¹ Browne, Eric C., ‘Testing Theories of Coalition Formation in the European Context’, Comparative Political Studies, III (1971), 391–410CrossRefGoogle Scholar; de Swaan, Abram, Coalition Theories and Cabinet Formation (Amsterdam: Elsevier, 1973)Google Scholar; Taylor, Michael and Laver, Michael, ‘Government Coalitions in Western Europe’, European Journal of Political Science, I (1973), 205–48.CrossRefGoogle Scholar

² Browne, Eric C., Coalition Theories: A Logical and Empirical Critique (Beverly Hills, Calif.: Sage, 1973).Google Scholar

³ De Swaan, , Coalition Theories, Chap. 8Google Scholar; Taylor, and Laver, , ‘Government Coalitions’, p. 225Google Scholar. There was considerable variation in the success of different theories from country to country, but this was the overall finding for all the countries examined.

⁴ Franklin, Mark N. and Mackie, Thomas T., ‘Measuring the Effects of Size and Ideology on the Formation of Governing Coalitions in Parliamentary Democracies’, Strathclyde Papers on Government and Politics, V (1983).Google Scholar

⁵ This is not the first time that a historical dimension has been introduced into the study of coalition formations, but other studies have been concerned with the satisfaction of partners in a previous coalition or with the sequence of events during actual negotiations to form a new coalition. See Laver, Michael, ‘Dynamic Factors in Government Coalition Formation’, European Journal of Political Research, II (1974), 259–89CrossRefGoogle Scholar; Brams, Steven J., ‘Bandwagons in Coalition Formation’, American Behavioral Scientist, XVIII (1975), 472–97CrossRefGoogle Scholar; Chertkoff, Jerome M., ‘Socio-psychological Views on Sequential Effects in Coalition Formation’, American Behavioral Scientist, XVIII (1975), 451–71CrossRefGoogle Scholar. These studies attempt to take account of rather more complex considerations than those which concern us here. The fact that coalition formation is not an ‘end of the world’ game was first pointed out in Browne, 's ‘Testing Theories’, p. 407Google Scholar, and this line of argument was taken further in the concluding paragraphs to Taylor, and Laver, 's ‘Government Coalitions’, p. 234Google Scholar. The present approach has its philosophical basis in these discussions. The notion of inertia as a force in coalition formations was suggested by Hinkley's study of coalitions over time in the US Congress. See Hinkley, Barbara, ‘Coalitions in Congress: Size in a Series of Games’, American Politics Quarterly, II (1973), 339–59.CrossRefGoogle Scholar

⁶ Browne, Eric C. and Rice, Peter, ‘A Bargaining Theory of Coalition Formation’, British Journal of Political Science, IX (1979), 66–88, p. 77.Google Scholar

⁷ In different countries, different conventions operate to determine whether a government will resign. For the most part, no allowance has been made for these differences in the present study, but correction has been made for one very gross difference. In some countries governments do not necessarily resign when elections are held. In order to ensure comparability, we assume a coalition forming situation to occur whenever an election is held, even when the government does not formally resign.

⁸ A good description of the specific differences between the various theories is to be found in de Swaan, , Coalition Theories, Chap. 4.Google Scholar

⁹ Budge, Ian and Herman, Valentine, ‘Coalitions and Government Formation: An Empirically Relevant Theory’, British Journal of Political Science, VIII (1978), 459–77.CrossRefGoogle Scholar

¹⁰ Even if parties originally came together because of ideological similarities, it would be hard to confirm this empirically, since new parties are rare and their policy preferences not necessarily well-documented.

¹¹ It should be borne in mind that the number of potential coalitions in a coalition-forming situation is determined by the number of parties represented in the parliament. With three parties represented, there are seven potential coalitions (counting single-party governments as ‘coalitions’ and also counting the grand coalition of all parties). With ten parties (not an extravagantly large number in European parliaments) there are 1,023 potential coalitions.

¹² Riker, William H., The Theory of Political Coalitions (New Haven: Yale University Press, 1962), pp. 32–8.Google Scholar

¹³ Gamson, W. A., ‘A Theory of Coalition Formations’, American Sociological Review, XXVI (1961), 373–82CrossRefGoogle Scholar; Leiserson, Michael A., ‘Factions and Coalitions in One-party Japan’, American Political Science Review, LXII (1968), 770–87CrossRefGoogle Scholar; Taylor, and Laver, , ‘Government Coalitions’, pp. 208–9Google Scholar; de Swaan, , Coalition Theories, Chap. 5.Google Scholar

¹⁴ None of these refinements have gone so far as to propose a combined requirement that might not be met by any potential coalition in some coalition-forming situation. For example, if one is trying to combine a preference for ideologically cohesive coalitions with a preference for coalitions that are small but winning, all combinations to date have opted for the smallest of the ideologically suitable coalitions. However, it might be the case that better predictions would be made by a theory that preferred coalitions that were both ideologically suitable and also as small as possible, even though there might be situations in which no small coalition was ideologically suitable. This sort of conjunction is essentially an interaction effect, and we shall see later in the present article that interaction effects do play a part in improving our ability to correctly predict formation outcomes.

¹⁵ An example of a disjunctive combination would be one that predicted formations that were either small and winning or ideologically suitable.

¹⁶ There can exist combinations that are conjunctive without being strictly multiplicative. We will see below that a multiplicative combination of size and ideology (an interaction term) differs from a corresponding conjunctive combination.

¹⁷ The only unusual thing about such a dataset is that is has very little variance. If a potential coalition is coded 1 where it is observed to form and 0 otherwise, all but one of the potential coalitions in any coalition-forming situation will in fact be coded 0 on the dependent variable (see footnote 11). This means that the prediction equation will be liable to produce biased estimates for the regression coefficients relating to such a dataset, although the corresponding multiple correlation and variance explained are not similarly affected. See Goodman, Leo A., ‘The Relationship between Modified and Usual Multiple Regression Approaches to the Analysis of Dichotomous Variables’, in David R. Heise, Sociological Methodology 1976 (San Francisco: Jossey-Bass, 1975), pp. 83–110.Google Scholar

¹⁸ If the independent variables are also dichotomies, the numerical value of the r² coefficient will be the same as would be obtained for the ø² coefficent (x²/N) for the corresponding contingency table, and R will be the same as ø. While the use of correlation coefficients has advantages over using significance tests in research of this kind (see footnote 20), because potential coalitions are imaginary a suitable weighting strategy is needed before these coefficients can be considered comparable with coefficients based on analysis of real entities (see footnote 24). However, within the realm of coalition studies the coefficients no doubt enable us to compare one theory with another, and measure improvements in explanatory power, just as in any other realm.

¹⁹ For example, in our own study a single country (Italy) generated over 40,000 notional cases corresponding to all the potential coalitions in all the coalition-forming situations falling within our time period (see p. 280). In the present research project these problems were overcome by considering as a single case all those potential coalitions (notional cases) with identical characteristics, and then weighting this physical case to reflect the number of notional cases it represented. Thus in one coalition-forming situation, all the potential coalitions that did not command a majority of seats and were not familiar would be recorded as a single physical case, as would those which did not command a majority but were familiar, and so on.

²⁰ De Swaan, , Coalition Theories, Chap. 6Google Scholar; Taylor, and Laver, , ‘Government Coalitions’, pp. 217–21Google Scholar. It is not in fact clear what meaning is to be ascribed to a result that achieves statistical significance in a dataset that is not a sample. Whatever the rate of correct prediction may be for some theory, this is the true rate for the universe under investigation. Whether it is a substantively important rate of correct prediction is a matter for judgement rather than statistics, and judgement is facilitated by knowing the degree of association as measured in some familiar fashion (such as by a correlation coefficient). By contrast, the level of statistical significance achieved by a theory depends heavily on the number of cases in the analysis. De Swaan's assumptions about what constitutes a coalition-forming situation result in a smaller dataset for analysis than is obtained on the assumptions employed by other researchers. De Swaan also obtains worse levels of significance than are obtained by Taylor and Laver. This might be a simple artefact of the smaller number of cases, or there might be other reasons for the different findings. Nor can the rules of statistical inference be used to generalize the findings to some larger universe, since the question then arises as to whether the larger universe can be expected to behave in the same manner as the smaller one, and this too is a question of judgement.

²¹ We know from previous research which theories give significant results, so relationships corresponding to these results can serve as threshold values above which statistical significance is assured (but see footnote 20).

²² Franklin, and Mackie, , ‘Size and Ideology’, pp. 19–26.Google Scholar

²³ Browne, , ‘Testing Theories’, p. 399Google Scholar; Taylor, and Laver, , ‘Government Coalitions’, p. 212Google Scholar. The countries included in our universe are thus Austria, Belgium, Denmark, Finland, Germany, Iceland, Ireland, Israel, Italy, Luxembourg, Netherlands, Norway and Sweden. The data are the same as those analysed in Franklin, and Mackie, 's ‘Size and Ideology’Google Scholar and are described in more detail in that publication, where the choice of countries and starting date is also discussed.

²⁴ Under this strategy, proposed (but not adopted) in Browne, 's ‘Testing Theories’Google Scholar, footnote 9, each potential coalition is weighted down by a sufficiently large fraction (the reciprocal of the number of potential coalitions) as to ensure that each coalition-forming situation is given a weight of 1 in the analysis. Another strategy was apparently first proposed by Mokken, Robert J., ‘A Simple Model for Testing Coalition Theories’ (University of Amsterdam Institute for Political Science mimeograph, 1972)Google Scholar. This was adopted by de Swaan, in Coalition Theories, p. 124Google Scholar, and adapted by Taylor, and Laver, in ‘Governing Coalitions’, p. 220Google Scholar, for their own purposes. This is to perform a separate statistical test for each interelection period, in which the number of successes scored by some theory is compared against chance, and the statistical probability associated with the result summed in an appropriate manner across situations. Finally, there is the strategy employed by Franklin, and Mackie, in ‘Size and Ideology’, p. 18Google Scholar, to achieve comparability with the Mokken test, in which each coalition-forming situation is given a weight of 2 in the analysis, i for the forming coalition and i for all others weighted down by identical fractions. This last strategy also gives us a firm basis for talking about the variance explained by variables corresponding to some theory of coalition formation (see footnote 25).

²⁵ Each of these approaches has its own deficiencies. Briefly, the problem with the Browne strategy is that no more weight is given to the actual historical coalition than to the myriad of imaginary coalitions that might have formed but did not, so that the importance of correct predictions varies from situation to situation with the number of parties. The problem with the alternative Franklin and Mackie weighting is over-compensation in the other direction. The imaginary coalitions that did not form are given so little weight that differences in the number of predictions made by some theory from the number made by some other theory count for too little compared with the importance of making a correct prediction. Thus theories that are parsimonious in their predictions are at a disadvantage compared to theories that make more predictions with the corresponding extra opportunity for one of them to be fulfilled. We prefer the Browne strategy because it permits us to say, for any situation or on average over all situations, what are the chances that a particular sort of potential coalition will form (one that is familiar, small or cohesive, or more than one of these) but this strategy does not yield any sort of absolute measure for the performance of a theory. Under Browne weighting the actual outcome is compared, not with anything real, but with everything that might have happened.

²⁶ It is not entirely clear why the Franklin-Mackie weighting strategy should give results comparable to those of the Mokken test. The correspondence was found by trial and error, rather than by theoretical derivation, and appears to be due to the practical difficulty of achieving intermediate results appropriate for summation across situations in the manner envisioned by Mokken, 's ‘Simple Model’Google Scholar. As both he and Taylor and Laver point out (‘Simple Model’, p. 3Google Scholar, ‘Governing Coalitions’, p. 219Google Scholar) the summation strategy assumes a continuous distribution; but with relatively few opportunities for confirmation between elections, a theory is liable either to be confirmed at a high level of probability or not at all. This means that the probabilities being carried forward for summation will in practice be very close to either 0 or 1. So the importance of the forming coalition is magnified in relation to the importance of non-forming coalitions, as it is with Franklin-Mackie weighting.

²⁷ Axelrod, Robert, Conflict of Interest (Chicago: Markham, 1970), Chap. 8Google Scholar. In fact, Axelrod proposed this theory in conjuction with one of the size theories, as Closed Minimal Winning Theory; but both de Swaan and Taylor and Laver later abstracted it from this combination and tested it as a separate theory in conjunction only with the winning requirement. In operationalizing this variable we employed the rankings used by Taylor and Laver and given in the appendix to their paper for all parties held in common by our universe and theirs. For other parties we used our own judgement.

²⁸ Franklin, and Mackie, , ‘Size and Ideology’, p. 27Google Scholar. There is nothing logically impossible about using a multiplicative combination as a component in an additive model, and indeed we do employ one such component (an interaction term) in our own model later in the present article. However, regression analysis performs best when the independent variables are uncorrelated, and thus bring to bear effects that are as different as possible from one another. Interaction terms may supplement or even replace the simpler components, but their presence is generally determined empirically, and indeed the interaction term we eventually employ is not the same multiplicative combination as we here eschew (see footnote 14 above).

²⁹ It is rare in social research to discover a preference for dichotomous variables over their interval-level counterparts, yet this discovery was made again and again in the course of the present investigation. Not only were interval-level variants operationalized for the concept of familiarity, but so too were interval-level variants of the concept of inertia (looking back beyond the immediate past for identical coalitions that had formed earlier in time) and of each of the size theories (counting the extent to which potential coalitions deviated from strict conformity with each theory). In every case, a dichotomous variable was found to make better predictions of formation outcomes. In retrospect it may be possible to explain this finding on the grounds that our dependent variable is a dummy variable that is overwhelmingly 0. Only 3 per cent of cases in the notional dataset represent forming coalitions. For this reason, ‘near misses’ in prediction on the basis of interval-level predictor variables do more harm than good.

³⁰ See footnote § to Table 2. The differences between findings generated under different weighting schemes is disturbing, particularly in the relationship between familiarity and formation. We have pointed out the major disadvantages of the alternative weighting strategy (footnote 25), notwithstanding its statistical credentials (footnote 26). In the present context it can be seen that the choice is also important in conveying the manner in which past experience appears to affect present outcomes.

³¹ Since the best of past results have been determined statistically significant, our results are also assured of this distinction, though it is not a distinction that we find particularly helpful (see footnote 20).

³² For a discussion of the ways in which the separate effects of different predictors can be shared in a multivariate analysis see Franklin, Mark N. and Mughan, Anthony, ‘The Decline of Class Voting in Britain: Problems of Analysis and Interpretation’, American Political Science Review, LXXII (1978), 523–34.CrossRefGoogle Scholar

³³ Draper, N. R. and Smith, H., Applied Regression Analysis (New York: Wiley, 1966), p. 175Google Scholar. Because the best predictor depends on the weighting scheme employed, we were not able to take the best predictor first in both cases. Instead we take the variables in the order of their average contributions under both weighting schemes.

³⁴ For discussion of the effect of taking a causal rather than a regression view of social processes, see Franklin, and Mughan, , ‘Decline of Class Voting’, pp. 530–2.Google Scholar

³⁵ Such a model has some of the features of a Markov process. See Coleman, S., The Logic of Collective Action (London: Heinemann, 1973), p. 14Google Scholar. In fact, the first coalition-forming situation will differ from later ones in the trivial sense that familiarity and inertia cannot be measured until some time has elapsed.

³⁶ D-Systems analysis was developed by Davis as an alternative to log-linear modelling for handling situations of this type while generating coefficients whose interpretation is the same as for regression estimates, and which can thus be used in causal models in the same manner as regression coefficients. See Davis, James A., ‘Analysing Contingency Tables with Linear Flow Graphs: D-Systems’Google Scholar, in Heise, (ed.), Sociological Methodology 1976, pp. 111–45Google Scholar. Cf. Goodman, , ‘Multiple Regression Approaches’.Google Scholar

³⁷ A multi-way contingency table contains an effect for each variable under every possible combination of circumstances indexed by the other variables in the table. These different effects have to be averaged in order to arrive at a single effect for each variable, and in the process it is easy to detect whether one of the coefficients being averaged is very different from the others. The d-coefficients, along with all other results reported in this article were calculated using the SPSS Conversational System. See Nie, Norman H., Hull, C. Hadlai, Franklin, Mark N. et al. , SPSS: An Introduction to the SPSS Conversational System (New York: McGraw-Hill, 1980)Google Scholar. The particular procedure employed is called CATFIT, and is not widely available except as part of SCSS.

³⁸ This is because the forming coalitions that occur overwhelmingly in the right-hand branches of the tree would be given so much more weight than the non-forming coalitions (see footnote 24).

³⁹ What this procedure amounts to is taking fifty forming coalitions at random and another fifty potential coalitions that did not form; and repeating the selection a very large number of times. On average in both groups, some seven in a hundred will be found having all the characteristics we have come to associate with formations, and Table 6 shows that, of these, 96·7 per cent will be forming coalitions. This example brings home the actual manner in which our alternative weighting scheme operates. It puts potential and actual coalitions on an equal footing, but the manner in which it does so strictly limits the utility of any resulting coefficients.

⁴⁰ Budge, and Herman, , ‘Coalitions and Government Formation’, pp. 459–77.Google Scholar