thus compromise policies shield the underlying explanations which do at least seem good to some faction from being criticized and abandoned.

  The system used to elect members of the legislatures of most countries in the British political tradition is that each district (or ‘constituency’) in the country is entitled to one seat in the legislature, and that seat goes to the candidate with the largest number of votes in that district. This is called the plurality voting system (‘plurality’ meaning ‘largest number of votes’) – often called the ‘first-past-the-post’ system, because there is no prize for any runner-up, and no second round of voting (both of which feature in other electoral systems for the sake of increasing the proportionality of the outcomes). Plurality voting typically ‘over-represents’ the two largest parties, compared with the proportion of votes they receive. Moreover, it is not guaranteed to avoid the populatio
n paradox, and is even capable of bringing one party to power when another has received far more votes in total.

  These features are often cited as arguments against plurality voting and in favour of a more proportional system – either literal proportional representation or other schemes such as transferable-vote systems and run-off systems which have the effect of making the representation of voters in the legislature more proportional. However, under Popper’s criterion, that is all insignificant in comparison with the greater effectiveness of plurality voting at removing bad governments and policies.

  Let me trace the mechanism of that advantage more explicitly. Following a plurality-voting election, the usual outcome is that the party with the largest total number of votes has an overall majority in the legislature, and therefore takes sole charge. All the losing parties are removed entirely from power. This is rare under proportional representation, because some of the parties in the old coalition are usually needed in the new one. Consequently, the logic of plurality is that politicians and political parties have little chance of gaining any share in power unless they can persuade a substantial proportion of the population to vote for them. That gives all parties the incentive to find better explanations, or at least to convince more people of their existing ones, for if they fail they will be relegated to powerlessness at the next election.