¹ Dahl, Robert, ‘The Concept of Power’, Behavioral Science, II (1957), 201–15, p. 201.Google Scholar

² Riker, William, ‘Some Ambiguities in the Notion of Power’, American Political Science Review, LVIII (1964), 341–9, p. 348Google Scholar; March, James, ‘The Power of Power’, in Easton, David, ed., Varieties of Political Theory (Englewood Cliffs, N.J.: Prentice-Hall, 1966), 39–70, p. 70Google Scholar; Dahl, , ‘The Concept of Power’, p. 201.Google Scholar

³ Riker, , ‘Some Ambiguities in the Notion of Power’, p. 343Google Scholar, claims that approaches 3, 4 and 5, which are presented below, are mutually exclusive, contradictory and have little in common. My paper should not be taken as a refutation of the many excellent points made in his article.

⁴ Dahl, Robert, March, James, and Nasatir, D., ‘Influence Ranking in the United States Senate’Google Scholar, paper given at the annual meeting of the American Political Science Association in 1956, and Dahl, ‘The Concept of Power’. The later work has theoretical definitions of power which need not be coterminous with the operational definition. For precursors see Lasswell, Harold P. and Kaplan, Abraham, Power and Society (New Haven, Conn.: Yale University Press, 1950)Google Scholar and Simon, Herbert, ‘Notes on the Observation and Measurement of Political Power’, Journal of Politics, XV (1953), 500–16.Google Scholar

⁵ March, James, ‘Measurement Concepts in the Theory of Influence’, Journal of Politics, XIX (1957), 202–26.Google Scholar

⁶ Thibaut, J. W. and Kelly, H. H., The Social Psychology of Groups (New York: Wiley, 1959).Google Scholar

⁷ Karlsson, George, ‘Some Aspects of Power in Small Groups’, in Caswell, Joan H., Solomon, Herbert and Suppes, Patrick, eds., Mathematical Methods of Small Group Processes (Stanford, Calif.: Stanford University Press, 1962), 193–202.Google Scholar

⁸ Shapley, L. S., ‘A Value for N-Person Games’, in Kuhn, H. W. and Tucker, A. W., eds., Contributions to the Theory of Games, Vol. II, Annals of Mathematics Studies 28 (Princeton, N.J.: Princeton University Press, 1953), 307–17Google Scholar; Shapley, L. S. and Shubik, Martin, ‘A Method for Evaluating the Distribution of Power in a Committee System’, American Political Science Review, XLVIII (1954), 787–92Google Scholar; Mann, Irwin and Shapley, L. S., ‘The A Priori Voting Strength of the Electoral College’, in Shubik, Martin, ed., Game Theory and Related Approaches to Social behavior (New York: Wiley, 1964), 151–64Google Scholar; Shapley, L. S. and Riker, William, ‘Weighted Voting: A Mathematical Analysis for Instrumental Judgements’, in Pennock, J. Roland, ed., Representation: Nomos X (New York: Atherton Press, 1968), 199–216.Google Scholar

⁹ Harsanyi, John, ‘Approaches to the Bargaining Problem Before and After the Theory of Games: A Critical Discussion of Zeuthen's, Hicks', and Nash's Theories’, Econometrica, XXIV (1956), 144–57.Google Scholar For an outstanding review of the literature see Cartwright, Dorwin, ‘Influence, Leadership and Control’, in March, James, ed., Handbook of Organizations (Chicago: Rand McNally, 1965), 1–47.Google Scholar This is also reprinted in an excellent collection of articles by Bell, Roderick, Edwards, David, and Wagner, R. Harrison, eds., Political Power (New York: Free Press, 1969).Google Scholar For a good recent discussion of power see Allison, Lincoln, ‘The Nature of the Concept of Power’, European Journal of Political Research, II (1974), 131–42.Google Scholar

¹⁰ Dahl either does not consider costs or implicitly assumes that the costs for each individual are identical. See Section D for an analysis when costs are considered. In this section the analysis of power is easier as there are no counter strategies. Power may be power over things.

¹¹ Weber, Max, The Theory of Social and Economic Organization, edited by Parsons, Talcott (New York: Free Press, 1947), p. 152.Google Scholar

¹² Nagel, Jack, The Descriptive Analysis of Power (New Haven, Conn.: Yale University Press, 1975).Google Scholar See also Alker, Hayward Jr., ‘On Political Capabilities in a Scheduled Sense: Measuring Power, Integration and Development’, in Alker, Hayward Jr., ed., Mathematical Approaches to Politics (San Francisco: Jossey-Bass, 1973).Google Scholar

¹³ The relationship between player reversal and restricted outcome has already been noted by Dahl, ‘The Concept of Power’, and March, ‘Measurement Concepts in the Theory of Influences’, when there is no negative influence. There is negative influence if when a person is for, the probability of passage is less than when he is against. If this were true a person would just pretend he was against when he was for.

¹⁴ March, in fact, did not consider zero-sum games.

¹⁵ Benn, Stanley, ‘Power’, The Encyclopedia of Philosophy, vol. 6 (New York: Collier-Macmillan, 1967), 424–7, p. 426.Google Scholar

¹⁶ This definition of amount and strength of power should not be confused with the definition presented by Harsanyi, , ‘Measurement of Social Power, Opportunity Costs and the Theory of Two-person Bargaining Games’, Behavioral Science, VII (1962), 67–80Google Scholar, and Dahl, , ‘The Concept of Power’.Google Scholar They define A's amount of power in terms of changes in B's behaviour, while I have defined amount of power in terms of changes in outcome. All the other approaches to power can also be broken down into strength and amount of power, e.g. if the probability that A could harm B was not 100 per cent as previously implicitly assumed but only 50 per cent, then the amount of power that A has is 50 per cent and the gross power is only half as much as originally given.

¹⁷ Harsanyi, , ‘Approaches to the Bargaining Problem’Google Scholar, uses threats as the initial bargaining point, but in zero-sum games this formulation is equivalent to Shapley's. Harsanyi does not use the word power for the outcome. This paper by Harsanyi should not be confused with Harsanyi, ‘Measurement of Social Power’, where he shows how power will lead to the outcome in his earlier paper. Other authors use the word power when referring to an outcome or set of solutions, e.g. Maschler, Michael, ‘The Power of a Coalition’, Management Science, 1 (1963), 8–29Google Scholar, and Shubik, Martin, ‘Games of Status’, Rand Corporation RM-5956-RC, Santa Monica, (1969)Google Scholar; but in two-person zero-sum games all these solution concepts equate the power of a player to the minimax value.

¹⁸ Shapley, and Shubik, , ‘A Method for Evaluating the Distribution of Power in a Committee System’Google Scholar; Mann, and Shapley, , ‘The A Priori Voting Strength of the Electoral College’.Google Scholar

¹⁹ Shapley, , ‘A Value for N-Person Games’.Google Scholar

²⁰ For other criticisms of the Shapley approach, see Coleman, James, ‘Control of Collectivities and the Power of a Collectivity to Act’Google Scholar, and Rae, Douglas, ‘Decisiveness in Election Outcomes’, in Lieberman, M. B.. ed., Social Choice (London: Gordon Breach, 1971).Google Scholar Note that for zero-sum games with side-payments it is not true in general that ability to harm yields the same power ranking as player reversal.

²¹ Harsanyi, , ‘Measurement of Social Power’.Google Scholar

²² Karlsson did not consider the costs of harming the other player.

²³ Harm and costs to players may be measured in terms of money, which is more measurable than elusive utility.

²⁴ A person can change the outcome without changing another person's behavior; e.g. in the following game without side-payments, Harsanyi's approach would not define B as being more powerful than A even though our intuitive notion would suggest so. In this case B is capable of changing his own behavior.

²⁵ There are, of course, significant measurement problems remaining. For example, some wins may be more important than other wins. For a discussion of these problems see Rae, Duncan Mac Jr. and Price, H. D., ‘Scale Positions and “Power” in the Senate’, Behavioral Science, IV (1959), 212–18.Google Scholar

²⁶ For examples of disagreement see Electing the President, Hearings before the Subcommittee on Constitutional Amendments of the Committee on the Judiciary, United States Senate – 91st Congress. Almost every senator is sure that the voters in his state are at a disadvantage in the Electoral College. Also see Longley, Lawrence D. and Braun, Alan, The Politics of Electoral College Reform (New Haven, Conn.: Yale University Press, 1972).Google Scholar

²⁷ Shapley, , ‘A Value for N-Person Games’Google Scholar; Coleman, James, ‘The Marginal Utility of a Vote Commitment’, Public Choice, V (1968), 39–58Google Scholar; Banzhaf, John III, ‘Weighted Voting Doesn't Work’, Rutgers Law Review, xix (1965), 317–43Google Scholar; Coleman, James, The Mathematics of Collective Action (Chicago: Aldine-Atherton, 1973).Google Scholar

²⁸ If we ignore the veto override, it makes no difference that the president cannot initiate legislation and can only accept or veto legislation passed by Congress. Since all three bodies must agree for legislation to be passed, it doesn't matter which initiates the legislation. This also holds when bargaining takes place.

²⁹ In turn, this is isomorphic to the following situation. There are two groups each with one vote, two votes being needed for passage. In order for the one group composed of the five permanent members to vote for, all the permanent members must vote for; in order for the group of six temporary members to vote for, two must vote for (that is, the groups have different internal rules). The word ‘isomorphic’ is used in a different way by Bartoszynski, Robert, ‘Power Structure in Dichotomous Voting’, Econometrica, XL (1972), 1003–19.Google Scholar

³⁰ The different approaches, however, will lead to different cardinal measures of power. See Coleman, , ‘Control of Collectivities’Google Scholar, and Rae, , ‘Decisiveness in Election Outcomes’Google Scholar, and Allingham, M., ‘Power and Value’, FELS Discussion Paper, No. 43 (Philadelphia: University of Pennsylvania, 1973).Google Scholar

³¹ This corresponds to the formation of subcollectives as presented by Coleman, , ‘The Marginal Utility of a Vote Commitment’Google Scholar, but is not exactly equivalent to the Electoral College as each state gets two votes regardless of size. See Brams, Steven, Games Theory and Politics (New York: Free Press, 1975)Google Scholar, who has many interesting insights into electoral games.

³² Two polar cases can serve to illustrate. If there are three states, two with three voters each and one with one voter, then four votes are needed to win and a voter in the one vote state is clearly a priori more powerful than the voter in a three vote state. If there are three states, one with three voters and two with one voter each, then three votes are needed to win and a voter in the three vote state has more power than a voter in a one vote state as the latter can have no effect on which side will win. See Wittman, Donald, ‘Power in Electoral Games’, in Hooker, C. A., ed., Foundations and Applications of Decision Theory (Dordrecht, Holland: D. Reidel, forthcoming).Google Scholar