¹ See Buchanan, James M. and Tullock, Gordon, The Calculus of Consent: Logical Foundations of Constitutional Democracy (Ann Arbor: University of Michigan Press, 1962), chaps. 7–12CrossRefGoogle Scholar; Tullock, , “A Simple Algebraic Logrolling Model,” American Economic Review, 60 (06, 1970), 419–426 Google Scholar; Coleman, James S., “The Possibility of a Social Welfare Function,” American Economic Review, 56 (12, 1966), 1105–1122 Google Scholar; Coleman, , “The Possibility of a Social Welfare Function: Reply,” American Economic Review, 57 (12, 1967), pp. 1311–1317 Google Scholar; Haefele, Edwin T., “Coalitions, Minority Representation, and Vote-Trading Probabilities,” Public Choice, 8 (Spring, 1970), 75–90 CrossRefGoogle Scholar; and Mueller, Dennis C., Philpotts, Geoffrey C., and Vanek, Jaroslav, “The Social Gains from Exchanging Votes: A Simulation Approach,” Public Choice, 13 (Fall, 1972), 55–80 CrossRefGoogle Scholar.

² Riker, and Brams, , “The Paradox of Vote Trading,” American Political Science Review, 67 (12, 1973), 1235–1247 CrossRefGoogle Scholar. Cf. Park, R. E., “The Possibility of a Social Welfare Function: Comment,” American Economic Review, 57 (12, 1967), 1300–1304 Google Scholar; Mueller, “The Possibility of a Social Welfare Function; Comment,” Ibid., pp. 1304–1311; Wilson, Robert, “An Axiomatic Model of Logrolling,” American Economic Review, 59 (06, 1969), pp. 331–341 Google Scholar; Koehler, David H., “Vote-Trading and the Voting Paradox: A Proof of Equivalence,” American Political Science Review, 69 (1975)Google Scholar; and, for a view which tends to support Riker and Brams but does not rule out the positions of Buchanan, Tullock and Coleman, see Bernholz, Peter, “Logrolling, Arrow Paradox and Cyclical Majorities,” Public Choice, 15 (Summer, 1973), 87–95 (but especially p. 94)CrossRefGoogle Scholar. But cf. the exchange among Tullock, Bernholz, and Riker, and Brams, in “Communications,” American Political Science Review, 68 (12, 1974), 1688–1692 Google Scholar. Also see Joe A. Oppenheimer, “Relating Coalitions of Minorities to the Voters' Paradox, or Putting the Fly in the Democratic Pie,” University of Texas mimeo, 1973. An earlier informal statement of the possible negative effect of vote trading is found in Niskanen, William A. Jr., Bureaucracy and Representative Government (Chicago: Aldine, 1971), p. 143 Google Scholar.

³ Riker, and Brams, , “The Paradox of Vote Trading,” p. 1240 Google Scholar.

⁴ Ibid., p. 1242.

⁵ On the distinction between “sincere” and “sophisticated” voting, see Farquharson, Robin, Theory of Voting (New Haven: Yale University Press, 1969), pp. 18 and 39–40 Google Scholar.

⁶ Riker, and Brams, , “The Paradox of Vote Trading,” pp. 1237–1238 Google Scholar.

⁷ Ibid., p. 1238.

⁸ This is Table 5, p. 1241.

⁹ Ibid., Table 6.

¹⁰ Ibid., Table 7.

¹¹ Ibid., Table 9.

¹² Ibid., pp. 1244 and 1242.

¹³ Ibid., p. 1242.

¹⁴ This amounts to accepting interpersonal comparisons of utilities. See Rothenberg, Jerome, The Measurement of Social Welfare (Englewood Cliffs, N.J.: Prentice-Hall, 1961), pp. 137–143 Google Scholar.

¹⁵ Riker, and Brams, , “The Paradox of Vote Trading,” pp. 1242 ff.Google Scholar

¹⁶ Buchanan, and Tullock, , The Calculus of Consent, pp. 85ff. and pp. 276–280 Google Scholar.

¹⁷ For a contrary point of view, see Pennock, J. Roland, “The Pork Barrel and Majority Rule,” Journal of Politics, 32 (08, 1970), 709–716 CrossRefGoogle Scholar.

¹⁸ Buchanan, and Tullock, , The Calculus of Consent, pp. 63–68 Google Scholar.

¹⁹ Riker and Brams (see the note to Table 10 on p. 1243 of “The Paradox of Vote Trading”) argue that their result is “strictly a Prisoners' Dilemma. …” If, however, we consider the general discussion of the dilemma, the preferred strategy of both players is cooperating (i.e., trading), but such a strategy is dominated by that of defecting (not trading). This is the case in our example; for a discussion of the dilemma, see Luce, R. Duncan and Raiffa, Howard, Games and Decisions (New York: John Wiley, 1957), pp. 88–113 Google Scholar. The Riker-Brams example is indeed a Prisoners' Dilemma, but not the classical type.

²⁰ Schattschneider, E. E., The Semisovereign People (New York: Holt, Rinehart and Winston, 1960), p. 35 Google Scholar. The paradox—at least as developed so far—follows rather directly from the fact that each voter can be said to be in either a winning coalition (for each motion or for the entire set of motions) or a losing coalition. Thus, the set of motions, J, constitutes a simple game: formally, “[a] game, v, in (0, 1) normalization is said to be simple if, for each S⊂N, we have either v(S) — 0 or v(S) — 1” ( Owen, Guillermo, Game Theory [Philadelphia: W. B. Saunders, 1968], p. 163 Google Scholar), where N is the set of players in the game, S is a coalition, and v(S) is the value of the coalition. Owen notes (ibid.) that “simple games … include voting ‘games’ in elections and legislatures”—precisely the situation we have. The result that at least one member of the total number of players must be excluded from some coalition follows from an extension of L. S. Shapley's comment that “[s]olutions that exclude one or more players … are found … in almost all simple games, and they probably exist in all constant-sum games.” See Shapley, , “On Solutions that Exclude One or More Players,” in Essays in Mathematical Economics in Honor of Oskar Morgenstern, ed. Shubik, Martin (Princeton: Princeton University Press, 1967), p. 57 Google Scholar.

²¹ Riker, and Brams, , “The Paradox of Vote Trading,” p. 1236 Google Scholar.

²² Cf. Rapoport, Anatol and Chammah, Albert M., Prisoners' Dilemma (Ann Arbor: University of Michigan Press, 1965), chap. 5CrossRefGoogle Scholar; and Lave, Lester B., “An Empirical Approach to the Prisoner's Dilemma Game,” Quarterly Journal of Economics, 76 (08, 1962), pp. 424–436 CrossRefGoogle Scholar. Nigel Howard has proposed an approach to the Prisoners' Dilemma in which cooperative strategies would be rational. In his “metagames,” players choose their strategies dependent upon those of the other players. See his Paradoxes of Rationality: Theory of Metagames and Political Behavior (Cambridge: MIT Press, 1971), pp. 44–48 and 55–60 Google Scholar. Note the strong criticism of this approach by Harsany, John C., in his review of Howard's book in the American Political Science Review, 67 (06 1973), 599–600 CrossRefGoogle Scholar. In experiments with repetitive play, individuals have tended to behave cooperatively. Rapoport and Chammah as well as Lave found that the proportion of cooperative responses approach seventy per cent. For an interesting argument that the Prisoners' Dilemma is not a genuine dilemma or even a paradox, see Cunningham, R. L., “Ethics and Game Theory: The Prisoners' Dilemma,” Papers on Non-Market Decision Making, 2 (1967) pp. 11–26 Google Scholar.

²³ The logical nexus between unanimity and majority voting rules vanishes once the deterministic nature of the argument is relaxed. The probability that an individual will agree to trade votes with one or more others is dramatically affected by the fact that, under the former decision rule, a single negative vote will suffice to defeat a motion. Under majority rule, however, the probability that a member will be able to affect the outcome of a collective decision is substantially reduced—and, indeed, is a function of the size of the voting body. Another factor which is also a function of the size of the voting body— and which seems to us to be more serious for a unanimity than a majority rule—is “transaction costs,” associated with the necessity of reaching a collective decision. Our discussion of pairwise vs. n-wise trading of votes below bears on this problem. We are grateful to T. Nicolaus Tideman and James M. Buchanan for calling this general problem to our attention, although we emphasize that our approach is not always consistent with theirs (in private communications). Cf. also note 34 below.

²⁴ Riker, and Brams, , “The Paradox of Vote Trading,” pp. 1245–1246 Google Scholar.

²⁵ Koehler, “Vote-Trading and the Voting Paradox.” See Appendix II.

²⁶ Riker, and Brams, , “The Paradox of Vote Trading,” p. 1239 Google Scholar.

²⁷ We are grateful to David H. Koehler (private communication) for suggesting the first example and to an anonymous referee for suggesting the second.

²⁸ Coleman, , “The Possibility of a Social Welfare Function,” especially p. 1107 Google Scholar.

²⁹ Riker, and Brams, , “The Paradox of Vote Trading,” p. 1244 Google Scholar.

³⁰ Cf. Buchanan, and Tullock, , The Calculus of Consent (p. 145)Google Scholar: “Potentially, the voter should enter into bargains until the marginal ‘cost’ of voting for something of which he disapproves but about which his feelings are weak exactly matches the expected marginal benefits of the vote or votes secured in return support for issues in which he is more interested.”

³¹ This condition together with the condition for vote trading introduced in the proofs below, is based upon situations in which (under majority rule) all motions will pass. While seemingly restrictive, the same logic underlying the three proofs can easily be extended to “corollaries” assuming that all motions will fail or that some will pass and some will fail. A less cumbersome approach, however, can extend the theorem we have presented rather easily. Let u(x_ij) represent the utility to legislator i of the majority position on issue j and represent the utility to i of the minority position on j (irrespective of the nature of the actual outcomes). Then the theorem we propose is completely general and subsumes not only our special case but all of the possible “corollaries” representing alternative cases.

³² See Riker, and Brams, , “The Paradox of Vote Trading,” pp. 1238ff.Google Scholar; Kramer, , “Sophisticated Voting over Multidimensional Choice Spaces,” Journal of Mathematical Sociology, 2 (07, 1972), 125–180 CrossRefGoogle Scholar; and Ferejohn, “Sour Notes on the Theory of Vote Trading,” California Institute of Technology, Social Science Working Paper Number 41 (June 1, 1974). We are particularly grateful to Steven J. Brams for his comments on this aspect of the present work.

³³ Riker, and Ordeshook, , An Introduction to Positive Political Theory (Englewood Cliffs, N.J.: Prentice-Hall, 1973), pp. 113–114 Google Scholar. Note especially their reference to Murphy, James, The Empty Porkbarrel (Lexington, Mass.: D. C. Heath, forthcoming)Google Scholar, which (according to Riker and Brams) argues that very little logrolling actualy takes place even on public works legislation. Also, cf. Mayhew, David R., Party Loyalty among Congressmen (Cambridge: Harvard University Press, 1966: ch. 6)CrossRefGoogle Scholar on the optimality of intra-party as opposed to inter-play cooperation. And, cf. the comments made in President Gerald R. Ford's inaugural speech to a joint session of Congress on August 12, 1974. Addressing in particular House Speaker Carl Albert (D., Okla.), Ford stated: “I have sometimes voted to spend more taxpayers' money for worthy federal projects in Grand Rapids [Michigan] while vigorously opposing wasteful federal boondoggles in Oklahoma” [cited in Congressional Quarterly Weekly Report, 32 (08 17, 1974), 2209]Google Scholar Also cf. the comments of Martin, and Tolchin, Susan (To the Victor [New York: Vintage, 1972], p. 218)Google Scholar: “… it was no surprise that when Representative Charles Joelson, a New Jersey Democrat, wrote a letter to each member of Congress asking ‘Where can we economize in your district?’ not one reply was returned to his office.” Many members probably thought that the letter was just another example of the humor for which Joelson was known (and to which the first-named author of this article can attest). Further evidence for the claim that legislators may be aware of the paradox of vote trading can be inferred from Asher, Herbert B., “The Learning of Legislative Norms,” American Political Science Review, 67 (06, 1973), 499–513 CrossRefGoogle Scholar. Asher notes (p. 508) that 68 per cent of a sample of 22 freshmen in the 91st House stated that they would be willing to trade votes with a colleague. While the effects of learning behavior do seem to be apparent in the House—more nonfreshmen (81 per cent of a sample of 21) expressed a willingness to trade than freshmen (p. 501)—the comment of a senior member is instructive (p. 503): “Yes, I'd trade votes, but this does not happen often. It is not a specific trade, but more a matter of good will. I never had a specific trade; this happens more often in the Senate.” (On the utility of trading through “good will,” or implicit logrolling, see Appendix II.)

³⁴ Murphy, , “Political Parties and the Porkbarrel: Party Conflict and Cooperation in House Public Works Committee Decision-Making,” American Political Science Review, 68 (03 1974), 179–180 CrossRefGoogle Scholar (first emphasis in original; second emphasis added). We thus find ourselves in the interesting situation of citing a single author on two different sides of the same question (cf. note 33 above). We await publication of his book before pursuing the interpretation of Riker and Ordeshook any further. For similar comments, see Fenno, Robert F. Jr., Congressmen in Committees (Boston: Little, Brown, 1973), pp. 58, 156-159, and 165–166 Google Scholar; and Mayhew, , Congress: The Electoral Connection (New Haven: Yale University Press, 1974), pp. 86–91 Google Scholar. For a somewhat different perspective on the effects of vote trading for porkbarrel projects from that presented in the text, see Ferejohn, , “Sour Notes on the Theory of Vote Trading,” p. 11 Google Scholar. For a statement which is more in accord with our view of the logrolling problem and also treats the problem of transaction costs (cf. note 23 above), see Mancur Olson, Jr., “The Optimal Allocation of Jurisdictional Responsibility: The Principle of ‘Fiscal Equivalence’,” in United States Congress, Joint Economic Committee, 91st Congress, First Session, The Analysis and Evaluation of Public Expenditures: The PPB System, I (Washington: Government Printing Office, 1969), pp. 321–331 Google Scholar. Olson argues (p. 326): “If all mutually advantageous bargains were struck, log-rolling would insure that all collective goods that it was Pareto-optimal to provide would be provided. But … especially where large groups of people are at issue, it will very often be the case that logrolling will not happen, and that there will not be a Pareto-optimal supply of public goods.” He adds (p. 326, n. 14): “In the United States Congress, logrolling probably leads to a greater expenditure when projects of a ‘pork barrel’ type are at issue. In most of these cases the projects are of a tangible, if not monumental type, and a Congressman is more likely to be identified in his district with such a project than with a general tax increase, which could not in any case usually be traced to any one package of local projects.” For an argument similar to that of Olson on government expenditures, see Davis, J. Ronnie and Meyer, Charles W., “Budget Size in a Democracy,” Southern Economic Journal, 36 (07, 1969), 10–17 CrossRefGoogle Scholar.

³⁵ Riker, and Brams, , “The Paradox of Vote Trading,” pp. 1235 and 1238 Google Scholar. Cf. Coleman, , “The Possibility of a Social Welfare Function: Reply,” pp. 1315–1317 Google Scholar.