POLITICAL STUDIES: 2005 VOL 53, 457–476 Duverger’s Law, Penrose’s Power Index and the Unity of the UK Iain McLean Alistair McMillan Dennis Leech Nuffield College Nuffield College University of Warwick As predicted by Duverger’s Law, the UK has had two-party competition for long periods in most electoral districts. However, there are different patterns of two-party competition in different dis- tricts and more than two effective parties in the Commons. Since 1874, parliament has always contained parties wishing to modify the Union and contesting seats only outside England. By cal- culating the Penrose power index for all parties in the House of Commons for all general elections since 1874, we identify when such parties were pivotal. We explain various legislative changes (for example the Crofters Act 1886, the first three Irish Home Rule Bills, the Parliament Act 1911) and non-changes (for example the failure to enact female suffrage before 1914) by reference to the Penrose index scores. The scores also explain how and why policy towards Scotland, Wales and Northern Ireland changed and did not change in the 1970s. A common misunderstanding of Duverger’s Law is that it produces two-party com- petition in legislatures using a plurality electoral system such as the lower houses of Canada, India and the UK. Because political historians have failed to charac- terise the UK party system correctly, they have, in general, failed to notice that Duverger’s Law is compatible with a multiparty legislature, in which parties from outlying parts of the Union may be pivotal. The best index of their pivotality is that which was proposed by Lionel Penrose in 1946. This article uses Penrose’s index to identify the occasions since 1874 on which the outlying parties have had the power to modify the Union or impose other policy priorities. The first and second sections of the article deal with Duverger and Penrose; the third and fourth deal with the power of the periphery since 1874. The conclusion examines the impli- cations of our findings and suggests further research. Duverger’s Law According to Duverger’s Law, ‘The simple-majority single-ballot system favours the two-party system’ (Duverger, 1954, p. 217). Duverger’s Law contains both mechanical and rational calculation components. The mechanical component is the responsiveness (Cox and Katz, 2002, pp. 32–7) of the plurality electoral system. An electoral system may be viewed as a mapping from votes to seats. Let the vote share of parties i and j be denoted by vi and vj and their seat shares by si and sj. Then the mapping may be written as: a si Ê vi ˆ = + e sj Ë v j ¯ © Political Studies Association, 2005. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA 458 IAIN McLEAN, ALISTAIR McMILLAN AND DENNIS LEECH where e is an error term, and the exponent a denotes the responsiveness of the system. If a = 1, the mapping is proportional representation. If a < 1, the system is relatively unresponsive. If a > 1, the system is relatively responsive. The plurality electoral system is highly responsive as between the two leading parties in nationwide, multidistrict competition. Early work (Parker Smith, 1910; Kendall and Stuart, 1950) assumed a = 3 for general elections to the UK House of Commons. Accordingly, Kendall and Stuart called the relationship the ‘cube law’. However, for some nineteenth-century general elections, a > 3 (McLean, 2001, pp. 87–112). For late twentieth and twenty-first-century general elections, 1 < a < 3 (Curtice and Steed, 1986). This high responsiveness usually guarantees that the largest single party wins more than half of the seats in the House of Commons and forms a single-party admin- istration. Of the 43 UK general elections since 1832 inclusive, 35 (81.40 percent) have resulted in a single party winning more than half of the seats. For a single party to win more than half of the vote is much rarer. In the 33 UK general elec- tions since 1874, the leading party has gained more than half of the vote only six times (that is, 18 percent of the time).1 Four of those six cases were in the nine- teenth century when there were still numerous unopposed returns, especially in Ireland. Only in 1931 and 1935 did the winning party indisputably win more than half of the votes. The joint probability that the largest party will win more than half of the seats and not win more than half of the vote in a UK general election exceeds 0.5. Usually, therefore, only two parties are capable of forming a UK government. Except between 1918 and 1924, no informed citizen has had any reason to doubt which two. By the rational calculation component of Duverger’s Law, agents realise that it is futile for parties to run and for electors to support any candidate except one of the top two in each district. However, this is a statement about what is ratio- nal at district level. Each elector should rationally ask Which parties are in contention in my district? not Which parties are in contention to form the next government of the UK? Since 1983, for instance, the pattern of party competition in England has featured two main sets of districts: those where the two leading parties are Labour and the Conservatives and those where the two leading parties are the Conservatives and the Liberal Democrats. In a few parts of urban England, there are Labour–Liberal Democrat contests. In Scotland and Wales, four parties are in contention: the three main UK-wide parties and the local separatist/nationalist party, the Scottish National Party (SNP) and Plaid Cymru (PC). In any one district, fewer than four parties are in contention, but the patterns of competition are such that in each country, all four have a reasonable chance. The Conservative Party won one seat in Scotland in 2001. Table 1 displays these patterns. The lower section of Table 1 lists the leading patterns of Duvergerian competition. There are 11 possible combinations of first and second parties.2 However, the table shows that five combinations account for 649 (98.5 percent) of the total contests in 2001: Labour–Conservative (414 seats), Conservative–Liberal Democrat (102 seats), Labour–Nationalist (58 seats), Labour–Liberal Democrat (57 seats) and the 18 seats in Northern Ireland (NI), uncontested by any Great Britain parties. In NI, there are two clearly delineated communities, of unionists and nationalists. Within DUVERGER, PENROSE AND UK UNITY 459 Table 1: Patterns of First and Second Parties’ Competition in the UK at the 2001 General Election Second Party Winning Party Con Lab Lib Dem Nationalist Other Total Con 0 106 58 1 1 166 Lab 308 0 49 54 2 413 Lib Dem 44 8 0 0 0 52 Nationalist 4 4 1 0 0 9 Other 0 1 0 0 18 19 Total 356 119 108 55 21 659 Battlegrounds Number of Seats % of Seats Con/Lab 414 62.8 Con/Lib Dem 102 15.5 Lab/Nat 58 8.8 Lab/Lib Dem 57 8.7 NI 18 2.7 All other patterns 10 1.5 Total 659 100 Notes: ‘Nationalist’ includes seats for both the Scottish and Welsh nationalists. All the Northern Ireland (NI) parties are listed as ‘Other’. Con stands for Conservative; Lab stands for Labour, and Lib Dem stands for Liberal Democrat. each community, there are moderate and extreme parties. The (Northern) Irish Unionists were allied with the Conservatives between 1886 and 1972. The UK parties which wish to weaken or dissolve the Union, are the SNP PC and the main NI nationalist parties, the Social Democratic and Labour party and Sinn Fein (SF). The Ulster Unionist Party and the Democratic Unionist Party wish to strengthen it. The operation of Duverger’s Law at constituency level therefore gen- erates multiparty competition at national level, with parties that wish to amend the Union being more likely to be pivotal than minor UK-wide parties. This situation is not new. The SNP and PC have been in contention since the Parliament of 1966–70 when each of them won a by-election. The Irish Unionists have been identifiable as a separate group since 1886; the Irish nationalists since 1832, albeit with a break between 1852 and 1874. Before 1918, Ireland elected 105 members of parliament (MPs). Where the local pattern of two-party competition differs from the standard English pattern, Duverger’s Law may fail to deliver the responsive and exaggerative properties for which it is famous. It may also produce one hegemonic party and 460 IAIN McLEAN, ALISTAIR McMILLAN AND DENNIS LEECH Table 2: The Effective Number of Parties in the House of Commons, 1874–2001 1874–1918 Mean 2.60 Median 2.52 Maximum 3.00 (1886) Minimum 2.22 (1900) 1922–2001 Mean 2.17 Median 2.14 Maximum 3.07 (1923) Minimum 1.27 (1931) 2 Source: Our calculations, using Laakso and Taagepera’s (1979) formula N = 1/ Spi where pi = the ith party’s propor- tion of seats. fragmented opposition parties, each with strength in only some regions of the country, as has frequently happened in Canada and India. A standard measure shows that the effective number in the House of Commons has always exceeded two except in the one-sided cases of 1931 and 1935. On average, Table 2 shows that the Effective Number (EN) score exceeded 2.5 when the whole of Ireland was in the UK, dropping to just over 2.1 on average in the subsequent period.