I

William H. Riker (1920–93) was a ‘once-in-a-century man’.Footnote 1 Three important books, reviewed here, have recently been published in his long shadow.Footnote 2 His work was cited in leading social science journals 144 times in 2005 and 2006.Footnote 3 It has already been evaluated twice in this Journal. Footnote 4 In the first review, Albert Weale pointed out that Riker’s ‘[c]onstancy of method is matched by variability of political interpretation’.Footnote 5 He began as a friend of federalism and of a strong two-party system. In the 1970s and 1980s he apparently lost interest in federalism. In Liberalism against Populism (1982; the target of all three books under review in this article),Footnote 6 he argued forcefully that democracy did not and could not mean the will of the people, but should mean the opportunity to throw the rascals out, even if it means replacing them by another set of rascals. The paradoxes of social choice ruled out any more inclusive conception of democracy.

By the end of his life, Riker had returned to federalism and retreated from some of the more extreme claims he had made about the nature of democracy. His method was not entirely constant, either – by the end of his life it was less mechanical and more interpretative than, for instance, in his Positive Political Theory, a textbook written with Peter Ordeshook in 1973.Footnote 7

Robin Farquharson introduced his Theory of Voting with the following advice from the mathematician C. A. Coulson:

The use of applied mathematics in its relation to a physical problem involves three steps:

i) a dive from the world of reality into the world of mathematics;

ii) a swim in the world of mathematics;

iii) a climb from the world of mathematics back into the world of reality, carrying the prediction in our teeth.Footnote 8

‘The third step is indispensable’, Farquharson added drily. Riker took all three. Unlike most political scientists in the 1950s, he was always fascinated by the possibility of exact, mathematical reasoning about politics. His most eminent forebear was Duncan Black, who might have remained an unknown economics professor in North Wales but for Riker’s promotion of his work. He has had so many descendants that some who feel threatened by him started an insurgent movement called ‘Perestroika’ in the American Political Science Association. However, the Perestroikans’ criticisms are properly directed against those who dive and swim, but fail to climb back with predictions in their teeth. Furthermore, Perestroikan writings typically do not acknowledge that diving and swimming are difficult or important. They are both.

Riker’s ‘size principle’, contained in his still frequently cited Theory of Political Coalitions,Footnote 9 predicted that governing coalitions will contain just the minimum number of parties to form a legislative majority, so that the spoils of office will be distributed among as few parties as possible. The size principle apparently sits awkwardly with Riker’s other work on federalism, history and heresthetics.Footnote 10 The one resolutely ignores ideology; the other revels in it. Riker used the chaos theorems of Norman Schofield and others to conclude that politics was in chronic disequilibrium, from which any policy outcome might follow. Herestheticians were the few people who could turn this chaos and disequilibrium to advantage by imposing a new structure on politics. The supreme heresthetician of American politics was Abraham Lincoln. The victory of Lincoln, with less than 40 per cent of the popular vote, in the presidential election of 1860 was important and improbable. So was the very existence of the United States, due to the highly improbable ratification of the Constitution by twelve of the thirteen original states – all bar Rhode Island. These were big events surrounded by big ideologies: federalism; democracy; human rights; states’ rights; territorial expansion; slavery.

However, both the size principle and Riker’s later work on heresthetics were about building, destroying and rebuilding governing coalitions. He is interested in ideology not so much for its importance as for its capacity to generate paradoxical and improbable political coalitions. Not so much, ‘Is slavery morally wrong?’ as ‘How did the compromise between slavers and anti-slavers come into existence in 1787, and why did it last until 1860?’ His detached quizzical approach seriously annoys Gerry Mackie.

The following sections of this article consider which parts of the Riker programme survive unscathed and which are fatally damaged by current assaults on them.

II

Norman Schofield has for some years brought the theorems of social choice, including his own, into mainstream comparative government and used them to illuminate extraordinary events. In the series of papers that culminates in Architects of Political Change he takes them into British and American history. He thereby makes a heresthetic turn in political history.

In the 1970s, Schofield and the late Richard McKelvey proved the ‘chaos theorems’ that may be read as a generalization of Arrow’s impossibility theorem of 1951. Arrow proved that no system of aggregating from individuals’ preferences to a social ordering could simultaneously satisfy five minimal conditions of fairness and rationality. In particular, any system that satisfies the conditions of universal domain (U), the weak Pareto principle (P), independence of irrelevant alternatives (I), and non-dictatorship (D) fails transitivity.Footnote 11 Such a system may report, for some {a, b, c}, the cyclical social ordering a > b > c > a, where the symbol > is to be read as ‘ranks socially ahead of’.

If opinion is single-peaked, no cycling arises, and therefore any ordering rule that satisfies U, P, I and D will give a transitive ordering. When there are two or more dimensions, however, cycling is always possible. Schofield and McKelvey showed that, if all individual rank orderings are equally likely, then the probability of cycling increases rapidly with the number of options and the number of voters. And there is no theoretical bound to where a cycle may go. In multidimensional space with a large number of heterogeneous voters, there is likely to be a global cycle – that is, for any {x, y}, there is a sequence of majority votes such that x >, …, > y >, …, > x.Footnote 12

This inspired Riker’s late work and all three of these responses to it. The first reaction was, ‘Why so much stability?’Footnote 13 Political outcomes do not cycle all over the place. In some polities, most notably the US Congress, we observe a stable pattern of distributional and ideological politics that is very rarely upset. Poole and Rosenthal report that on only ‘two occasions – both before the Civil War – stable voting patterns break down entirely, and “chaos” results’.Footnote 14 The two chaotic occasions are the 1820s and the 1850s, when an alignment based on slavery overtook the earlier economic interest-based alignments in the years leading up to the Civil War, splitting the party of Jefferson and Jackson into its northern and southern wings.

Riker was interested in herestheticians – those few politicians who can manipulate the inherent instability of multidimensional politics to produce a surprising outcome. His stories involve turning-points in American political history. Some involve well-known politicians such as James Madison and Abraham Lincoln; others, now-forgotten figures such as Gouverneur Morris (who is, however, enjoying a minor revival),Footnote 15 David Wilmot and Chauncey DePew. All relate to the great cross-cutting issue in American politics – race.

In Architects, Schofield first defines a constitutional quandary as ‘a situation of great uncertainty in the electorate’ (p. 16) – uncertainty about fundamental constitutional or existential matters. Can our state survive? Is it time to break away and form a new state – or to prevent other people from doing so? Those who can frame such decisions are the ‘architects of political change’ of Schofield’s title – a subset of Riker’s herestheticians. Brooding over them all, in the cover picture, is Abraham Lincoln. Schofield discusses the founding of the United States, from the Quebec Act 1774 to the victory of Thomas Jefferson in the election of 1800. He also considers Lincoln and the origins of the Civil War; Lyndon B. Johnson and the critical alignment of 1964; and J. M. Keynes and the ‘Atlantic Constitution’, i.e., ‘the set of political, economic and social beliefs common to … the polities on the Atlantic littoral’.Footnote 16

According to the Declaration of Independence, the British had abolished:

The Quebec Act (14 Geo. III, c. 83) extended the territory of ‘Quebec’ (modern Quebec and southern Ontario, as won by the British from France after Wolfe’s fatal victory at the Heights of Abraham in 1759). The new boundary ran through Niagara Falls, round the western edge of Pennsylvania, and down the Ohio River valley to its confluence with the Mississippi, where the enlarged ‘Quebec’ met the Hudson’s Bay Territory, another British possession. The ‘arbitrary government’ was the restoration of French civil law and recognition of Roman Catholicism in this enlarged ‘Quebec’.

The Quebec Act is surprising. Why should the British put extra territory under the laws, customs and religion of the nation they had just defeated? According to Schofield, it was a device to keep the Anglophone colonists of the thirteen mutinous colonies in their place – a place limited by the Appalachians and the Ohio River to the west, and Spanish and French territory to the south. With the Ohio in British hands and New Orleans in French or Spanish, land west of the Appalachians would be of little value to the colonists (because their trade could not freely move up or down river, and there were no roads). Land west of the Ohio, in which George Washington and Benjamin Franklin had freely traded, would be entirely lost to them.

Although Schofield does not make the connection, the Quebec Act bears the stamp of Adam Smith, whom his friends described as ‘very zealous in American Affairs’ at this time. A confidant of several British government ministers, Smith thought that the colonists were free-riding on defence of their western frontier against the French and the Native Americans, provided for them at no cost by the British Army. He may have suggested the Quebec Act to one of these confidants, Lord Shelburne. I cannot prove that, but it is consistent with his startling advice to another, Alexander Wedderburn: Britain should ‘restore Canada to France and the two Floridas to Spain; we should render our colonies the natural enemies of those two monarchies and consequently the natural allies of Great Britain’.Footnote 18

The Quebec Act threatened all the colonists’ hopes. But how could they challenge the world’s leading military power? Was not the Declaration of Independence a hopelessly risky venture? Not if the colonists were promised external support. The only possible supporter was France. Schofield therefore believes that Benjamin Franklin received a secret signal before 4 July 1776 that the French would come to America’s aid, as they first did by shipping arms to the West Indies in charge of P-A. de Beaumarchais, the author of Figaro and French government spy, in the fall of that year.Footnote 19

Perhaps; but the evidence is mixed. I have found two possible signals before 4 July 1776. One, mentioned by Schofield, is that in December 1775 the Committee of Secret Correspondence, including Franklin, engaged C. W. F. Dumas as their agent in the Netherlands, to sound out in secret the governments of Europe on whether they would support the rebellion against Britain. In May 1776 Dumas reported, partly in cipher (for which the key was explained in the same letter, referring to books that Dumas had sent earlier and therefore resistant to interception), that the French government would ship arms to the West Indies. Also, and not mentioned by Schofield, the French government had sent a spy, de Bonvouloir, to Philadelphia in 1775 with the message that France would trade with them if they became independent. But there are problems with both ‘signals’. The Bonvouloir signal was noisy, as the colonists could not be sure that he was who he said he was. The Dumas signal is not known to have reached Philadelphia until late September 1776. The fact that Dumas’s letter of May itself contains the cipher to be used in secret communications proves that there was no earlier letter.Footnote 20

Schofield can be accused of reasoning: ‘The odds against American success in a war were so overwhelming that no rational person would have taken them on; the delegates to the Continental Congress were rational; therefore they must have received a secret signal of French help’. The signal is inferred from Congress’s behaviour. Perhaps, though, Franklin and a few other delegates, like Winston Churchill in 1940, just believed in the chances of winning against all the objective evidence. That may have sufficed to induce the belief cascade that Schofield plausibly sees as a tipping mechanism leading to 4 July.

Another controversial claim comes in the next chapter. Previous work has investigated whether Condorcet’s work was known in America, and whether it played any role in the design of the Constitution.Footnote 21 Schofield seizes on a phrase of James Madison’s in Federalist No. 10. Arguing that an ‘extended republic’ is the best place in which to try out representative government, Madison says: ‘If the proportion of fit characters be not less in the large than in the small republic, the former will present a greater option, and consequently a greater probability of a fit choice.’Footnote 22 Madison sounds Condorcetian. So does his co-author Alexander Hamilton, extolling the Electoral College in Federalist No. 68: ‘This process of election affords a moral certainty that the office of President will seldom fall to the lot of any man who is not in an eminent degree endowed with the requisite qualifications’.Footnote 23 For Condorcet, the task of constitution-writers was to design institutions that would maximize the probability of coming to the correct decision. Did either Madison or Hamilton get the idea from him?

Schofield thinks that Madison did. But Madison probably saw Condorcet’s voting theory only after writing the Federalist; and he disapproved of Condorcet’s unicameralism. He read it in 1788, when Filippo Mazzei vainly asked him to translate it. The Schofield window could have been open between April 1787, when Madison wrote ‘Vices of the Political System of the United States’ as an aide-memoire to the Virginia delegation at the Constitutional Convention, and November 1787, when he started writing his numbers of The Federalist. However, none of Jefferson’s letters from Paris that reached Madison in this period, nor any of Madison’s acknowledgements discusses either probability or voting theory. True, Madison’s letters to Mazzei and Jefferson in September–October 1787 acknowledge receipt of work by Condorcet and the Encyclopedists. But this reduces the window for the Schofield hypothesis to about two months.Footnote 24

Although much of Federalist No. 10 is in ‘Vices of the Political System’, the section on ‘the probability of a fit choice’ is not. ‘Vices’ ends with the ‘desideratum’ of ‘such a process of elections as will most certainly extract from the mass of the Society the purest and noblest characters which it contains’.Footnote 25 Schofield’s alternative hypothesis is that, in between the two versions of his argument, Madison learnt about Condorcet’s probabilism from Franklin, who would have heard about it in Paris between 1776 and 1785. Armed with probabilistic reasoning, he was able to show that the proportion of fit characters in the legislature would be higher in a large country than in a small one. However, neither Schofield nor I can produce evidence in support of this.

Unlike Riker, Schofield does not think Abraham Lincoln forced Stephen Douglas into an impossible choice in their famous Debates of 1858. Nor does he believe in the cycle which Riker thought he had detected in the four-candidate presidential election of 1860. But that election was in two dimensions – and it is in two dimensions that herestheticians come out to play. Schofield’s Lincoln was a heresthetician, because of the two belief cascades that he triggered, beginning when he won the Republican nomination in 1860. One cascade was in the North, to the effect that Lincoln was more moderate than other potential Republican nominees, and therefore capable of winning the election. The other was in the South, where he was viewed as an implacable opponent of slavery, so that secession suddenly seemed attractive. As a result, the new alignment that even Poole and Rosenthal find in their congressional data for the 1850s was set in stone – and blood.Footnote 26 After Lincoln, Republicans hid the diversity of their coalition by ‘waving the bloody flag’ to remind all their supporters about the Civil War. When the Southern rebels regained the franchise in 1876, the Democratic party became the strange coalition of southern racists and agrarians from the rest of the country that endured until Franklin D. Roosevelt.

Schofield next reports a profound trivium. Take the notable presidential elections of 1896, 1960 and 2000: knowing which party carried a state in 1896 does not predict its choice in 1960; nor does 1960 predict its choice in 2000. However, If you predict that a Republican state in 1896 would be Democratic in 2000, and vice-versa, you get all but six states correct. Footnote 27 For this reversal to be possible, US national politics must have been truly two-dimensional all the time. The dimensions were economic and social. Social liberalism means support for civil rights (and therefore opposition to slavery), and the galaxy of other rights (gender rights, abortion rights and so on) that have joined civil rights since the 1960s. Social conservatism is the converse. The social liberal/conservative dimension has for two centuries included slavery, race and religion. Now we can see how states that supported (opposed) the bible-thumping socially conservative Kansas Democrat, William Jennings Bryan, in 1896,Footnote 28 supported (opposed) Republican George W. Bush in 2000. The Riker–Schofield account of American national politics is more parsimonious than the Poole–Rosenthal account. It explains everything they explain plus some things that they do not explain, such as this inversion from 1896 to 2000.

Chapters 7 and 8 of Schofield’s book involve a giddying tour d’horizon of twentieth-century mathematics and physics. J. M. Keynes believed that, whereas commodity markets operate under risk, asset markets operate under uncertainty. Under risk, an agent can calculate the odds of each of the possible outcomes and hedge so as to minimize risks. Under uncertainty, the set of possible outcomes is unknowable, because it depends on the interaction of all rational agents in the world in an N-person game, where N is the adult population of the world. Schofield traces Keynes’s thought from his Treatise on Probability (1921), where he makes this distinction, to the General Theory (1936), with its famous remark that the stock market was a sort of meta-beauty contest in which everyone tries to guess what everyone else thinks (everyone else thinks …) makes the most beautiful woman in the world. Finally, Keynes was involved in the creation of the ‘Bretton Woods institutions’ in 1944 – the World Bank, the International Monetary Fund and what has more recently become the World Trade Organization. In Schofield’s view, these organizations have failed to move the global economy from a world of uncertainty to a world of risk. If social scientists can be divided into prophets of equilibrium (neoclassical economists; Downsian political scientists; Poole and Rosenthal) and prophets of chaos (Keynes, Arrow, Riker), then Schofield is for chaos.

Architects is thus a model of heresthetical history. Like his mentor, though, Schofield sometimes overclaims ‘facts’ and it would be nice if they were true.

III

Gerry Mackie’s provocative book has already been carefully reviewed.Footnote 29 Jointly with another work, it won the American Political Science Association’s Gladys M. Kammerer Award 2004 for the best political science publication in the previous calendar year in the field of US national policy. It brought comfort and joy to the Perestroikans:

But a book should be neither damned nor praised by association. What matters is not who likes Mackie’s book but whether his arguments work. Some of them do. I deal with only two: the reasonability of Arrow’s Independence of Irrelevant Alternatives axiom and the status of Riker’s ‘canonical’ stories.

In Liberalism against Populism, there are ‘liberal’ and ‘populist’ democratic theorists. Populists say: ‘Find out the will of the people, and follow it.’ The classic populist was Rousseau, but he has many modern followers. The classic Rikerian ‘liberal’ is Schumpeter: ‘democracy does not mean and cannot mean that the people actually rule in any obvious sense of “people” and “rule”. Democracy means only that the people have the opportunity of accepting or refusing the men who are to rule them’.Footnote 31 Populism, says Riker, is incoherent. Arrow’s theorem proves that the ‘will of the people’ does not exist. Therefore, by elimination, only Schumpeterian liberalism is viable.

Riker’s argument commits the fallacy of the excluded middle. The failure of Rousseau does not of itself validate Schumpeter. Mackie’s attack does not take this easy line, however. He takes aim squarely at two of Arrow’s conditions: unrestricted domain and independence of irrelevant alternatives (U and I in the notation introduced earlier). Condition U is unnecessarily demanding, according to Mackie, because if there are no cycles, as, he says, there usually are not, the median voter theorem ensures that majority rule is well behaved. This argument occupies Chapters 5 and 9–15 of Mackie’s long book. Condition I is normatively unjustified and has been frequently (including by Arrow himself) confused with a different condition, namely expansion-consistency and contraction-consistency. This argument occupies Chapter 6, and I deal with it first.

Riker claims that Condition I ‘seems a fundamental requirement of consistency and fairness to prevent the rigging of elections and the unequal treatment of voters’.Footnote 32 Mackie insists that it is nothing of the sort. Many of the stories that are made up to justify Riker’s claim depend on confusing Condition I with a consistency condition known as IIA (RM) in the literature.Footnote 33 IIA (RM) insists that, for all {i, j, k}, if an alternative i wins in the set {i, j, k}, it must still win if the set is contracted to {i, j}. Also, if i beats j in the set {i, j}, it must still rank ahead of j if the choice set is expanded to {i, j, k}. Riker shows that a popular choice rule, the Borda count, violates IIA (RM). The Borda count is used in the Eurovision Song Contest and some US sporting awards. Each voter ranks each option, giving a points to the worst option and b for each interval (a ⩾ 0, b > 0). Adding up the scores gives a winner. If we can overlook the Borda count’s violation of IIA (RM), then maybe we avoid Arrow’s impasse.Footnote 34

However, that is too easy. The Borda count, characterized differently, does violate Condition I, and Condition I is a reasonable one to impose. When the Borda and Condorcet rules lead to different results, Mackie, like other scholars such as Michael Dummett and Donald Saari, seems to prefer the Borda rule on the ground that it takes intensity of preference into account.Footnote 35 But this is dangerous. It violates the economist’s rule that you cannot strictly measure interpersonal comparisons of utility from what people say or do. You have to work around the idea in a cumbersome way, such as ‘von Neumann–Morgenstern (VNM) cardinalization’. I say I like spaghetti sauce with garlic 100 times more than spaghetti sauce without garlic. My wife then says she prefers spaghetti sauce without garlic 1,000 times more than spaghetti sauce with garlic. No voting mechanism which gave her 1,001 votes to my 101 would be stable, or free from manipulation. The Borda rule is a bad surrogate for true revelation of intensity of preference. Even if one can measure intensity of preference via VNM means, Borda still does not measure it. And furthermore, precisely because it is the most open and straightforward of all voting rules, it is the most manipulable.

There is a more basic problem. A long time ago, Amartya Sen proved that ‘For any collective choice rule neutrality (N) implies independence of irrelevant alternatives’.Footnote 36 When we are talking about normatively desirable rules, we are almost always talking about collective choice rules, not social welfare functions. We have to choose a president or a champion figure skater. The ordering of other candidates is at best an interesting subsidiary, and for presidents it is not even that. Almost nobody in the literature has ever challenged Kenneth May’s axioms for choice rules: universal domain, anonymity, neutrality and positive responsiveness.Footnote 37 Neutrality demands that a procedure be neutral between options. If a choice rule prefers x over y, given the preferences of individual voters between x and y, and if the preference profile between w and z is identical to that between x and y, then a neutral rule must choose w over z. That seems a pretty basic normative requirement. Only very special circumstances (for instance, jury decision, where it may be judged less harmful to acquit a guilty suspect than to convict an innocent suspect) would warrant us in dropping neutrality. Majoritarianism implies the neutrality axiom among others. Well, if you like neutrality, you have a hard time arguing against I.

Mackie discusses the manipulability results of social choice in four places,Footnote 38 but he does not bring out their full force because they are damaging for his attack on Condition I. The manipulability results apply to all determinate social choice rules (i.e., not to lotteries or similar ways of making a choice). In 1975, Mark Satterthwaite proved that a non-manipulable determinate choice rule must respect transitivity and Arrow’s Conditions U, P and I. But then, by direct application of Arrow, it must violate Condition D – must, in other words, be dictatorial.Footnote 39 Condorcet rules satisfy U, P, I and D but violate transitivity. Therefore they are manipulable by herestheticians. Borda rules satisfy transitivity, U, P and D but violate I. Anybody can manipulate them. Ask the Greek and Turkish jurors in the Eurovision Song Contest. They do not have to be herestheticians to see how to do one another down. Therefore, Condition I really is a basic condition of fairness and non-manipulability.

In his two books about heresthetic, Riker produced a dozen stories about cycling in American politics, and Mackie sets out with gusto to demolish most of them.Footnote 40 Like other critics, I find this the most impressive part of the book.Footnote 41 His demolitions mostly succeed; and yet there are some rugged foundations sticking out among the rubble. None of Riker’s stories, as reinterpreted from the primary sources by Mackie, reveals an actual voting cycle: yet the main run of them reveals disequilibrium, confirming Riker’s original intuition. That main run is all about the second dimension in American politics.

It begins in the Constitutional Convention of 1787, in a story that Mackie does not discuss. On slavery, the delegates were divided three ways. The Deep South (Georgia and the Carolinas) wanted to continue importing slaves. Virginian delegates had enough slaves already. They did not want to get rid of those they had, but importing more would reduce their value. Northern states were against slaves. How then did the Constitution, as reported out from the Convention, contain an article allowing slave imports to continue until 1808 and one requiring fugitive slaves to be returned to the state from which they had escaped – positions preferred by only three to five of the twelve states at the Convention, which voted by states?Footnote 42 It was because of an explicit log-roll, attested by the assiduous James Madison and by the South Carolina delegates in their ratification assembly. In return for New England support for these two clauses, the Deep South offered support for a navigation act to protect the commercial interests of New England. Both were carried. Heresthetics, or what?Footnote 43

Two of Riker’s heresthetic moves, a century apart, forced the second dimension under the noses of Congress. They are the Wilmot Proviso (1846–47) and the Powell Amendment (1956–57). In 1846, President Polk (southern Democrat) proposed to bribe the Mexican army to make peace with the United States and hand over northern Mexico at the same time. David Wilmot (D–PA) proposed the addition:

The Wilmot Proviso would have destroyed the Missouri Compromise, whereby slave and free states were admitted to the Union in pairs, so that the Senate remained balanced between them and no anti-slavery constitutional amendment could be carried there.

In 1956 President Eisenhower (Republican) proposed a bill for federal aid to schools to the Democratic-controlled Congress. The leading African-American Representative, Adam Clayton Powell (D–NY), proposed an amendment to the appropriation bill to restrict the federal aid to unsegregated schools. In both the Wilmot and the Powell cases, the House voted to add the amendment to the bill; and the amended bill then failed. In both cases, there was strategic voting, wrongly characterized by Riker and (I believe) correctly by Mackie. In both cases, politics was seriously two-dimensional. That is what gave herestheticians their purchase for strategic voting.

Mackie demonstrates that the election of 1860 did not feature the cycle among three of the four candidates that Riker says it did. But under most electoral systems (including the Borda count that Mackie likes), not Abraham Lincoln but Stephen Douglas, who was also the Condorcet winner, would have become president in 1860.Footnote 45 How then did Lincoln win? The Electoral College worked as well as it could have done for him, and as badly as it could have done for Douglas. But the Electoral College was a known institution with known effects. Rikerian historiography explains how and why two more candidates entered in spite of these effects. The Constitutional Union party, whose presidential candidate was John Bell, was created, according to Mackie,Footnote 46 in the expectation that the Republicans would nominate the ‘extreme’ anti-slavery William Seward rather than the ‘moderate’ Lincoln. Bell was the first candidate in the field. Bell and Douglas would then slug it out for the median voter, leaving the extreme Republicans and splintered Southern Democrats on the fringe. If Bell did not win, he could at least expect to deny anybody a majority of electors and hence throw the election into the House of Representatives. This was a reasonable hope until the southern Democrats put forward a fourth candidate.

The Democrats had earlier changed the rules of their nominating convention, at Southern insistence, so that the nominee must win two thirds of the votes. In 1860 that was impossible, and the delegates from the Deep South left the deadlocked convention to nominate the sitting vice-president, John Breckinridge. Breckinridge, according to Riker and Mackie, was a Condorcet loser. In pairwise comparisons he would have lost to each of the other three candidates. So why did he stand?

An easy (and possibly true) explanation is that some politicians in the nascent Confederacy already anticipated secession; they wanted to make a gesture and, if that gesture led to the hated Lincoln becoming president, so much the better. But, on the latest calculations, Breckinridge’s intervention interacted with the constitutional rules for selecting a president in a dramatic and unpredictable way. If no candidate for president won an absolute majority of the votes in the Electoral College, then the top three candidates would go to the second stage, in which the House of Representatives would vote, with each state having one vote; again, an absolute majority of states is required. Lincoln was bound to be one of the top three; and the Poole–Rosenthal nominate program makes it possible to locate each member of the 36th House (and each senator, to whom the choice would go if the House was deadlocked) on slavery, and assign his vote to the nearest candidate. The results are truly chaotic. In the House, either Douglas would have won, or the House would have been deadlocked. Bell would have won almost no states with sincere voting, but might have been elected president with strategic voting. The single representative from Oregon might well have been in a position to dictate who became the president of the United States. The Senate fall-back makes prediction even tougher.Footnote 47

Thus Mackie the historian is quite convincing, but his demolition of Riker is not as complete as he thinks. The idea that politics may be multidimensional, and thus give space for herestheticians to operate, survives. Mackie the normative theorist is less convincing. He seems at times to aim at proving Riker to be some kind of fraud. Angered that Riker justifies just one (Schumpeterian) conception of democracy, Mackie wishes to argue for a broader conception. But his willingness to pooh-pooh Condition I is hard to square with normative democratic theory. As Sen pointed out, you can – indeed must – reject I if you are a utilitarian, but not if you are a democrat.Footnote 48

IV

We have seen that in a sense the US Constitution caused the Civil War. Would a different constitution, perhaps one with fewer veto groups and more proportional representation, have held the country together? Anthony McGann, like Mackie, disputes Riker’s claim that cycling makes strong claims to democracy meaningless. His own strong claim is that ‘Political equality logically implies … proportional representation at the electoral stage, and simple majority rule without checks and balances at the … parliamentary stage.’Footnote 49 Furthermore, deliberation cannot be expected to lead to consensus.

This is bold. ‘Seat allocation rules are used to choose representatives, who subsequently make social decisions.’Footnote 50 This may seem to beg the question. The proponent of first-past-the-post need only say, ‘Oh no, they’re not.’ Fortunately, McGann has a second argument for proportional representation (PR): namely, that equality between voters entails and is entailed by anonymity and neutrality, as defined by May in 1952 for binary choice.Footnote 51 A series of May-like theorems then proves that only pure list PR satisfies these criteria for non-binary choice. Later in the book, McGann specifies that he means closed-list PR.Footnote 52 This seems perverse – it puts elected politicians at the mercy not only of their constituents but of their party whips. McGann here loses sight of the citizen. The fundamental argument for open-list systems is that they put the citizen, not the politician, in control. It is true, as McGann says, that open-list can help to cause factionalism and worse.Footnote 53 In the Italian First Republic, the Mafia used it to check that each of its clients had actually voted the requested ticket. But this only helps to show that the best voting system is one that citizens control as far as possible, and politicians control as little as possible. Additional Member and Single Transferable Vote fit that description, but McGann has little to say about them.

McGann, unlike Mackie, believes that binary independence of irrelevant alternatives is a reasonable condition to impose on voting rules, because a rule that violates binary independence is open to flagrant manipulation by introducing new alternatives. He goes on to argue that a legislature (itself selected by PR) should decide by simple majority rule, because requiring qualified majorities violates neutrality. This comes at a price. Majority rule among more than two alternatives violates transitivity and can produce cyclical outcomes. But McGann argues that this is good: rather have losers expect to overturn the outcome than have them resort to secession or civil war. He contrasts his position to Riker’s; but actually Riker the historian celebrates this very possibility. It is Riker the democratic theorist, the butt of McGann’s as well as Mackie’s attacks, who says that disequilibrium invalidates all strong theories of democracy.

McGann attacks the ‘tyranny of the supermajority’. If each citizen is in the majority for about the same proportion of the time, it is elementary to show (as McGann does) that simple majority rule protects minority rights better than any system of qualified majority (supermajority) rule. But the hard case is that of permanent minorities. American history shows many examples. The (misnamed) Madisonian system of checks and balances, including supermajorities, failed to protect slaves. However, the supermajoritarian and anti-majoritarian provisions in the original US Constitution protected southern whites and, when (culminating in Lincoln’s election) they saw those protections being swept away, they felt that the only option was Breckinridge, followed by secession. Likewise, a Northern Ireland Catholic might doubt whether majority rule outlaws the tyranny of the majority.

But that is the true Madisonian message. McGann is absolutely right to insist that the true Madison is the author of Federalist No. 10, whereas the author of Federalist No. 51 is a strategist propounding arguments in which he does not believe himself. The purpose of publishing eighty-five arguments for ratifying the Constitution in the New York newspapers was to persuade sceptical New Yorkers to vote that way.Footnote 54 No. 10 (reworked in a hurry from Madison’s earlier ‘Vices’) states that ‘an extended republic’ is the best antidote to the tyranny of the majority. It is No. 51 which recommends both horizontal and vertical checks and balances – for example, those states’ rights that would have prevented a legislature dominated by Virginia and Pennsylvania from imposing their will on the middle-sized New York State.

Madison’s argument in favour of an extended republic began life in an argument about religion between the close friends David Hume and Adam Smith. They both wanted to preclude the sort of religious despotism that had terrorized Scotland until the generation before their birth. Hume’s remedy is a state church, where the state monitors dangerous theology. Smith’s remedy is to let society be ‘divided into two or three hundred, or perhaps into as many thousand small sects, of which no one could be considerable enough to disturb the publick tranquillity’.Footnote 55 Madison adapted Smith’s argument, first in the Virginia legislature, then in his notes for the Virginia delegation to the Constitutional Convention (‘Vices’), and finally in Federalist No. 10, to apply to both religious and political factions. He piloted through Congress what became the ‘Establishment Clause’ of the First Amendment to the US Constitution: ‘Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof.’

Smith and Madison may have been right to suppose that in an extended republic no religious faction would dominate the whole (in which case the Establishment Clause would have been unnecessary). But it is a stretch to believe the same of Madison’s more political concerns. The division of society into debtors and creditors, which worried him, is binary. One of the two is likely to be dominant. So some minority protection as recommended in Federalist No. 51 is probably desirable even in an extended republic. But any such constitutional guarantee is anti-majoritarian. It would take more than a simple majority of both houses of Congress to repeal the First Amendment. No doubt some Congresses wanted to – for instance during the anti-French panic of 1798, or the McCarthyite panic of the early 1950s. McGann is sceptical about supermajoritarian rules in constitutions as a form of rights protection. But he is caught in a paradox. He wants to protect PR for legislative elections. The Irish Constitution guarantees the Single Transferable Vote (STV), which is a sort of PR. Twice, the hegemonic party in Ireland, Fianna Fáil, has tried to repeal that clause in its partisan interest. Twice, it has failed. I assume McGann is glad that STV is sufficiently embedded in the Irish Constitution to be immune from majoritarian tinkering.

McGann next takes a pot-shot at deliberative democrats. The claim that deliberation provides a way out of Arrow’s theorem is vacuous. Nothing a priori says that deliberation will make cycles less likely to happen. It may equally make them more likely to happen. If and when deliberation fails to result in consensus, there has to be a vote; and all the paradoxes of social choice apply with undimmed force. However, McGann makes some of the deliberative democrats sound more naïve than they really are. Dryzek and his co-authors argue that deliberation may induce metaconsensus. If individuals come to agree on the dimension along which they array their positions, even if they continue to disagree violently on the subject in hand, the chance of single-peakedness increases. There is a small empirical literature, some of it unfortunately not yet printed, on whether deliberation induces single-peakedness. List et al. found that, on a remote issue (utility regulation options in Texas), it did. On a salient issue (the republic referendum in Australia), it changed the structure of opinion. If everyone in Australia had attended a deliberative poll, they might have voted for a republic.Footnote 56

McGann’s pin-up countries are the ‘consensual democracies’ of Western Europe. These have been widely studied since the Dutch political scientist Arend Lijphart drew attention to their kinder, gentler properties.Footnote 57 However, McGann argues, Lijphartian consensualism needs to be split into its two dimensions. The first dimension – the degree of proportionality in the electoral system – does not, he argues, lead to Tsebelian ‘veto points’,Footnote 58 because parties in a majority coalition are not really veto players. The second dimension (federal/unitary), however, does create true veto players. When there is more than one elected chamber (say, a chamber of the people and a chamber of the territories), a supermajority is required to enact legislation; likewise when lower-tier bodies, such as states or provinces, have concurrent powers with the centre. McGann’s set of consensual democracies without supermajority rule is therefore Austria, Belgium (that poster-boy for consensual democracy), Ireland and the Nordic countries.

If McGann’s argument is right, there must be something especially kind and gentle about the consensual democracies. What is that? On the one hand, they are redistributive and respectful of minority rights (though Belgium achieves that only by being all but split into three countries, a situation McGann does not discuss); on the other hand, they are not over-stable, and therefore do not arbitrarily protect some minorities at the expense of other minorities, or of the majority.

Stability is not an unambiguously good thing.Footnote 59 A regime may be too stable in at least two different ways. One is supermajoritarianism; but another is protected by the kinder gentler proportionality that Lijphart and now McGann celebrate. Recent work by two eminent pairs of political economists contrasts the stability of PR regimes with the decisiveness of plurality (miscalled majoritarian) regimes such as those of the ‘Westminster model’.Footnote 60 On the one hand, plurality regimes spend less and have smaller budget deficits – presumably because it is easier for them to take tough decisions, as McGann notes in passing.Footnote 61 On the other, PR regimes have a larger welfare state and are more redistributive. Plurality regimes are generally run by the centre-right, and PR regimes by the centre-left. Which is better? Contra McGann, the political scientist cannot say, because any answer makes a normative trade-off between budget discipline and social security.

McGann’s conclusions seem to be in tension, but it is a Rikerian tension. He puts them in small capitals:

1. political equality implies majority rule and proportional representation …

2. majority rule is the decision rule offering most protection to minorities …

7. cycling is essential to the understanding of democracy.

Conclusion 7 is slightly surprising in the light of the others, but I think it is true.

V

Most fashions in political science fail to outlive their proponents (think of systems theory or structural-functionalism). Like W. S. Gilbert’s Ko-Ko, most of us also have a little list of current fashions that would not be missed. The second review of Riker’s work in this Journal queried whether the Riker agenda had a future.Footnote 62 The work reviewed here shows that it does. Even when Mackie and McGann disagree with Riker, his work is the reference point without which theirs cannot be understood.

What these second-generation studies have done is to separate the essential from the inessential Riker. Riker the normative theorist has faded. Riker the analytic historian and peerless student of rhetoric and heresthetic survives. His normative claims did not follow from his analysis. His historical claims do not all survive Mackie’s onslaught. But some of them do. Both his claims and his method inspire Schofield and Mackie to go and write heresthetical histories of their own.Footnote 63

None of this would be possible if politics was always single-dimensional. This is where Schofield’s comparison of the 1896, 1960 and 2000 US presidential elections is so telling. As Riker insisted, the second dimension of US national politics has always been there. It is embedded in the very Constitution. It caused the Civil War. It is perfectly possible that it will generate a further spin such that the Democrats become the party of ‘cosmopolitanism’ (social liberalism combined with free-market economics) and economic liberalism in the redistributive New Deal sense) and the Republicans the party of both social and economic conservatism. Both of these pairs are uncomfortable if you prefer to have a united party.Footnote 64