¹ Hatsell, John, Precedents of Proceedings in the House of Commons (2 vols, London, 1781Google Scholar; I have been able to examine only the four volume edition published by Luke Hansard in London in 1818 and therefore I cite it), Vol. 2, pp. 207–08. This aphorism has had a long, semi-independent existence. Jefferson quoted it in the second paragraph of his Manual of Parliamentary Procedure (written between 1797 and 1801 and reprinted biennially in The Rules of the House of Representatives) and recently Eric Taylor quoted it in the beginning of the second chapter of his excellent essay, The House of Commons at Work (Harmondsworth, Middlesex, 1951)Google Scholar. It might well be pointed out that Hatsell's comment occurs in connection with his discussion of methods of voting, which methods are the subject of the present essay as well. Hatsell further said (II, 208); “If the maxim, ‘Stare super vias antiquas’ has ever felt any weight, it is in those matters, where it is not so material, that the rule should be established on the foundation of sound reason and argument, as it is that order, decency, and regularity, should be preserved in a large, a numerous, and consequently sometimes tumultuous assembly.” If voting, certainly the central activity in a legislature, need not be established on the “foundation of sound reason and argument,” then there is apparently no need whatever for “sound reason” in parliamentary law.

² Edwards, Ward, “Probability Preferences in Gambling,” American Journal of Psychology, Vol. 66 (1953), pp. 349–64CrossRefGoogle ScholarPubMed, reports an experiment in which subjects were asked to choose between bets of differing probabilities. The coefficient of consistence, which is

was in the beginning of the experiment .76 and .73 but became in the end of the experiment .84 and .82 for different kinds of bets respectively. The improvement can, presumably, be attributed to an attempt by the subjects to rationalize their orderings of preference. The remaining inconsistence (i.e., intransitivity) derived, so Edwards suggested, from the fact that the subjects chose bets by different standards. On the one hand, they tried to maximize their winnings (i.e., to choose the bet with the highest expected value). On the other hand, they clearly displayed a preference for bets with a 4/8 probability as against bets with a 6/8 probability, regardless of the expected value of the paired bets. When these two standards of choice came into conflict, the subjects tended to be inconsistent. May, Kenneth O., “Intransitivity, Utility, and the Aggregation of Preference Patterns,” Econometrica, Vol. 22 (1953), pp. 1–13CrossRefGoogle Scholar, reports an experiment with similar results. He offered subjects choices which they might order by three different standards and found a some-what greater degree of inconsistency than Edwards. Examining this result with a mathematical analysis, May proved that, if there are three or more things to be chosen and three or more criteria by which they may be chosen, then it is possible for one chooser to order his preferences intransitively unless one criterion dominates all others. From his analysis, one infers that the primary intransitivity is among the criteria of choice rather than among the ordering of things to be chosen. If the criteria for determining preference can be arranged in a transitive order of dominance, then, apart from human error, preference is transitive. But if the criteria cannot be so arranged, then the preferences of an otherwise unerring chooser may be ordered intransitively. Thus, May has discovered an analogue to the paradox of voting, an analogue which, however, is in a single person rather than a group. To distinguish the two paradoxes, the analogue will here be called a paradox of preference. It may be described thus: Let the criteria, α, β, …, γ, by which a chooser orders his choices be themselves ordered by a relation of dominance, D, such that if aPb with respect to α and if bPa with respect to β and if in fact aPb, then αDβ. So defined, D is a transitive relation, which, however, may calculate to an intransitivity if γ ≧ 3 in just the same way that the paradox of voting may appear in a group. Unlike the paradox of voting, which, as this essay suggests, can be broken only by the crudest practical measures, the paradox of preference exists inside one person and so is wholly under his control: once brought to his attention, he can break it by an analysis of his own emotions, if he chooses to do so. Hence, although the paradox of preference might appear to demonstrate the ineradicable irrationality of human behavior, an irrationality so pervasive that it calls the whole notion of rationality into question, still the fact that this irrationality may be eradicated when raised to the conscious level encourages us to conclude that we may continue to use the transitivity of preference as a standard of rational behavior. (Further discussion of these and other experiments along with an elaborate bibliography may be found in Edwards, Ward, “Theory of Decision Making,” Psychological Bulletin, Vol. 51 (1954), pp. 380–417CrossRefGoogle ScholarPubMed. See also Papandreou, Andreas G., “An Experimental Test of an Axiom in the Theory of Choice,” Econometrica, Vol. 21 (1953), p. 477Google Scholar (abstract).

³ May, Kenneth O., “Intransitivity, Utility, and the Aggregation of Preference Patterns,” Econometrica, Vol. 22 (1953), pp. 1–13CrossRefGoogle Scholar. May argues that preference, P, ought to be regarded as the probability, p, that one of a pair of alternatives, a and b, will be chosen over the other in a particular set of circumstances, E. Thus, he defines preference:

(1) aPb=p(a⃒a, b; E).

He then points out that, on the basis of this definition, it is false to write:

(2) for any E, if aPb and if bPc, then aPc.

Statement (2) can be translated into:

(3) for any E, if p(a⃒a, b) > p(b⃒a, b) and if p(b⃒b, c)p(c⃒b, c), then p(a⃒a, b) > p(c⃒b, c).

and statement (3) may easily be false because p(a⃒ a, b) and p(c⃒ b, c) are calculated with respect to the non-identical sets, {a, b} and {b, c}. But preference may be, and in most circumstances of real life undoubtedly is regarded as a choice of one element of several. So regarded preference is defined, not as in statement (1), but thus:

(4) aPb ≡ p(a⃒a, b, …, n; E).

Using statement (4) instead of (1), statement (2) is translated into:

(5) for any E, if p(a⃒a, b, …, n)> p(b⃒a, b, …, n) and if p(b⃒a, b, …, n) > p(c⃒a, b, …, n), thenp(a⃒a, b, …, n) > p(c⃒a, b, …, n).

And statement (5) is valid by the rule of syllogistic inference. Hence, if the set of possible choices is identical for all the choices in the chain, then P is a transitive relation. May's argument is valid, therefore, only in the case of a series of binary choices and, as has often been observed, truly “either-or” situations seldom arise in real life.

⁴ (New York, Wiley, for the Cowles Commission, 1951.) See also May, Kenneth O., “A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision,” Econometrica, Vol. 20 (1952), pp. 680–84CrossRefGoogle Scholar.

⁵ See also McGarvey, David G., “A Theorem on the Construction of Voting Paradoxes,” Econometrica, Vol. 21 (1953), pp. 608–10CrossRefGoogle Scholar.

⁶ Black, Duncan, “On the Rationale of Group Decision Making,” Journal of Political Economy, Vol. 56 (1948), pp. 23–34CrossRefGoogle Scholar. See also two other important essays by Black on the subject of transitivity of preference: “The Decisions of a Committee Using a Special Majority,” Econometrica, Vol. 16, (1948), pp. 262–70CrossRefGoogle Scholar; and “The Theory of Elections in Single Member Districts,” The Canadian Journal of Economics and Political Science, Vol. 15 (1948) pp. 158–75Google Scholar.

⁷ If m₁, m₂, …, m_n motions are ordered in numerical sequence of the subscripts and if each m_i is paired with each m_i that appears after m_i in the sequence, then m₁ appears first in n – 1 pairs, m₂ is first in n – 2 pairs, …, m_n–1 is first in one pair. The sum, r, of the pairings is the sum of the terms of an arithmetic progression: r = 1 + … + … (n–2) + (n – 1) = ½12; (n – 1) (1+n–1) = n(n – 1)/2.

⁸ “On the Rationale of Group Decision Making,” passim.

⁹ A note of warning is appropriate here: The kind of irrationality discussed in this essay has nothing whatever to do with the behavior of individual persons. It is, of course, true that individual Congressmen may order their preferences intransitively. A multitude of studies in social psychology and political behavior have long since made us fully aware of the potentialities of unreason in every man. Despite the revelations from the time of Freud onward, we have nevertheless held onto our eighteenth century faith in our ability to create rational institutions. What the paradox of voting so devastatingly reveals, however, is that our institutions of vote counting may have implicit defects. Even if we charitably assume, as I do in this essay, that the preferences of individuals are transitively ordered, still the institutions of voting may bring about intransitive results. Since this is a study of just this aspect of institutions and not a study of persons, it is only concerned with how votes are counted. Specifically, it is not concerned with how people make up their minds. Whether they think carefully to a decision or simply vote on a nod from a leader or lobbyist, or on a trade for other votes on other matters, their internal thoughts are a matter of complete indifference in this analysis, for it is concerned with external behavior, not motives.

¹⁰ Although in this classification only events occurring in the Committee of the Whole are considered, an analogous classification can be constructed for events occurring when a bill on the House Calendar is read for amendments, or when the House considers Senate Amendnents to a House bill, or when the Senate considers a bill reported from committee, or when it considers House amendments to a Senate bill.

¹¹ Arrow, op. cit., p. 10.

¹² It may be argued that there is no irrational behavior if the bill as a whole is accepted in spite of an intransitivity over one clause. Such an argument, however, denies any significance to the whole process of reading for amendment.

¹³ Black, , “On the Rationale of Group Decision Making,” p. 33Google Scholar. I have computed, by a method to be reported elsewhere, the proportion of possible results of adding preferences which are transitive or partially transitive. When n = 3, 3/4 of the possible outcomes are transitive. When n = 5, only 5/16 of the possible outcomes of addition are such that one choice is preferred to all others.

¹⁴ See Congressional Record, Vol. 98 (1952), pp. 4713–31Google Scholar.

¹⁵ Possibly of course, the intransitivity, if it existed, was not a paradox of voting, but several paradoxes of preference. In the ensuing discussion of this section it will be assumed that the members themselves were rational and that this is an instance of the paradox of voting. At the end of section IV, the event will be briefly discussed with this assumption removed.

¹⁶ The potentialities of deceit in the method of successive elimination can be visualized from the Soil Conservation incident. Defining the “least popular” choice as the one able to defeat the fewest of the others, we eliminate the O'Toole amendment. Dropping the fifth row and fifth column of Figure 1 results in Figure 2. Since no one motion in Figure 2 can beat all the rest, the Javits amendment next must be eliminated, whereupon the result is Figure 3. This is identical with Figure 2 except that the second row and second column are eliminated. Since no one motion in Figure 3 can defeat all the rest, it is again necessary to eliminate one. Assuming that the least popular choice in Figure 3 is the one with the smallest sum in its row, then the original paragraph must be eliminated. Then the Andersen amendment wins. But the Whitten supporters, foreseeing this result, can ensure victory for the original paragraph (their second preference) by voting against themselves at the stage represented by Figure 3. In effect they reorder their preferences thus: 1P4, 4P3, and hence, 1P3. This results in the matrix of preferences of Figure 4. Thus, by falsifying their preferences, they can profitably change the outcome—assuming, of course, that other factions do not also falsify their preferences.

Similar examples are discussed in Arrow, op. cit., pp. 80–81. This is substantially the maneuver described in Majundar, Tapas, “Choice and Revealed Preference,” Econometrica, Vol. 24 (1956), pp. 71–3CrossRefGoogle Scholar.