¹ Whereas the idea of the core originated with von Neumann and Morgenstern, the term ‘core’ has been adopted by other authors. See Luce, R. D. and Raiffa, H., Games and Decisions (New York: Wiley, 1957).Google Scholar

² For a detailed discussion, see Riker, William H. and Ordeshook, Peter C., An Introduction to Positive Political Theory (Engelwood Cliffs, N.J.: Prentice-Hall, 1973).Google Scholar

³ See Leiserson, Michael A., ‘Game Theory and the Study of Coalition Behavior’, in Groennings, S., Kelley, E. W. and Leiserson, M. A., eds., The Study of Coalition Behavior: Theoretical Perspectives and Cases From Four Continents (New York: Holt, Rinehart and Winston, 1970), pp. 255–72.Google Scholar

⁴ The quotations are from Hardin, Russell, ‘Hollow Victory: The Minimum Winning Coalition’, American Political Science Review, LXX (1976), 770–87Google Scholar, passim. Brams, Steven J., ‘Positive Coalition Theory: The Relationship Between Postulated Goals and Derived Behavior’, in Cotter, C. P., ed., Political Science Annual: An International Review, Vol. 4 (Indianapolis: Bobbs-Merrill Co., 1973), pp. 3–40Google Scholar, makes much the same point when he states: ‘While a number of “reasonable” sets of constraints have been proposed [they] have not led to solutions that seem particularly useful in the study of the behavior of real coalitions.’ (p. 19).

⁵ Riker, William H., The Theory of Political Coalitions (New Haven, Conn.: Yale University Press, 1962), p. 32.Google Scholar

⁶ de Swaan, Abram, Coalition Theories and Cabinet Formations (Amsterdam: Elsevier, 1973), p. 56Google Scholar. For the definitions of ‘simple’ and ‘normalized’, see von Neumann, J.Morgenstern, and O., Theory of Games and Economic Behavior (Princeton, N.J.: Princeton University Press, 1953), para, 49·3 and 49·4.Google Scholar

⁷ Hardin, , ‘Hollow Victory’.Google Scholar

⁸ De Swaan, , Coalition Theories, p. 57Google Scholar. His extended argument is found on pages 52–65.

⁹ Gamson, William A., ‘A Theory of Coalition Formation’, American Sociological Review, XXVI (1961), 373–82, p. 376.CrossRefGoogle Scholar

¹⁰ See Schelling, Thomas, The Strategy of Conflict (New York: Oxford University Press, 1963), Chaps. 3 and 4.Google Scholar

¹¹ Browne, Eric C. and Franklin, Mark N., ‘Aspects of Coalition Payoffs in European Parliamentary Democracies’, American Political Science Review, LXVII (1973), 453–69CrossRefGoogle Scholar. For an extension of this analysis using an expanded data set, see Browne, Eric C. and Frendreis, John P., ‘A Reexamination of the “Parity Norm” for Ministerial Payoffs to Parties in Cabinet Coalitions’ (University of Wisconsin, Milwaukee, mimeo, 1978).Google Scholar

¹² de Mesquita, Bruce Bueno, Strategy, Risk and Personality in Coalition Politics: The Case of India (New York: Cambridge University Press, 1975).Google Scholar

¹³ For example, parity may be achieved in the Indian case if those actors with a greater-than-proportional share of the ‘important’ ministries receive a less-than-proportional number of portfolios, and vice versa.

¹⁴ It may also be the case that the value of a payoff share is established by trading across dimensions such that underpayment in terms of the quantitative receipt of payoff units is compensated by overpayment in terms of more valuable units. In examining this possibility, Browne, and Franklin, (‘Aspects of Coalition Payoffs’, p. 468)Google Scholar conclude that no cross-dimensional trading of this sort characterized distributions in their universe.

¹⁵ Notable exceptions to this generalization for European cabinet formation have occurred in the French Fourth Republic (excluded from the Browne and Franklin data set) and, less frequently, in Finland. Both of these settings have also had a history of instability at the cabinet level.

¹⁶ Perhaps the most promising of recent work attempting to extend game-theoretic treatments of coalition formation in pure conflict settings has been pursued by Steven J. Brams and his associates, using presidential nominating conventions as an empirical referent. See Brams, Steven J. and Riker, W. H., ‘Models of Coalition Formation in Voting Bodies’, in Herndon, J. F. and Bernd, J. L., eds., Mathematical Applications in Political Science, VI (Charlottesville, Va.: University Press of Virginia, 1972), 79–124Google Scholar; Brams, Steven J. and Garriga-Pico, J., ‘Deadlocks and Bandwagons in Coalition Formation: the 1/2 and 2/3 Rules’, American Behavioral Scientist, XVIII (1975), 472–96CrossRefGoogle Scholar; and Brams, Steven J. and Heilman, J. H., ‘When to Join a Coalition and With How Many Others Depends on What You Expect the Outcomes To Be’, Public Choice, XVII (1974), 11–25.CrossRefGoogle Scholar

¹⁷ See for example Axelrod, Robert, Conflict of Interest (Chicago: Markham, 1970)Google Scholar; De Swaan, , Coalition TheoriesGoogle Scholar; and Taylor, M. and Laver, M., ‘Government Coalitions in Western Europe’, European Journal of Political Research, 1 (1973), 205–48.CrossRefGoogle Scholar

¹⁸ Although a very large number of all actual formations were closed in this sense, the accuracy of such predictions is compromised by the very large solution sets generated by the proposition. Moreover, de Swaan has observed statistical support for a variant of the closed coalition, ‘closed minimal range theory’, which holds that no partners will be added to a winning coalition that extends its ideological range further on a unidimensional scale. This finding, however, is conditional upon a stable political situation as this quotation shows: ‘Historical coalitions tend to be closed coalitions and in times of normalcy these coalitions will be of minimal range also. Or, in the conjunction of minimal range theory and the closed coalition proposition, closed minimal range theory is the explanation of coalition behavior in normal times; in times of crisis, this component, minimal range theory, loses its validity but the other component, the closed coalition proposition, remains valid on its own’ (De Swaan, , Coalition Theories, p. 156Google Scholar, italics in original text).

¹⁹ De Swaan, , Coalition Theories, p. 288.Google Scholar

²⁰ The assertion that virtually all parliamentary actors are motivated to participate in the government is intended to be taken literally, and it does not imply that each actor is expected to adopt the same strategy in pursuing this goal. Rather, we expect that the willingness of an actor to accept membership in a coalition cabinet (or, perhaps more important, to be found acceptable for membership) will be conditional upon the extent of its actual (or perceived) disposition to share governmental office on a more or less continuing basis. Hence, to the extent that electoral outcomes do confer near parliamentary majorities upon coalition actors, there may indeed, as Bueno de Mesquita has suggested, be an incentive for other actors to oppose them through exclusion from a cabinet coalition (e.g. recent Irish and Norwegian cabinets). Here, exclusion of ‘dominant’ parties may be seen as an attempt by smaller parties to preserve their own opportunities for continuing to participate in the government. Additionally, parliamentary actors characteristically identified with ‘extreme’ ideological positions (both left and right) are typically found to be excluded from cabinets (usually by groups of moderate bourgeois parties, which together command a parliamentary majority). It has been usual to attribute to such parties the goal of attempting to achieve exclusive and permanent control of governmental decision making by subverting the democratic processes of the established constitutional order, thus threatening the ability of ‘system-orientated’ actors to continue as active players. Actors identified with extreme ideological positions, having historically been faced with the prospect of remaining in permanent opposition in bourgeois-dominated parliaments, have responded with a splintering (particularly on the left) within ranks, with traditional socialist parties migrating ideologically toward the right in search of legitimation as a responsible political alternative and Communist parties remaining somewhat behind to articulate a more militant Marxist position. This phenomenon, frequently commented upon in the literature, amounts to a weakening of resolve regarding the attributed goal of radical transformation of society (although not necessarily its abandonment) in favour of attempting to join the established game of parliamentary coalition politics.

²¹ See Ullman-Margalit, E., The Emergence of Norms (New York: Clarendon Library of Logic and Philosophy, Oxford University Press, 1978)Google Scholar, and Schelling, , The Strategy of Conflict.Google Scholar

²² Dependence on a conventional norm can be dropped by appealing to a prisoners' dilemma supergame and arguing that appropriate discounting of expected utility makes co-operation the individually rational strategy. We feel, however, that this kind of argument relies too heavily on simple rationality in a complex situation, which we have elsewhere in the paper argued against.

²³ See Nash, J. F., ‘Two-Person Cooperative Games’, Econometrica, XXI (1953), 128–40.CrossRefGoogle Scholar

²⁴ Sven Groennings, , ‘Patterns, Strategies and Payoffs in Norwegian Coalition Formation’Google Scholar, in Groennings, S., Kelley, E. W. and Leiserson, M. A., eds., The Study of Coalition Behavior, pp. 60–79.Google Scholar

²⁵ The utility to a party of a division of ministries is here calculated as the sum of the utilities it holds for the ministries it receives. This process of addition ignores such matters as the adverse effect on Party A if Party B gets A's favourite ministry, or the possibility that two ministries together may be worth more than the sum of their individual utilities.