¹ Most of these theories were not stated by their authors as equilibrium theories, explicitly based on preferences and predicting that undominated coalitions form. But they are essentially equilibrium theories. This is assured by the requirement in these theories that ‘coalition choices must be reciprocated’, i.e., a coalition, if it is to be amongst the predicted coalitions, must be the first choice of all its members.

² This is of course a strong assumption. The left and right wings of a non-extreme party are likely to have different preferences amongst coalition partners.

³ See, for example, Maschle, Michael, Playing an n-Person Game: An Experiment, Econometric Research Program RM 28 (Princeton, N.J.: Princeton University, 1965)Google Scholar; Leiserson, Michael, Coalitions in Politics, Ph.D. Dissertation, Yale University (New Haven, Conn., 1966)Google Scholar Chapters 4 and 5; and Riker, William H., ‘Bargaining in a Three-Person Game’, American Political Science Review, LXI (1967), 642–56.CrossRefGoogle Scholar

⁴ Leiserson, Coalitions in Politics, Chapter 7.

⁵ An excellent introduction to these solution concepts is given by Rapoport, Anatol, N-Person Game Theory: Concepts and Applications (Ann Arbor: The University of Michigan Press, 1970)Google Scholar, although he does not discuss the extension of the theory to the no-side-payments case, which is probably more common in politics than the ‘classical’ side-payments case.

⁶ This is how Leiserson (Coalitions in Politics, Chapter 7) uses the kernel in his work on cabinet coalitions.

⁷ Rapoport, , N-Person Game Theory, p. 287.Google Scholar

⁸ This assumption is ‘unrealistic’ in two respects. First, in the theories to be discussed, it will follow from this assumption that minority governments will never form. In practice, of course, these sometimes occur. Out of 196 governments in 19 parliamentary democracies between 1945 and 1969,37 were minority governments. See Taylor, Michael and Herman, V. M., ‘Party Systems and Government Stability’, American Political Science Review, LXV (1971), 28–37.CrossRefGoogle Scholar These theories can never account for such occurrences; but the introduction of ad hoc assumptions (for example, that in certain countries any coalition is ‘winning’ if it has at least 40 per cent of the seats) should perhaps be postponed until these more parsimonious theories have been extensively tested. Second, certain parties (such as Communist Parties) may sometimes prefer to stay out of the government (even if they would be invited to join).

⁹ If ‘xRy’ means ‘x is preferred or indifferent to y’, then R is a binary (preference) relation. A binary relation R is said to be reflexive if, for all x, xRx; complete if, for all x and y, xRy or yRx; and transitive if, for all x, y and z, xRy and yRz implies xRz; where x, y, z, … are, in this case, alternative coalitions. A binary relation which is reflexive, complete and transitive is called an ordering.

¹⁰ For a discussion of ‘payoffs’ to government coalitions and illustrations from France, Italy, Sweden and Japan, see Leiserson, Coalitions in Politics, Chapter 7, and ‘Factions and Coalitions in One-Party Japan: An Interpretation Based on the Theory of Games’, American Political Science Review, LXII (1968), 770–87.Google Scholar

¹¹ Riker, William H., The Theory of Political Coalitions (New Haven: Yale University Press, 1962)Google Scholar; Gamson, William A., ‘A Theory of Coalition Formation’, American Sociological Review, XXVI (1961), 373–82CrossRefGoogle Scholar; Leiserson, Coalitions in Politics, Chapter 7, and ‘Factions and Coalitions’.

¹² Leiserson, Coalitions in Politics, Chapter 7.

¹³ Leiserson, Coalitions in Politics, Chapter 7.

¹⁴ Exactly half of all the possible coalitions must be winning, since the complement of every winning coalition is a losing coalition and conversely. Thus, if there are n parties, there are alway 2n-1 winning coalitions.

¹⁵ ‘EMW makes the same predictions as a theory proposed by Leiserson ('Factions and Coalitions'), which will be discussed later.

¹⁶ I am most grateful to Nicholas Miller for pointing out to me this last set of identities.

¹⁷ A preference ordering of a set of alternatives is said to be strong if no alternatives are tied (indifferent).

¹⁸ In many situations it is this property which makes the lexicographic assumption far too strong: an alternative (P, say) which is close to an actor's first preference (L) on his most important criterion but far away (possibly infinitely far away) on all succeeding criteria, is preferred to an alternative (Q) which is ‘infinitesimally’ close to L on all the succeeding criteria yet only a little farther away from L on the first criterion than P. This objection to the lexicographic principle would be reasonable if there were infinitely large differences between some of the coalitions on some criteria (which, it should be remembered, are variables taking numerical values), but loses much of its force when (as here) the set of alternatives is finite and the variables representing the criteria have finite bounds, so that no two coalitions can be ‘infinitely different’ on any criterion.

¹⁹ The notion of non-lexicographic preferences — or a utility function involving a ‘trade-off’ between criteria — could be made more precise if we first represented the coalitions as points in a multi-dimensional space. This, however, would involve a considerable digression and some mathematics not needed elsewhere in our analysis. Compare the references given in footnote 20.

²⁰ The results of this section are strikingly similar to some theorems recently proved in the theory of collective decision-making, and might appear to be merely applications of those theorems. The coalitions can be represented as points in a multi-dimensional space (which we might call a coalition space), whose dimensions are the various Criteria of Choice, assumed to be numerical variables. This space is formally similar to the ‘commodity space’ of economics or the ‘policy space’ of the theory of collective decisions. Each actor has an ‘optimum’ point in this space - the location of his most preferred coalition. Of any two elements of the set of winning coalitions of which he is a member, he prefers the one which is ‘closest’ to his optimum (as defined by his utility function, or some other assumption - such as the lexicographic - about how he combines the criteria). However, a coalition not in this set might be ‘closer’ to his optimum than one in the set, yet be less-preferred; furthermore, this set varies from actor to actor. This prevents any straightforward application of the theorems or equilibrium in policy spaces (in the theory of collective decisions) to equilibrium in coalition spaces. Nevertheless, we note the very suggestive similarities: equilibrium points in policy space always exist when all the actors use the lexicographic principle with the same salience ranking of the policy dimensions, but may not exist if the salience ranking differs between actors (see Taylor, Michael, ‘The Problem of Salience in the Theory of Collective Decision-Making’, Behavioral Science, XVI (1970), 415–30);CrossRefGoogle Scholar if preferences are not lexicographic, equilibrium points exist only for very special utility functions (Rae, Douglas and Taylor, Michael, ‘Decision Rules and Policy Outcomes’, British Journal of Political Science, I (1971), 71–90)CrossRefGoogle Scholar or for very unlikely distributions of the optima in the policy space (as shown by Plott, Charles R., ‘A Notion of Equilibrium and its Possibility under Majority Rule’, American Economic Review, LVII (1967), 788–806).Google Scholar For a survey of these and related results, see Taylor, Michael, ‘Mathematical Political Theory’, British Journal of Political Science, I (1971), 339–82,CrossRefGoogle Scholar Sections 5 and 6.

²¹ As before, we are not construing theories of n-person games as theories of coalition formation.

²² Actually, the assumption of only one criterion is a special case of the assumption of several criteria used lexicographically.

²³ Gamson, ‘A Theory of Coalition Formation; Riker, Theory of Political Coalitions; Leiserson, Coalitions in Politics, Sections 2.1 and 3.6; Axelrod, , Conflict of Interest (Chicago: Markham, 1970), Chapter 8;Google ScholarDe Swaan, Abraham, ‘An Empirical Model of Coalition Formation as an N-Person Game of Policy Distance Minimization’, in Groenings, Sven, Kelley, E. W. and Leiserson, Michael, eds., The Study of Coalition Behavior (New York: Holt, Rinehart and Winston, 1970) pp. 424–44.Google Scholar

²⁴ Axelrod, , Conflict of Interest, pp. 150–1.Google Scholar

²⁵ The difficulties of measuring ideological positions at the cardinal level are widely recognized. But even though assumptions of the form ‘Parties A and B (in some country, in a certain period) are 1.7 times as far apart as Parties C and D’ may be too heroic, most students could probably agree on such statements as ‘Parties A and B are farther apart than, but not more than three times as far apart as, Parties C and D’ I suspect that a collection of assumptions of this form would suffice to make the necessary conflict-of-interest comparisons in a given party system.

²⁶ Miller, Nicholas, unpublished memorandum, University of California, Berkeley (02, 1971).Google Scholar

²⁷ This discussion of Axelrod is concerned only with the connection between conflict of interest and mcw coalitions. The absence of a precise connection does not, of course, imply that either part is not an interesting theory in its own right.

²⁸ Gamson, ‘A Theory of Coalition Formation’; Leiserson, ‘Factions and Coalitions’.