Voting Systems, Honest Preferences and Pareto Optimality Author(S): Richard Zeckhauser Source: the American Political Science Review, Vol

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Voting Systems, Honest Preferences and Pareto Optimality Author(s): Richard Zeckhauser Source: The American Political Science Review, Vol. 67, No. 3 (Sep., 1973), pp. 934-946 Published by: American Political Science Association Stable URL: https://www.jstor.org/stable/1958635 Accessed: 27-05-2020 22:08 UTC

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This content downloaded from 206.253.207.235 on Wed, 27 May 2020 22:08:36 UTC All use subject to https://about.jstor.org/terms Voting Systems, Honest Preferences and Pareto Optimality*

RICHARD ZECKHAUSER

Harvard University

"In a capitalist democracy there are essentially A Desirable Voting Scheme two methods by which social choices can be made: In the market, an individual indicates his pref- voting, typically used to make 'political' decisions, erences through his purchases and sales. In and the market mechanism, typically used to make general, these tracings are insufficient to define an 'economic' decisions."' To economists, the freely individual's complete preference mapping, but functioning competitive market mechanism has from the standpoint of efficiency this creates no alluring structural properties.2 If each individual difficulties. With merely the limited amounts of follows his own self-interest in making his market information fed to it, the market leads to a decisions, an efficient outcome is achieved. Pareto-optimal outcome. (It should be noted that What of the possibility of constructing a voting no individual has an incentive to disguise his mechanism that would achieve a collective politi- preferences.) cal outcome that would have desirable properties A voting system in theory could operate in the that parallel those of the competitive market out- same decentralized fashion. Each individual come? Can a voting system be impersonal and would feed in some indication of his preferences, decentralized, allow each individual to act on his and the processing mechanism would employ this own behalf, and guarantee that a Pareto-optimal information to yield an outcome. To see whether outcome will be achieved? such a procedure should be established as an * Kenneth Arrow, Robert Klitgaard, Pamela Memi- ideal, we must determine how it can perform along shian, Howard Raiffa, Thomas Schelling, Michael a number of significant dimensions. Spence, Milton Weinstein, and a referee for the Ameri- can Political Science Review provided me with helpful Major Characteristics of Social comments. This research was supported by NSF grants GS-28626x and Gr-58. Decision Procedures I Kenneth Arrow, Social Choice and Individual Values The flow diagram below provides the context (New York: John Wiley and Sons, Inc., 1964), p. 1. for the major topics discussed in this paper. (At a 2 It is with some discomfort that I leave aside the problem of distribution in this paper. The issue is at- later juncture, it will serve to summarize its princi- tacked directly in a later effort, "Risk Spreading and pal results.) These topics are indexed A through F. Distribution," Kennedy School of Government, Discus- Social decision problems are first classed as sion Paper No. 10 (August, 1972). There I argue that a economic or political. Economic decisions are profound belief in the efficacy of the outcome of a perfect market gives one insights into distributional ques- handled by the market mechanism. Under the ap- tions. One can consider the problem of drawing the propriate competitive conditions, the market out- social contract equivalent to that of a group of future come will meet some preassigned standards of ac- citizens starting in some initial position of equal poten- ceptability. The principal standard to be met is tial (they face the same lottery on future possibilities). efficiency. The measuring rod for this standard is They draw up a contingent claims agreement, the mag- nitudes of payment to depend on their future fortune in Pareto optimality. securing endowments of goods and capabilities (that is There are myriad situations in which the very the states of the world). In retrospect, we may decide restrictive competitive assumptions are not met. that it is our duty to redistribute in the manner that For these situations, much the subject of current would be prescribed by the hypothetical agreement that antedated our present position. For a philosophically economics debate, the unhindered play of the compatible, but perhaps policy inconsistent view, see market will not lead to an attractive outcome. John Rawls, The Theory of Justice (Cambridge, Mass., Perhaps the establishment of new areas of prop- Harvard University Press, 1971). erty rights, or innovative additions to present 'Thomas Schelling has commented at great length, interpreting and questioning the significance of Pareto market mechanisms will set things right. What- optimality as a criterion for social choices. He states, ever, this is not material to the present discussion. 'the significance of failure to achieve Pareto-optimality These situations would be of import for this merely means that the situation is improvable . . . it is paper if they were treated as political decisions. the degree of improvability that determines how impor- They then would be carried down the lower tant the nonoptimality is. Pareto optimality is not like virginity or justice. . . . Small departures are of small branch of the flow diagram, where they could interest, large departures are of large interest." Personal serve as further exemplars for the analysis of that communication, July 12, 1972. branch. Economists have long recognized that Pareto optimality by itself is a hopelessly incomplete guide to norma- Political Decisions and the tive and descriptive considerations of social choice. An Individualist Approach indication of this author's response is given by the conclusion to this paper. In evaluating outcomes in the economic arena, 934

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Market decentralized, ordinal, unique, _ ___ X

U mechanism |[ . / ~~~~~~~~~~~~~~~~~~~~~~~~ ~ ~ ~~~~~~~~Dictatorial |Inefficient Democratic ]

Cr\ r D~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~effcet

Context and Summary Diagram we rely on consumer sovereignty. Each individual success. Yet Kenneth Arrow, in a much celebrated is entitled to judge what is best for himself, work, and has demonstrated that there is no general these individual judgments form the basis for procedure as- to pass from individuals' preferences sessments of social welfare. This paper carries to social orderings that meets certain standards of over this individualist ethic to the evaluation of attractiveness.4 The power of the Arrow result de- all social choices, including those that are con- rives from the fact that the standards he imposed ventionally considered to be of a political nature. were both innocuous in appearance and intui- Thus individuals' preferences form the basis for tively appealing. the social decisions considered here. In social de- Arrow sets forth four standards. In unrigorous cisions more than one individual is involved. language, they are: (1) The procedure must in- Together, social decision and consumer sover- clude all logically possible combinations of indi- eignty exclude from consideration any form of viduals' orderings. (2) It must lead to Pareto- paternalistic rule. optimal outcomes. (3) The choice between any two The ground rules then are set. We are searching alternatives cannot be influenced by the presence for a procedure that enables us to proceed from or nonpresence of a third alternative. (4) No indi- information on individuals' preferences to a social vidual can always secure his choice regardless of outcome that meets some conditions of accept- the preferences of others. A later expositor has ability. The ultimate objective is to discover a de- provided a mnemonic for these properties:' un- centralized procedure, one which consciously or restricted domain, U; Pareto principle, P; inde- automatically takes the (perhaps incomplete) in- pendence of irrelevant alternatives, I; nondictator- formation provided by individuals' ballots and ship, D. Dealing with the simplest case of interest yields an acceptable social outcome. (at least two individuals and at least three alternatives), the famous Arrow result is that there is no Centralized Decision Procedures social welfare function that satisfies U, P, I, and D. For purposes of comparison, I shall begin with Let us stop for a moment to get our bearings a look at the success or nonsuccess of centralized in the flow chart. We should like all of our social procedures. With centralized decision, a single de- decisions to apply over an unrestricted domain. cision authority with accurate knowledge of in- Condition U is imposed at all nodes in the flow dividuals' preferences selects an outcome. diagram and throughout this paper. Next observe

Centralized Decision, Ordinal Preferences (A) 4 For the conditions, see K. Arrow, Social Choice, pp. 22-33. At first glance it would seem that centralized 5Amartya K. Sen, Collective Choice and Social Wel- decision procedures hold out promise of great fare (Holden-Day, Inc., San Francisco), pp. 41-42.

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that the "Arrow result" box lies on the ordinal this paper. Thus far we have been merely setting branch. The independence of irrelevant alterna- the stage and filling in the complementary boxes. tives condition, I, requires that the centralized Decentralized Decision Procedures decision procedure rely on no more than individuals' ordinal preferences. The impossibility of The remainder of this paper will follow out the simultaneously satisfying all four Arrow condi- implications of and possible standards of per- tions is indicated by the paths of the arrows ema- formance for a variety of decentralized decision nating from the box. They are stopped either by procedures. It will index them, as it did the cen- inefficiency (nonsatisfaction of P) or dictatorship tralized procedures, by the capital letters in the (nonsatisfaction of D). flow diagram. The analysis that follows is largely nonmathe- Centralized Decision, Cardinal Preferences (B) matical and thus almost by necessity, unrigorous. Somewhat different formulations of the central- Other theorists, no doubt, will be able to reconsti- ized social choice problem will lead to success tute these results in more formal, perhaps stronger stories. A number of analysts, some of whom sub- contexts. The purpose of this paper is to sketch stantially predate Arrow, have proposed that ac- some new methods and results, and the framework ceptable centralized decision procedures can relax in which they apply. one or another of the conditions he imposed. With centralized decision, each voter's prefer- Perhaps the most frequently cited suggestion is ences are assumed to be known by the social that the independence of irrelevant alternatives choice mechanism. With decentralized decision, condition, I, be softened or eliminated. The ob- knowledge of preferences is gained only indirectly. jective underlying this relaxation is to secure some Under decentralized procedures each voter fills cardinal measurement on individuals' intensities out a ballot; he is assumed to follow as best he can of preferences. his own self-interest. These ballots are processed By way of example, more than two centuries by some mechanism to produce a social choice. ago, Borda proposed the rank-order method of Depending upon the particular choice mechanism election, a method now frequently employed to that is in operation, and the situation in question, establish national rankings of college athletic a voter's ballot may or may not be a complete and teams. Under this system, preassigned weights are truthful indicator of his preferences. This obser- given to voters' first, second, third choices, etc. vation provides a hint about where we are going. Sum each candidate's scores over all voters: high- A whole new class of problems arises when we est aggregate score wins. The procedure is neither move from centralized to decentralized decision dictatorial nor inefficient.' Reference to the flow procedures. Essentially they revolve around get- diagram shows that its emanating arrow pene- ting individuals to reveal their preferences. The trates to the rightmost rectangle. unfortunate result, as we shall see later, is that the Are rank-order systems ethically appealing? very structure of social choice schemes that elicit Neither economic theory nor any other axiomatic true preferences on ballots leads to inefficiencies in system can give us any insight into this question. outcomes. Each analyst of social choice must decide for him- An abstract representation of a decentralized self whether in making the choice between A and choice procedure will facilitate discussion. There B, it is of concern how each fares with the voters are not only with respect to the other, but also with respect to C. m voting individuals indexed over i, and We need dwell no further on the appropriate n alternatives indexed over j. constraints for centralized procedures; decentral- An individual, i, fills out xi as the ballot indicating ized procedures represent the principal focus of his preferences. The voting system processes the 6 Following Pareto optimality, efficiency requires that received ballots, X= (xi, * , Xi, , Xm) to a switch away from the selected candidate would lower arrive at a social choice. A social choice is the welfare of at least one voting individual. If the se- represented by the probability vector P(X) = lected candidate had the highest total score, then there (p1(X), . . - , pj(X), - . , pn(X)) whose elements could be no other candidate who received an equally indicate the probability that an alternative is high individual score from each voter. A switch from the man with the highest total would therefore involve selected. In the language of mathematics, a voting a loss for at least one voter. mechanism maps X into P. Special difficulties arise if lotteries on alternatives are Most of the familiar voting procedures are permitted; and rank-order schemes need not lead to deterministic. Given X, a single alternative is Pareto-optimal outcomes. See Richard Zeckhauser, "Majority Rule with Lotteries on Alternatives," The selected with certainty; one element of P is unity, Quarterly Journal of Economics, 83 (November, 1969), the rest zero. 696-703. Later sections of this paper discuss voting pro-

This content downloaded from 206.253.207.235 on Wed, 27 May 2020 22:08:36 UTC All use subject to https://about.jstor.org/terms 1973 Voting Systems, Honest Preferences and Pareto Optimality 937 cedures that may lead to randomized outcomes. To conduct an analysis of situations of this sort These outcomes are analyzed and evaluated be- requires that individuals define what are called fore the final results are known, before the lottery von Neumann-Morgenstern utilities for each pos- is conducted. Randomized outcomes may lead to sible outcome.9 The utility values for individual i gains in efficiency. More important for the pur- are indicated by the via's; they are normalized with poses of this paper, they may help insure that in- a value of 1 assigned to the most preferred out- dividuals mark their ballots with their true come, and 0 to the least. preferences. The voting mechanism, however, may seek to To evaluate probabilistic outcomes, a voter elicit something much less than the individual's must be able to evaluate outcomes in a non- full cardinal preference structure. Majority rule deterministic world. Even if the world is deter- procedures, for example, merely ask the individual ministic, there may not be full information, in which is his most preferred alternative. which case the voter may be making a decision It is useful to classify decentralized decision under uncertainty. Fortunately, there is a well- procedures on a couple of dimensions. Decentral- developed body of literature on prescriptive de- ized procedures, like those that are centralized, cision making under uncertainty. We shall follow can rely on cardinal or merely ordinal indications the major track of that literature.7 of individual's preferences. The two alternatives A rational decision maker under uncertainty is are indicated in the first branching in the flow assumed to assign a utility value to each possible diagram. outcome. The paradigm decision problem requires that he select one among alternative lotteries Unique Voting Schemes where each lottery is described by the probability The second branching reflects a consideration vector it offers for the different outcomes. In that does not arise with centralized procedures; choosing among lotteries, the rational decision for with them, individuals' preferences are as- maker selects the one that offers him the highest sumed to be known. With decentralized proce- expected utility value. dures, however, each individual ballots in his own Notice that judgments of welfare are made here self-interest. This raises the possibility that with on an ex ante basis, before lotteries are resolved. some of these procedures and some situations an Thus certain social choices will be ruled non- individual working in his own behalf will find it Pareto optimal, even though they will eventually desirable to disguise his preferences. This may be lead to situations that considered in isolation are unfortunate. It may be important for the appa- Pareto optimal. An example displays the proper- ratus that is processing the votes to be able to ties of this before-the-fact evaluation procedure. determine the exact preferences of an individual Two risk-averse individuals each have $10,000 in from his markings on the ballot. the bank. They are given the opportunity to gam- If an individual will mark his ballot the same ble their total savings on the flip of a coin. Both way for two or more preference structures, then would decline as they both prefer their present the ballot is not unique. situation to participation in the lottery. If they were forced to participate in the lottery, it would Definition: A voting system is called unique if be a non-Pareto-optimal situation. This despite a voting individual will mark his ballot in a the fact that if the lottery were conducted and one different manner for every possible configura- man ended up with $20,000 and the other with tion of his preferences. nothing the outcome would be Pareto-optimal.A Decentralized Decision, Ordinal Preferences, I For a good introduction to this subject, see Howard Unique (C) Raiffa, Decision Analysis (Reading, Mass.: Addison- If merely ordinal preferences are required, it Wesley, 1968). 'The ex-ante expected utility approach can be em- would seem the task of constructing a unique ployed to make evaluations of social welfare. See Rich- scheme would be somewhat simpler. In many real ard Zeckhauser, "Determining the Qualities of a Public world situations things are simpler still. Individ- Good-A Paradigm on Town Park Location," Western uals are asked to indicate only a small portion of Economic Journal, 11 (March, 1973), 39-60. For an application of this principle in the social decision con- their ordinal preference structure, their first-place text, see Richard Zeckhauser, "Majority Rule With Lot- choice. This is the customary procedure in the teries on Alternatives." Peter Fishburn formalizes this United States when we are selecting a single indi- argument in "Lotteries and Social Choices," Journal of vidual for an office: the presidency, a mayorship, Economic Theorv, 5 (October, 1972), 189-207. Peter C. the local dog catcher. Ordeshook employs the principle to define candidates' mixed strategies as lotteries on social states should they get elected. See "Pareto Optimality in Electoral Com- 9 Appropriate historical attribution would give much petition," The American Political Science Review, 65 of the credit for this utility concept to Frank P. (December, 1971), 1141-1145. Ramsey.

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A traditional majority-rule decision procedure tic. If an alternative j is marked as first choice on will lead to honest revelation of preferences if k ballots, pj will equal k/m. That is, a candidate there are but two alternatives. But when the com- will have a probability of election that is propor- petitors number three or more such honesty can tional to the number of first place votes he re- no longer be assured. For example, if in a three- ceives. The pj's will take on fractional values (ex- man race your preferred candidate A is likely to cept in the special case where there is a unani- receive little support, you may choose to vote for mous first choice); the system will not be deter- your second choice rather than waste your vote ministic. on a lost cause. If your ordering is A-B-C, you We shall see here, and again later with cardinal will vote for B. You would vote the same way if preferences, that the consideration of probabilistic your ordering were B-A-C or B-C-A. The voting outcomes is essential if a voting scheme is to be procedure is not unique, but this should have been unique. Consider the voter's behavior in the ran- expected. There are six possible orderings and dom dictator model. He assigns utility values VA, only three possible ballot markings. There is no VB, and ve to the three candidates. Why should he way to have a unique marking for each ordering. vote for his favorite, A? Assume that instead he What is disturbing is that the system is not unique had voted for B. This would reduce the prob- with respect to the first-place preference, and it is ability that A gets elected by 1/m; it would in- information on that preference that the system crease the probability that B got elected by 1/m. was attempting to solicit. Considerations such as The net change in the expected utility to the voter these lead us to a modified concept of uniqueness, would be (l/nM)(vB-VA). By assumption, VB < VA; one that applies in a more restricted context. the voter would lose by switching; he also would lose if he decided to choose C. The key element Definition: A voting system is called unique here is that when the individual switches his vote with respect to a property of a preference the probability gain for the switched-up candidate ordering if a voting individual will mark his is just equal to the probability loss for the ballot in a different manner for every possible switched-down candidate. This implies that the configuration of that property. probability of election of the third candidate re- mains constant when a vote is switched between For example, a voting system is unique with the other two. This absence of third-candidate respect to first-place preferences if a voting indi- effect is essential. Otherwise, the decision on vidual will mark his ballot differently depending whether to vote for A or B might depend on how upon which candidate he likes best.'0 Majority the vote affects the probability that C gets rule does not qualify. elected.'1 A formal generalization of this argu- Does a voting system exist that is unique with ment requires a new concept. respect to first-place preferences? Yes, there is a type of system that meets this requirement. It is Definition: A voting system will be said to be most easily understood with the aid of a mechani- probabilistically linear if it is of the following cal example that has the required intrinsic proper- form. Let S be the matrix of votes, where ele- ties. ment sij= 1 if voter i submitted his ballot for j, and 0 otherwise. The vector P that gives The Random Dictator System: A Voting System the probability that each candidate is elected That is Unique with Respect to First-Place is calculated P = wS+e. Here e is a row vector Preferences giving exogenous contributions to the election probabilities, and w is a row vector of Each individual writes the name of a candidate on a ballot. The voters' ballots are collected and non-negative weights summing to 1-E i e3 placed in a revolving drum. After shuffling, a bal- Theorem I: If a decision procedure that solicits lot is chosen at random. The name on the chosen individuals' first place preferences is to be ballot is the elected candidate. In effect, the voter unique, it must be probabilistically linear. whose ballot is chosen dictates the outcome, hence the name of the system. (Matters would be just the Note that probabilistic linearity is equivalent to same if a voter were first selected at random, and the following property: For any voter i and for then the selected voter were asked to choose the any pair of candidates D and E, switching his vote elected candidate.) from D to E must increase the probability that The random dictator system is not determinis- candidate E gets elected by the amount wi, de-

10 If we extend this concept to the market setting, "Say VA is only slightly greater than VB, but much we discover that market procedures are unique with greater than vc. If C had less probability of being respect to the information required for efficiency: in- elected if the individual voted for B rather than A, dividuals' marginal rates of substitution and producers' it might be reasonable for him to vote contrary to his marginal rates of transformation among valued goods. true preferences.

This content downloaded from 206.253.207.235 on Wed, 27 May 2020 22:08:36 UTC All use subject to https://about.jstor.org/terms 1973 Voting Systems, Honest Preferences and Pareto Optimality 939 crease the probability that D gets elected by the third, etc., with q>r>s. The selection procedure amount wi, and leave all other election probabili- is random as before. ties unchanged. Thus we find that only variants of the random The proof of the theorem can be demonstrated dictator system will elicit ballots unique with re- with a three-man example. For notation, let spect to individuals' (portions of individuals') AD to E represent the change in the probability ordinal preferences. How well does the random that F is elected if voter i switches his vote from dictator system perform? An example tells the D to E. Consider the previous example where the sad news. present vote is for A. Necessary conditions for With the values shown in the box, Voter 1 will probabilistic linearity are that mark his ballot with A and Voter 2 will inscribe a C. A B A C AA to B = - AA to B = AA to C = AA to C Utility Valuations in Two Voter World

VA VB VC and that Voter 1 1 .90 0

C B Voter 2 0 .90 1 AA to B = \AA to C = 0. The resulting social choice when the random dictator system is employed will have the probability Given that the sum of the mutually exclusive prob- vector P=(.5, 0, .5). The expected utility values abilities of election must add to 1 as they did when for each of the voters will be .50. But this outcome the vote went to A, there are the conservation is not Pareto optimal. Both voters would have equations preferred the social choice P= (0, 1, 0). A B C The difficulty is apparent. The random dictator AA to B + AA to B + AA to B = 0 system does not take into account intensities of and preference. (Indeed, no information is gathered on second and further preferences, but even if it A B C were, this would not cover the intensities problem. AA to C + AA to C + AA to C = 0. With the same ordinal patterns, both voters could To see that AC to B must equal 0, assume it were have assigned a value of .10 to VB. In that case the negative. The voting system must handle any set social choice of the random dictator device would of preferences. Thus, let the voter strongly dislike be Pareto-optimal, as would be any lottery on C, say VA- vc= k, where k is a large number, and A and C.)12 let him slightly prefer A to B, VA - VB = E. Despite To sum up, it is interesting to observe that there this preference, the voter will switch his ballot to exists a class of unique ordinal voting schemes."3 B, as his net utility gain will be -AC to B(k -) 12 For further discussion of this matter, see R. Zeck- a positive number. Keep the same ordinal prefer- hauser, "Majority Rule with Lotteries on Alternatives." ences, but let the gap between VA and VB be k, 13 This paper is addressed to situations where a single with the gap between B and C being the minimal E. outcome is selected, and all individuals vote to help Then the individual will choose A over B, the determine that outcome. The model does not apply, for example, to the selection of a legislature. (With utility gain once again being - AC to B(k -E). a stretch of the imagination, we might construe voters Thus, unless AcA to B =0, the voter may ballot to be in choosing among all possible legislatures, but two different ways, despite the fact that his ordinal usually an individual voter ballots to help determine preferences are the same in both instances. This only a small part of that body.) Strict proportional representation is the closest (but still imperfect) leg- contradicts the assumption that the voting system islative equivalent to the random dictator system. The is unique. Therefore ACA to B = 0 as claimed. number of seats secured is proportional to the number An identical argument shows that zB to C=O0 of votes received. These results together with the conservation equa- Proportional representation will guarantee honest tions show that A to B -AA to B and AA to C balloting if two conditions are satisfied. (1) Each voter's utility for the legislature must be an additive A -A to C- To show that A to B must equal separable function of his utilities for each elected rep- A to C, merely apply the argument above to show resentative. (Otherwise, his preferred vote might de- that whether you ballot for B or C cannot in- pend on the candidates others were voting for.) (2) fluence the probability of election for A if a voting All ballots cast in the first round must count in the determination of the final legislature. This requires that system is to be unique with respect to ordinal representation be broken down into arbitrarily small preferences. The conditions of the theorem are units, and that there be no elimination of candidates. thus demonstrated. (Condition (2) can be violated if there are strict party This theorem and the random dictator system interests, and if each voter is indifferent among the candidates of his party.) See Mark Thompson and can be extended to deal with further places in Richard Zeckhauser, "Proportional Representation," voters' ordinal rankings. Provide each voter q bal- mimeo (1971). A referee suggests that Kramer also lots for his first choice, r for his second, s for his has unpublished work on this subject.

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We now know that such schemes perform terribly that enables me to achieve my other results. The inefficiently. Leave aside the adverb, and this reader may come up with other definitions that lesson could have been derived directly from the serve equally well. Arrow results. Dictatorial Outcomes-What They Are and Decentralized Decision, Ordinal Preferences, Why They Should Bother Us Nonunique (D) Most discussions of social choice are concerned This next class of social choice procedures with the selection of a candidate in a single elec- merits discussion primarily because it is the one tion, or an outcome in a particular set of circum- most frequently in use in the United States. Ma- stances. Given the restricted domain of such a jority and plurality rule both fall under this cate- social decision, it might be perfectly acceptable to gory. The Arrow results tell us that these schemes reach an outcome that catered heavily to one in- cannot meet his four standards either. dividual's or group's preferences and little or not One difficulty with these procedures, as is well at all to those of others. (Surely, this provides the known, is that they lead to intransitive choice pat- normative justification for such practices as log- terns for certain combinations of individuals' rolling.) preference orderings. Transitivity is required if we Other social decisions can be of much greater are to have a social ordering. The only way around consequence. Indeed at the most general level, we the problem is to rule out patterns of individuals' can conceive of a grand social decision as selecting preferences that do not satisfy the single-peaked- a single point from the entire space of social states. ness condition." To exclude in this way would It is with reference to momentous social decisions be to violate Arrow's condition U. of this sort that we impose distributional require- Now that we are working on a decentralized ments on the outcomes of social choice processes. basis, further problems may arise. The vote We should like to observe outcomes that demon- processing mechanism may not even be able to strate that the preferences of more than one indi- divine individuals' true preference orderings. It vidual have been taken into account. turns out, fortuitously, that single-peakedness is Even if a series of social choices is made sequen- also sufficient to eliminate any incentive for a tially or in a variety of separate arenas, we might voter to disguise his preferences. Getting rid of conceive of the combination of the voters' ex- one problem automatically eliminates the other. pressed preferences on all issues and the social Whenever the structure of voters' preferences is choice mechanism as selecting one point in the such that majority rule can be guaranteed to work space of social states. Thus even if a great number on a centralized basis, it can be guaranteed to of social choices is to be made on a one by one work in a decentralized format as well. basis, we should like to see the total social outcome reflect the fact that all individuals have par- Standards for Decentralized Schemes ticipated in the selection process. One lesson we have learned thus far is that One indication that a group had been excluded ordinal preferences do not provide sufficient in- from positive consideration in the social choice formation to enable social choices to meet Ar- procedure would be that it selected an outcome row's standards. For centralized procedures, we that was most unattractive for that group. A more saw that information on cardinal preferences is precise understanding of what is meant by un- sufficient to guarantee an efficient, nondictatorial attractive is given by the term Pareto-terrible. outcome. The key question for the remainder of Definition: An outcome is called Pareto-terrible this paper is whether we can find decentralized for a group of individuals if it is a strictly procedures that perform satisfactorily when the constrained minimum: if there exists an out- balloting system elicits cardinal indications of the come that is preferred by all members of the voters' preferences. group, and if there is no outcome that is Satisfactoriness, like beauty, is in the eyes of the worse for some member without being better beholder. My criteria, consistent with the earlier for some other member. analysis, are Pareto optimality and nondictator- ship. The first concept is unambiguous; the latter An outcome is Pareto-terrible for a one-man is subject to a variety of interpretations. I shall group, an individual, if it is his least preferred out- provide one definition of dictatorial outcomes come. With lotteries permitted there will always 14 See A. Sen, Collective Choice, pp. 166-168 for a be outcomes that are not Pareto-terrible for any discussion of single-peakedness where individuals' pref- reference group. erences are defined over a single dimension. There is I shall employ the Pareto-terrible concept as a no equivalent to the single-peakedness criterion if two or more dimensions enter individuals' preferences; cycli- cornerstone of my definition of dictatorial out- cal majorities will be the rule. comes.

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Definition: A voting scheme is dictatorial for a tween two situations which require different out- set of voting individuals preferences if it comes to be Pareto-optimal. selects an outcome that is Pareto-optimal for The 1, 0, 0 and 0, 0, 1 vectors are ruled out be- one nonempty subset of individuals and is cause they are dictatorial. They allow one voter Pareto-terrible for the complementary non- to get his first choice at the expense of the other empty subset of individuals.'5 who receives the outcome that for him is worst. This would be particularly objectionable if, as is With this definition in tow, we can proceed on quite possible, the voting scheme were selecting a our search. Exposition is simplest if we step single point from the space of social states, i.e., slightly out of order to the bottom of the Context if all social decisions were being simultaneously and Summary Diagram and address first the resolved."6 question of honest balloting. To achieve Pareto optimality, any voting scheme that is nondictatorial for this set of prefer- Decentralized Decision, Cardinal, Nonunique (F) ences must be able to distinguish between FIRST Nondictatorial, Pareto-optimal social choices and SECOND configurations of individual I's pre- cannot be made unless, in general, the voting ferences. An equivalent "counterexample" demon- scheme is unique with respect to individuals' car- stration can be arranged for any indistinguishable dinal preferences. In particular instances non- pair of individual l's preferences. The three- uniqueness may not matter. For every nonunique alternative case covers all essential complexities. ballot marking for individual 1, however, there First consider the case where individual 1's in- will be a set of preferences for individual 2 such distinguishable configurations have different alter- that Pareto optimality cannot be assured. natives as the most preferred. Let individual 2 be A three-candidate example will reveal the dif- indifferent between the two sometimes-most- ficulties. Assume that individual 1 marks his bal- preferred-by-i alternatives, with the third alterna- lot x' whether his preferences are (1, .60, 0) or tive less attractive. Which of two alternatives is (1, .70, 0) for candidates A, B, and C respectively. Pareto-optimal depends on which of individual l's Individual 2 has preferences (0, .35, 1). If indi- preference configurations applies; there is no way vidual l's FIRST set of preferences hold, candi- to tell. date B will have zero probability for all Pareto- Matters are a bit more complicated if individual optimal choices. With individual l's SECOND 1's indistinguishable orderings rearrange his sec- preferences, all Pareto-optimal probability vectors ond and third candidates or merely change their have either A or C receiving zero probability.

Pareto Optimality of Different Probability Vectors Configuration of Probability Vectors Individual l's (A+ indicates positive probability) Preferences 1,0,0 0,1,0 0,0,1 +, +, + +, +,0 +,0, + 0, +, + FIRST yes no yes no no yes no SECOND yes yes yes no yes no yes

Except for the first and third listed vectors, no probability vectors arerelative scores.Pareto-optimal To construct the counterexample for both of individual 1's possible preference configurations. first normalize the two individuals' utilities to 1 The problem is simple. The mechanism that and 0. Add together l's and 2's utilities for each processes the votes is unable to distinguish be- of the three candidates. The candidate with the lowest total must receive zero probability in any '5The concluding theorem of the paper employs the nondictatorial Pareto-optimal outcome. (If the dictatorship concept negatively. It states that no system can simultaneously be nondictatorial and at the " On a less grandiose level, there are a number of same time guarantee that other desirable standards are ways that single social-choice decisions can be formu- met. The more restricted the definition of dictatorship, lated to deal with a large number of issues. Issues can then, the stronger my final result. This result would be compounded. The issues at hand might be whether still be achievable if the term "Pareto-optimal" in the Smith or Jones should be mayor and whether or not above definition were replaced with "unanimously the new high school should be built. This can be most preferred." The argument for the definition in viewed as a single decision with four possible out- the text is that it captures more of what we may comes. Alternatively, we might amalgamate issues by mean by dictatorship in a general context. employing a lottery. A coin will be flipped to see It should be noted that Pareto-terrible is not a per- whether a voting mechanism will select a mayor or fect counterpart to Pareto-optimal. There are situa- decide on a high school. Here too there are four pos- tions where all outcomes are Pareto-optimal and none sible outcomes, each composed of two contingent de- Pareto-terrible. The converse is not true. cisions.

This content downloaded from 206.253.207.235 on Wed, 27 May 2020 22:08:36 UTC All use subject to https://about.jstor.org/terms 942 The American Political Science Review Vol. 67 two voters have opposite orderings, then the ex- Decentralized Decision, Cardinal, Unique (E) treme candidates may tie for lowest total. Then, To continue the thrust of our major investiga- at least one of these candidates will be excluded tion, we should like to know the properties of from any Pareto-optimal outcome.) Given the voting schemes that are unique with respect to two different utility vectors for 1, select a utility cardinal preferences. In essence, we are asking: vector for 2 so that (a) his first place choice is not What voting schemes will get individuals to mark the same as that of 1, and (b) the candidate re- their true cardinal preferences? Unique schemes ceiving the lowest total will differ depending on require one-to-one correspondences between bal- which of l's vectors applies. Such a selection will lot markings and preferences. If the mapping always be possible.17 process is understood, we can say that in effect It is true, of course, that for certain patterns of individuals are marking their true preferences.'9 preference for 2, particular instances of non- The required structural characteristics can be uniqueness for 1 will not matter. This raises the easily delineated in our three-candidate example. question whether a voting individual can monitor Before marking his ballot, the individual voter other individuals' preferences or ballot markings will consider his possible effects on the probability to help select his own ballot marking. Such moni- outcome vector. Matters are simplest if we start toring raises no problems so long as each indi- with the case of one individual. vidual's markings are unique, conditional on the What are the structural characteristics of a de- ballot markings of others. For with this informa- cision process that gets a single individual to re- tion, all individuals' accurate cardinal preferences veal his true cardinal preferences? The individual can be deduced. is assumed to understand how the scheme works. A definitional problem arises with the concept The scheme is an announced function, s, that of uniqueness if voters monitor each others' bal- transforms the individual's ballot into a prob- lots and only employ the same marking for two ability vector outcome. This process can be different preference orderings if they discover that represented as it will not affect the set of Pareto-optimal outcomes. Notice that this monitoring would have to P = s(x). be most accurate, for as long as there is any positive probability that the other individuals) have The individual, knowing the form of s, will a "counterexample" set of preferences, Pareto select x to maximize his expected utility. Repre- optimality cannot be assured. In discussing senting his utility vector as V, and his optimal bal- uniqueness for the remainder of this paper, I shall lot as x*, this can be expressed assume either (a) that the required accurate moni- max S(X) VT is achieved at x*. toring is not possible, or (b) if it is possible, an x individual who has an incentive to mark his ballot nonuniquely will sometimes do so in cases where We wish to discover the properties of the function other individuals have counterexample-relevant s that will lead sets of preferences."' This slightly tortuous argument leads to a x*--V. general result that can serve as a building block That is, we wish to construct a function s such that for later arguments. it is in the individual's best interest to vote his true Theorem II: For nondictatorial voting schemes, cardinal preference. if the ballot processing mechanism is always First, the function must obviously assign probabilities in the same rank order as the utilities that to lead to Pareto-optimal outcomes, the voting individuals' ballots must be unique are provided through the x vector. (For simplicity, with respect to their cardinal preferences. It may be possible to construct voting schemes that are not unique for each individual taken alone, 17 Another numerical example may be helpful. As- but are unique for all individuals taken together. This sume that 1 marks his ballot the same way whether would require that individuals have some capability to his cardinal utility values are 1,.5,0 or 1,0,.2. Then monitor each others' preferences and ballot markings. if individual 2 has preferences 0,1,.9, for example, a For example, individual 1 might mark his ballot the nondictatorial Pareto-optimal outcome cannot be as- same way when he has preferences V1' and individual sured. The third candidate can never be included for 2 marks X2' as he does when he has preferences V," individual l's first set of preferences and must always and individual 2 marks X2". The processing system be included for his second set of preferences. can take 2's markings into account in deciphering in- "I strongly believe that it is possible to substantiate dividual l's preferences. the arguments in the remainder of this paper without If each individual's cardinal preferences are to be these assumptions. But the whole area is most elusive, revealed, it is sufficient that given other individuals' and I have not been able to discover a convincing markings there be a one-to-one correspondence be- proof for this assertion. tween an individual's markings and his preferences.

This content downloaded from 206.253.207.235 on Wed, 27 May 2020 22:08:36 UTC All use subject to https://about.jstor.org/terms 1973 Voting Systems, Honest Preferences and Pareto Optimality 943 we consider the case without ties.) Otherwise, the The first requirement tumbles out of the second- individual would have an incentive to misreport order conditions that guarantee that the individual which alternative he liked best and which least. is at a maximum, not a minimum. The second and With no loss of generality, we can normalize third requirements insure that the probability and assign a utility value of 1 to the most preassigned to the highest value outcome is greatest, ferred alternative and 0 to the least. Assign the the second is second, and the lowest is lowest. value y to the intermediate alternative; thus We should also like to have V= (1, y, 0). The key question then is, what additional prop- f(l) = (1) erties must s(x) have so that g(0) = 1 - f(0) - g(0).

If alternatives are ranked the same, they should receive the same probability assessment.2" max s(1, z, 0) y is achieved at z = y There is an infinite number of schemes that \0/ meet these requirements. One satisfactory scheme has the functions for all y, 0

The sum of the probabilities of the outcomes pi = f(z) = .6 - .z2/2, must equal 1. Thus, the function s(x) is fully de- P2 = g(z) = .25 + .lz, scribed by detailing the effect of the choice of z on pi and p2. Represent these relationships and

Pi = f(z), and P3 = 1-P --P2 = .15 +.(z2/ 2-z).

P2 = g(Z)- Here g'(z) =.1; and to satisfy requirements The individual's expected utility is f'(z) = -zg'(Z) = -.lz iP1 + YP2 + 0(1 - pt - P2)- for z = y. For all values of z, there must be the ordering p >p2>p3. This three-alternative scheme He will select z to maximize is illustrated in the figure below. f(Z) + Yg(Z); 1 the third term vanishes because of the zero multi- plicand. Probability I will deal with the situation in which the f and Alternative g functions are continuous and differentiable; my Is Chosen results can easily be extended to noncontinuous pi cases. First-order maximization procedures ap- plied to the preceding expression reveal that

f (Z) = -Yg'(Z).

If it is to be in the individual's self-interest to 0 ~~~~~~~~~p3 select z=y, a significant property of f and g is 1 dictated.20 For all z Value of z

f g' = z. A] Figure 1. A scheme that leads an individual to express his true cardinal preferences. Additional required properties are that for all z, Value of z

f "(Z) + Zg"(Z) < 0, 0 .2 .4 .6 .8 1 g(Z) < f(z), and pi .6 .598 .592 .582 .568 .55

g(Z) > 1 -f(Z) - g(Z). [B] P2 .25 .27 .29 .31 .33 .35

'The uniqueness requirement is equally well satis- P3 .15 .132 .118 .107 .102 .10 fied so long as there is a one-to-one mapping between z and y. Any scheme that induces such a mapping has the key structural aspects that are discussed below in 21 I am indebted to Kenneth Arrow for this point relation to this scheme. among others.

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The procedure just employed to construct a P= (z, y), unique voting scheme for cardinal preferences can and be readily extended to the many-alternative case. The computations get somewhat messy as the P2 k(z, y). number of alternatives increases, but no new con- ceptual problems arise.22 The efficiency condition that parallels [A] is for all z, Pareto Optimality of the Unique with Respect to Cardinal Preferences Scheme ji(z, y)- -zkl(z, y) 23 [C] The scheme presented is unique with respect to There are no simply-stated conditions that in- the individual's cardinal preferences. Unfortu- sure the individual will write down his preferences nately, it is not Pareto-optimal. Pareto optimality in their true order. What is needed boils down to would require that pi= 1. This misfortune, alas, is the tautological requirement that for all -y, the x representative. With rare exceptions, schemes that that is the maximizer of are unique with respect to information required to make an efficient decision are by their very con- P+xP2, 0 < X < 1 struction not Pareto-optimal. The general difficulties that are encountered can is of the form (1, z, 0), where z too is a value be understood in the context of the simplest case between 0 and 1.24 where the difficulties matter, the case of two voters Consider a scheme that meets all requirements and three alternatives. (The one-voter case is not and is unique with respect to the individual's car- disturbing; Theorem II applies only to nondicta- dinal preferences. For such a scheme an individual torial voting procedures. With one individual, with preferences (1, y, 0) will select z=y for all dictatorship is acceptable; the cardinal preferences possible values of y. From condition [C], given required by Theorem II are not needed to achieve that z is less than 1, the increase in P2 as z in- a desirable outcome. Merely ask the individual creases is greater than the corresponding decrease which alternative he prefers and give it to him.) in pi. The slack can only be taken up in P3, (pl+p2+p3= 1), which therefore must be de- Voting Schemes with Two Voters and creasing over the whole range of z. This implies Three Alternatives that P3 must have positive probability (except in With two or more individuals, as Theorem II the case when the individual is indifferent between tells us, efficient voting procedures that are demo- his first and second alternatives).25 This observa- cratic require cardinal preferences as inputs. The tion suggests another general result. procedure must be unique with respect to individuals' cardinal preferences. Theorem III: A voting scheme that is unique The insights we have derived from considera- with respect to individuals' cardinal prefer- tion of the one individual situation are readily ex- ences must lead to probability vector out- tended. Look at the matter from the standpoint comes all of whose elements are positive. of voter 1. Assume that the vector giving voter 2's true cardinal preferences, y, has been fed to the 23 This analysis could be carried out equally well decision mechanism, and that voter 1 knows for a world where other individuals' ballot markings this y. were uncertain parameters. A subjective probability The social outcome now is a function of both 'y distribution f(y) would be placed over that parameter. and x. Represent this relationship as Thus, efficiency condition [C] would become

P = r(x, -y). f js(z, y)f((T)d-y = -zf k 1 (z, y)f (T)dy. If honesty is to be insured, whenever Other conditions would be modified equivalently; the basic results would be the same. Where there is per- x* is miaximizer of r(x, y)VT, then fect information on other individuals' preferences and ballot markings, the case considered in the text, = v. f(-y) = 1 for y = .y*, otherwise 0. Paralleling our procedure from before, with z 24 One scheme that elicits true preferences for this being the utility value of the in-between alterna- two-individual case allows either individual to allocate tive, represent the function r fully by 50 per cent of the outcome probability. Each individual takes half his probability as given and deals 22 To deal with discontinuous cases, merely replace with the rest as he would in Figure 1, with the vertical (f[x + Ax] - f[x)/Ax wherever f'(x) appears, and sim- scale relabeled with .5 placed where there is now a 1. ilarly for g'(x). Second derivatives can be handled in 21 If y - z - 1, then p3 can be 0. As z can no longer parallel fashion. increase, p3 need not further decrease in value.

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Shortcomings of Schemes That Are Unique utility vectors of the excess alternatives are lin- with Respect to Cardinal Preferences early dependent (lie on the same straight line in the two-individual case) on those in some set of n. Theorem III has a very unfortunate implication. No democratic voting schemes that are unique Theorem IV: In an n-individual world, assign- with respect to individuals' cardinal preferences ing positive probability to more than n alter- will guarantee Pareto-optimal outcomes. (We saw natives whose vectors of individuals' utilities this previously when there was but one indi- are linearly independent will result in a non- vidual.) With two individuals and three alterna- Pareto-optimal outcome. tives, say, all three alternatives will have positive entries in the probability outcome vector. The Major Theorem diagram below illustrates why, in general, such a The motivation at the start of this paper was to probability vector cannot represent a Pareto- discover a voting system that leads to Pareto- optimal outcome. In the diagram, each alternative optimal outcomes. Individuals were assumed to is represented by its vector of individuals' utilities. vote in their own self-interests. It was recognized that depending on the situation and the voting system, they might or might not vote honestly. The vote-processing system would then lead to an 1 A outcome. For at least some situations we should like to require that the outcome be democratic. Voting Voting ~ in PointsPareto Optimal Individual PB It was hoped that the outcome would be Pareto- optimal as well. The objective was to find systems 1's Utility \ that could guarantee these properties. Unfortu- nately, no such system exists.

Theorem V: No voting system that relies on 0 individuals' self-interested balloting can 1 guarantee a nondictatorial outcome that is Pareto-optimal. Voting Individual 2's Utility

Figure 2. Pareto optimality and vectors of Proof individuals' utilities for alternatives. Consider the case of two individuals with strict preferences among three alternatives whose vec- The northeast frontier represents all Pareto- tors of individuals' utilities are linearly indepen- optimal points. These points can be achieved dent. By Theorem II, "if the ballot processing through linear weighting of alternatives A and B mechanism is to lead to Pareto-optimal outcomes, or a linear weighting of alternatives B and C. If all the voting individuals' ballots must be unique three alternatives are given positive weight, a with respect to their cardinal preferences." By point strictly interior to the triangle ABC will be Theorem III, this voting scheme "must lead to the outcome. Every such interior point is domi- probability vector outcomes all of whose elements nated by a point on the northeast frontier; there- are positive." But by Theorem IV, assigning posi- fore it cannot be Pareto-optimal. Only in the case tive probability to all three alternatives in this where all three alternatives lie on the same two-individual world will lead to a non-Pareto- straight line will it be possible for a Pareto-optimal optimal outcome. The three theorems together scheme to give positive probability to all three prove that no satisfactory voting system can be alternatives. (In such a case, the concept of effi- found for this case. An equivalent demonstration cient choice loses much of its meaning. All possi- can be provided for any number of voters. ble probability outcome vectors are Pareto- optimal.) Summary The conclusion, therefore, is that in general The recent literature of social choice has pro- when there are but two individuals, only two vided rather pessimistic conclusions about the alternatives will be involved in Pareto-optimal possibility of passing from individuals' ordinal outcomes. Converting the geometry to n dimen- preferences to socially desirable outcomes. Most sions, we find that with n individuals a maximum voting mechanisms, the most common social of n alternatives will need to be given positive choice procedure where political outcomes are probability weight in order to achieve all Pareto- involved, elicit no more extensive information optimal outcomes. Furthermore, assigning posi- than individuals' ordinal preferences, and those tive probability to more than n alternatives will preferences may not be honestly &0vealed. lead to non-Pareto-optimal outcomes unless the This paper has confronted these problems.

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First, it showed that procedures are available that The results of this paper contribute to a muffled will elicit individuals' honest ordinal and cardinal but growing plea: Let us not rely on market-based preferences. With honest cardinal preferences as standards of behavior and performance when possible inputs to a decentralized voting proce- examining those important areas of the world dure, there was a flicker of hope for finding an that in no way resemble the model of perfect com- efficient social choice mechanism. The flicker was petition. Rather, we should describe and evaluate extinguished when it was discovered that any those areas employing terms and theories devel- social choice mechanism that elicits honest pref- oped to meet their specific needs. erences automatically leads to non-Pareto-optimal Where might this lead us? It might force us to outcomes. Theorem V presents the disheartening undertake some study of the role of preference conclusion.26 No voting system that relies on in- revelation. With the competitive market outcome, dividuals' self-interested balloting can guarantee it is each man for himself. No one cares about a nondictatorial Pareto-optimal outcome. anyone else's preferences, and that is the reason that no one has an incentive to disguise or distort information about his own. In all other situations, Conclusion we care very much about the structure of everyone No decentralized mechanism can guarantee a else's preferences. Self-interested people become satisfactory social choice. This result is particu- appropriately self-conscious about revealing their larly unsettling to us economists, for we are ac- preferences, and find that they have an incentive customed to the paradigm of the efficient market to distort them. For example, the essence of a mechanism functioning in a world of perfect voting problem is a common or shared decision. competition." I shall try to influence your vote. Not only that, Perhaps we should not ask despairingly, Where the way I shall vote myself will depend very much do we go from here?, but rather inquire, Have we on my assessment of the voting patterns of others. been employing the appropriate mind-set all I cannot be relied upon to provide the information along? that any vote-aggregating mechanism will wish to We economists, to the extent that we are scien- elicit. Decentralized decision procedures cannot tists, have fallen prey to the natural tendency to be expected to function effectively. Recognizing concentrate our attentions on those phenomena this, we might ask, Is a fully decentralized deci- for which our theories have the greatest power. sion procedure what we should really seek? (In- The land of perfect competition has been our deed, to the extent that consumer sovereignty homeground, and most times when we have ven- relies on individuals' expressed preferences, we tured out, it has been to consider areas that differ might even question its universal desirability.) In in but one or two aspects from the place we know political contexts, this would come down to best. Our prescriptive theories for situations with asking, Is voting the appropriate way to make externalities and public goods with rare exceptions social decisions ?28 have been developed as if the "aberrant" case we This brings us full circle from the quote that are considering is the sole deviation from the norm opens this paper. That observation suggests in- of the competitive model. By following this track voking a geometric analogy for a concluding we have achieved some useful results. But this suc- theme. Geometers may find that they can derive cess has only been achieved at a cost. We have their most powerful results if they analyze and constricted ourselves to view and evaluate new deal with circles, for circles are representations of lands in the same terms that we have been able perfect closed curves. But the more general cate- to apply profitably to familiar territory. gory is the class of closed curves. It deserves at- tention too. Reference to the exceptional perfect 26New variants of Theorem IV could be developed case, the full circle in this instance, may be a type by altering the strength of different definitions, and of analogy that is regularly overdrawn. interchanging existential and universal modifiers. I suspect that Pareto improvements may be possible: 28 Many observers of the political scene would argue stronger results with the same definitions, or stronger that our present social choice procedures are very far definitions leading to the same results. from aggregative processes that rely on individuals' 27 I am indebted to Thomas Schelling, who suggested preferences. See Edward Banfield, "'Economic' Anal- that I emphasize this theme in my conclusion and pro- ysis of 'Political' Phenomena; A Political Scientist's vided me with helpful insights. This theme reiterates Critique" (mimeo.), 1967, and a forthcoming joint the central message of my volume, Studies in Inter- piece by Banfield and Zeckhauser on the same sub- dependence. ject.

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