JOURNAL OF ECONOMlC THEORY 13, 217-228 (1976)

Manipulation of Social Choice Functions

PETER GARDENFORS

Department of Philosophy, Vniversity of Lund, Lund, Sweden and Department of Philosophy of Science, Vmed University, Umerf, Sweden

Received October 22, 1975

1. INTRODUCTION

It is well known that for most group decision functions used in practice, it is possible to manipulate the outcome of the function in the sense that an individual (or a group of individuals), by misrepresenting his prefer- ences, secures an outcome he prefers to the outcome of the function which would have obtained if he had expressed his sincere preferences. However, it is only recently that the possibilities of finding group decision functions which are not manipulable have been investigated. As expected, it is found that most functions are manipulable and those which are not must be rejected for other reasons.

As an introductory example, take the method where each individual has one vote and where the alternative which gets the greatest number of votes wins. Suppose we have a voting situation where there are two alternatives x and y, each of which is considered best by 45 ‘A of the voters, and an alternative z, which is considered best by the remaining 10%. Now it may be possible for those who prefer z to manipulate the outcome of the voting by misrepresenting their preferences. It is clear that one of the alternatives x and y will win, so, by voting for the best of these instead of voting for z, it is possible to manipulate the outcome in a favorable direction.

This example shows that manipulability is not necessarily something to be avoided. Those who would vote for z, if voting sincerely, conclude that such a vote would have no effect on the outcome and the only way for them to influence the outcome is to misrepresent their preferences. However, a group decision method which is not manipulable is appealing, since it makes needless all strategic considerations, and thus makes voting simple from a game-theoretical point of view.

Recent research on the manipulability of voting methods has mainly been devoted to social choice functions, which, in all decision situations,

217 Cc prright 0 1976 by Academic Press, Inc.

All rights of reproduction in any form reserved.

218 PETER GhDENFORS

select a single alternative as the winning alternative. We call such functions resohte social choice functions. It has been shown independently by Gibbard [5] and Satterthwaite [8] that all such functions with at least three possible outcomes are either dictatorial or subject to individual manipulation. This result is, however, dependent on the assumption that the outcome of a resolute social choice function consists of a single alternative.

In this paper we will mainly discuss a more general class of social choice functions which do not necessarily select only one alternative in all situations, but merely a (nonempty) subset of the set of alternatives. In connection with manipulability, this kind of voting methods has been studied by Pattanaik (see, e.g., [6, 71).

For social choice functions in general, in contrast to resolute functions, it is not a trivial problem to define manipulatiliby, since one has to compare outcomes consisting of several alternatives, when the only information available is individual preference orderings of single alternatives. We shall formulate some conditions which are sufficient for manipulability, but which are weak in comparison to some other attempts. Using this concept of manipulability, we then show that most democratic social choice functions, among them all majority functions, are manip- ulable. This result is, however, dependent on the fact that individual preference orderings are allowed to contain ties. Some examples show that if individual preferences are restricted to linear preference orderings, then there are non-trivial functions which are not manipulable. These functions are unfortunately very undecisive in most situations.

2. NOTATION

In this section, we introduce the notation and the formal frame for the group decision functions.

The set of ahmatives is denoted A and is assumed to contain m elements, m > 3. Single alternatives are denoted X, y, z,..., and nonempty subsets of A are denoted X, Y, 2 ,... . The set of voters is denoted V and is assumed to contain n elements, II > 1. Single voters are denoted 1,2,..., n, and as variables we use i, j, and k.

A binary relation R is called a weak ordering if it is transitive and connected. Weak orderings will be used to represent preference orderings of the alternatives. If i is a voter, Ri will denote his preference ordering. For every preference ordering R, we define the strict preference relation P and the indifference relation I in the usual way, i.e., xPy iff xRy and not yRx; and x1y iff xRy and yRx. A weak ordering is called linear iff for all

MANIPULATION OF SOCIAL CHOICE FUNCTIONS 219

alternatives x and y, if x # y, then either xPy or yPx. R denotes the set of all weak orderings of A. Similarly, P denotes the set of all linear orderings of A. A situation is an element in R”, and will be denoted (Ii1 , R, ,..., R,,). We will use a, b, c,... to denote particular situations. Ri, will denote voter i’s preference ordering in the situation a. Situations will be described in the following self-explanatory manner:

a: 1. yxz

2. X(YZ)

3. (XYZ)

Preference is indicated by position where the ordering is from left to right, except for elements enclosed in parentheses, which are ties.

A social choice function (SCF) is a function F: R” - 2A - 4, where 2A denotes the set of all subsets of A. A resolute social choice function (RSCF) is a function F: R” N A. Hence, a social choice function selects a nonempty subset of A in each situation, while a resolute social choice function selects only one alternative. If we identify the one element subsets of A with the elements, we see that every RSCF is a SCF, but the converse is not true.

3. MANIPULATION OF REZSOLUTE SOCIAL CHOICE FUNCTIONS

For resolute social choice functions the following definition of manipulability is the most natural:

DEFINITION 1. A resolute social choice function F is manipulable by i at (R, ,..., Ri ,..., R,) iff there is an ordering Ri, such that F(R, ,..., Ri ,..., R,J P,F(R, ,a.., Ri ,,.., R,). We say that F is non-manipulable or stable iff F is nowhere manipulable.

We will now state the Gibbard-Satterthwaite theorem for resolute social choice functions.

DEFINITION 2. A resolute social choice function F is dictatorial iff there is a voter i such that, for every situation a and every alternative y in the range of F, F(a) RS y. The voter i is called a dictator for F.

THEOREM 1 (Gibbard-Satterthwaite). If F is a resolute social choice function which is stable, and if the range of F contains at least three elements, then F is dictatorial.

Theorem 1 may give the impression that any search for a reasonable

220 PETER G;iRDENFORS

stable decision function is a hopeless enterprise. The proof of the theorem is, however, crucially dependent on the assumption that the function is a resolute social choice function. For such functions, ties between two or several alternatives are never allowed as the outcome of the decision function. This is a rather restrictive and unnatural assumption, since, for many democratic group decision methods, there are situations where the outcome is a tie which is then broken by some chance procedure to obtain the winning alternative. We therefore turn our attention to the entire class social choice functions.

4. A GENERALIZED DEFINITION OF MANIPULABILITY

In order to be able to define manipulability, one needs a criterion on how the voters value the outcomes of the choice function. Since the outcomes of resolute social choice functions are single alternatives, one obtains, for each voter, a valuation of these outcomes directly from his individual preference ordering. If we consider social choice functions in general, matters become more complicated, since manipulability has to be defined from the voter’s valuations of different subsets of A, and the only information available is the preference orderings which are orderings of the elements of A.

Given an individual preference ordering Ri of A, we shall now define a partial ordering >i of the nonempty subsets of A. The main idea when judging a subset as definitely better than another subset is a sure-thing principle; if some alternative has been added, it should be at least as good as all tbhe other alternatives, and if some alternative has been deleted, it should be worse than the remaining alternatives.

DEFINITION 3. Let R be a weak ordernig of A. Let X and Y be non- empty subsets of A, Then X > Y iff one of the following conditions is satisfied:

(i) XC Y, and for all x E X and y E Y - X, xRy, and there exist xEXandyE Y- XsuchthatxPy.

(ii) Y C X, and for all x E X - Y and y E Y, xRy, and there exist XEX- YandyEYsuchthatxPy.

(iii) Neither X C Y nor Y C X nor X = Y, and for all x E X - Y and y E Y - X, xRy, and there exist x E X - Y and y E Y - X such that xPy.

EXAMPLE 1. Let A consist of x, y and z, and let R be the preference

MANIPULATION OF SOCIAL CHOICE FUNCTIONS 221

ordering determined by xPy and yPz. It is then easy to check that {x,Y) > iv>, {xl > k ~1, ix, Y> > {x9 4, and lx, Y> > {Y, 4. It is, however, not true that {x, z> > {y},

EXAMPLE 2. Let A be as above, and let R be the preference ordering determined by xPy, XPZ and y1z. Then {x} > {x, y}, {x, y} > {y, z}, and {x, Y> > ix, Y, 4, but not (Y> > (Y, 4.

The fact that {x, y} > (x, y, z} in the example above may appear somewhat unmotivated. However, if the final choice from an outcome consisting of several alternatives is made by a random mechanism which assigns equal probability to all alternatives in the choice set, then {x, y} will be a better outcome than {x, y, z>, if one’s preferences are as in the example above, since the best alternative x has a greater chance of winning in the first outcome than in the second.

We need not confine ourselves to this interpretation of the outcomes when we show that most social choice functions are manipulable. The cases we need in the proofs will all be obvious examples of manipulation.

It is possible to show that, for every weak ordering R, the corresponding relation > is a partial ordering, i.e., transitive and irreflexive. The proof presents no difficulties but is rather tedious, so we omit it.

DEFINITION 4. A social choice function F is manipulable by i at (4 ,...> R,J iff there is an ordering Ri’ such that F(R, ,..., Ri ,..., R,) >i F(R, )...) Ri )s..) R,), where >i is the ordering derived from Ri . F is non-manipulable or stable iff F is nowhere manipulable.

This definition obviously reduces to Definition 1 when restricted to resolute social choice functions.

We next give two simple examples of stable social choice functions.

EXAMPLE 3. Let Fl be the social choice function which is defined by F(a) = A, for all situations a. It is an immediate consequence of the definitions that Fl is stable.

EXAMPLE 4 (Gardner [4]). Let F2 be the social choice function which is defined in the following way: x E F,(R, ,..., R,) iff there is some Ri such that for all y, xR,y. F,(a) thus consists of all alternatives which are top- ranked in some voter’s preference ordering in a. It is easy to verify that F, is stable.

This example shows that Theorem 1 cannot be extended to social choice functions in general, since the range of F2 is A and F2 is non-dictatorial.

In Definition 4 we have chosen a weak concept of manipulability such that there is no doubt that i prefers the outcome X to the outcome Y,

222 PETER G;IRDENFORS

if X >i Y. It will therefore be a strong result if one is able to show that all functions in a certain class of social choice functions are manipulable. On the other hand, showing that a particular function is stable in the sense of Definition 4 is a comparatively weak result, since, firstly, it is possible that a stronger concept of manipulability is more natural and, secondly, showing that a certain function is not manipulable by a single individual does not in general imply that the function is not manipulable by a group of individuals. We will return to this topic in connection with Theorem 4.

In [6] and [7], Pattanaik uses a maximin relation to define an ordering of the subsets of A. He obtains a connected ordering of the subsets, and thus every two subsets of A are comparable with respect to manipulability. Pattanaik’s concept of manipulability is therefore stronger and seems less natural than ours.

5. MANIPULATION OF SOCIAL CHOICE FUNCTIONS

In this section we will show that, according to our definition of manipulability, most democratic social choice functions are manipulable. The following two conditions are basic for such functions.

DEFINITION 5. A social choice function F is anonymous iff whenever two situations a and b are identical except that Ri, = Rib and Rja = Rib , for some voters i and j, then F(a) = F(b).

DEFINITION 6. A social choice function F is neutral iff whenever two situations a and b are identical except that x and y have changed places everywhere, then x E F(a) iffy E F(b) and y E F(a) iff x E F(b).

In simple words, a social choice function is anonymous if it treats every voter in the same way, and neutral if it treats every alternative in the same way.

THEOREM 2. Let F be a social choice function which is defined for three alternatives and three voters. Let a be the following situation:

a: 1. zyx

2. xyz

3. xzy.

If F is anonymous and neutral, and ifF(a) = {x}, then F is manipulable.

ProoJ Besides a, consider the following situations:

MANIPULATION OF SOCIAL CHOICE FUNCTIONS 223

b: 1. zyx c: 1. zyx

2. (XY)Z 2. yxz

3. xzy. 3. xzy.

Situation c is a variant of the so called “voting paradox.” By the assumption that F is anonymous and neutral, one can show that F(c) = {x, y, z} (cf. [3, pp. 11-121). We will now show that whatever nonempty subset of A we choose as F(b), F will be manipulable. We consider four cases.

(i) z EF(~). In this case, it is easy to check that F is manipulable by 2 at the situation b, since, for any set Z such that z E Z, we have F(a) &, Z (‘OZb” is the ordering of the subsets of A which corresponds to Red.

(ii) F(b) = {x, y}. In this case, F will be manipulable by voter 2 at the situation c, since {x, y} >zo {x, y, z>.

(iii) F(b) = {x}. Consider the following situation:

d: 1. yzx

2. (XY)Z 3. xzy.

If z E F(d), then F is manipulable by 1 at b, as is easily verified. If z $ F(d), then since F is anonymous and neutral and x and y have symmetrical positions in d, we conclude that F(d) = {x, y}. But then F is manipulable by 1 at b since {x, y} >la {x}.

(iv) F(b) = {y}. Consider the following situation:

e: 1. zyx

2. (XY)Z 3. zxy.

If F(e) = {z}, then F is manipulable by 3 at b, since {z} >3b { y}. If F(e) # {z}, then x E F(e) or y E F(e). Now consider the following situation:

t 1. zyx

2. xyz

3. zxy.

This situation can be obtained from a by permuting x and z and inter-

224 PETER GWRDENFORS

changing l’s and 2’s preference orderings. Since F(a) = {x} and F is anonymous and neutral, we conclude that F(f) = {z}. But then F is manipulable by 2 at f since for any nonempty subset X of A which contains x or Y, 1 bf h>.

These four cases exhaust all possible ways to choose F(b), and we have shown that F is manipulable in all cases. This proves the theorem.

The assumption that F is defined for three alternatives and three voters only is introduced in order to simplify the proof. We next show that the theorem can be extended to cover most social choice functions used in practice.

DEFINITION 7. A social choice function F satisfies the Concorcet criterion iff whenever there is an alternative x in a situation a such that, for every alternative y # x, the number of individuals who strictly prefer x to y is greater than the number of individuals who strictly prefer y ot x, then F(a) = {x}. Such an alternative is called a majority alternative in the situation a.

THEOREM 3. Let F be a social choice function which is defined for at least three voters. If F is anonymous, neutral and satisfies the Condorcet criterion, then F is manipulable.

ProoJ: If A and V both contain three elements, then the theorem follows immediately from Theorem 2, since any function which satisfies the Condorcet criterion selects {x} in the situation a. If A contains more than three alternatives, then the situations in the proof of Theorem 2 may be augmented with dummy alternatives which are ranked after x, y, and z, in some fixed ordering in every preference ordering. Similarly, if V contains more than three voters, these situations may be augmented with dummy individuals who all are indifferent between x, y, and z. The arguments of the proof of Theorem 2 are not affected by these additions, as is easily checked. This completes the proof of the theorem.

In the literature there occur several types of social choice functions where the outcomes are determined from the sums of points assigned to the different positions in the preference orderings. Functions of this kind have been called summation social choice functions by Fishburn [2], represent- able functions by Gardenfors [3], point systems by Smith [9], and social choice scoring functions by Young [IO]. As soon as the first position in a preference ordering is assigned the greatest number, and the corresponding function is neutral, such a function will select {x} as the choice set in the situation a in Theorem 2, which is what is needed to conclude that the function is manipulable.

MANIPULATION OF SOCIAL CHOICE FUNCTIONS 225

6. MANIPULABILITY WHEN VOTERS' PREFERENCES ARE LINEAR

Theorem 2 does not leave much room for useful stable social choice functions. However, the proof of the theorem exploits the fact (in situation b) that the voters are allowed to have ties in their preference orderings. The following example will show that if the domain of a social choice function is restricted to situations containing only linear orderings, i.e., situations in P”, then there are interesting stable functions.

EXAMPLE 5. Let F3 be the social choice function which is defined for situations in P” in the following way: F3(a) = (x}, if there is a majority alternative x in the situation a, and F,(a) = A otherwise.

THEOREM 4. F3 is stable, anonymous, neutral, and satisfies the Condorcet criterion.

Proof. We show that F3 is stable. The remaining properties are imme- diate consequences of the definition of the function. Suppose there exists a situation (PI ,..., P,), a voter i, and a preference ordering Pi’, such that F,(P, , . . . , Pi’,. . ., P,) > i F3(P, ,. . . , Pi , . . . , P,). Let a = (PI , . . ., Pi ,. . . , P,) and a’ = (P, ,..., Pi’ ,..., P,). We divide the proof into three cases.

(i) F,(a’) = {x} and F,(a) = {y}, for some alternatives x and y. Since F,(a’) >i F,(a) we conclude that xPi y. Since F,(a) = (y}, y is a majority alternative in a, and since xPfy, x can never become a majority alternative in a’, no matter how Pi’ is chosen. Hence this case is impossible.

(ii) F,(a’) = (x: and f;,(a) = A, for some alternative x. Since F,(a’) >i F,(a), it follows, from the definition of >i and the assumption that all preference orderings are linear, that xPi y for all alternatives y # x. If x is not a majority alternative in a, it is not a majority alternative in a’, no matter how Pi’ is chosen. Hence this case is impossible too.

(iii) F,(a’) = A and F,(a) = {x>, for some alternative x. Since F,(a’) >i F3(a), it follows in the same way as in case (ii) that yP,x for all alternatives y # x. So, if x is a majority alternative in a, it will be a majority alternative in a’, no matter how Pi’ is chosen. This shows that also the third case is impossible.

These three cases are the only possible ones, if it is assumed that F3 is manipulable, according to the definitions of F3 and >i . We have thus shown that F3 is stable and the proof is complete.

As we remarked earlier, this kind of theorem is rather weak since we use a concept of manipulability which includes as little as possible, and since we only allow one individual to misrepresent his preferences. Theorem 4

226 PETER G;iRDENFORS

can be strengthened, however, since it is possible to show that not even a group of individuals can manipulate the outcome of F3 . Here, we define a social choice function F to be manipulable by a group J of individuals, J C V, at a situation a iff there is a situation b where the individuals in J have misrepresented their preferences such that, for all individuals j in J, F(b) >i F(a). The proof of the fact that F3 is stable under manipulation by groups runs along the same lines as the proof of Theorem 4, changing the statements about the individual preference ordering to statements about the orderings of all individuals in the group.

The function F3 is completely undecisive in situations where there are no majority alternatives, and thus not suited for practical use. We can construct a somewhat more decisive function in the following way.

EXAMPLE 6. We say that an alternative x is Pareto dominated in the situation (R, ,..., R,) iff there exists an alternative y such that yPix for all i. The set of all Pareto dominated alternatives in a situation a is denoted pd(a). We now define a social choice function F4 by F,(a) = F,(a) - pd(a), for all situations in P”. As is easily checked, F4 is anonymous, neutral, weakly Pareto-optimal, and satisfies the Condorcet criterion. It can also be shown that F4 is stable. The proof, which follows the same lines as the proof of Theorem 4, reduces to a number of subcases. Each of these cases is rather simple, but the proof still becomes long winded, so we omit it.

F4 is still very undecisive, but we have not been able to find any more decisive function which is stable and satisfies minimal requirements on democratic decision functions. Our conjecture is that all such functions are too undecisive to be of practical interest.

7. CONCLUSIONS

This paper has shown that when defining and investigating manip- ulability of group decision processes there are several factors which have to be taken into account.

Firstly, the type of the outcome of the group decision function is important. We have here studied two types of social choice function, where the resolute functions form a subclass of the more general class. Defining manipulability for resolute functions is straightforward, while, for social choice functions in general, there are several possible ways to draw the line between what is manipulation and what is not. Here, we have chosen a definition of manipulability based on a sure-thing principle.

Another type of group decision functions, which is not dealt with in this paper, is “social welfare functions” as defined by Arrow [l]. The outcome

MANIPULATION OF SOCIAL CHOICE FUNCTIONS 221

of a social welfare function in a decision situation is an ordering of the alternatives instead of a choice of a subset of them. Gardner [4] has introduced a concept of manipulability for this kind of functions based on measures of the degree of similarity between the preference ordering which is the outcome of the function in a given situation and the preference ordering which expresses the sincere tastes of a voter in that situation.

Secondly, the kind of orderings used to represent voters’ preferences are relevant when determining which decision methods are manipulable. We have shown that if ties are allowed in the voters’ orderings, then all democratic social choice functions which satisfy the Condorcet criterion are manipulable. However, if ties are not allowed, i.e., if all preference orderings are linear, then there exist democratic functions which satisfy the Condorcet criterion and are stable.

Taking together the results in this and other recent papers on manipulation of group decision processes, one finds that it is impossible to find a decision method which is democratic, decisive, and stable. If a function is decisive in the extreme sense that it selects only one alternative in every situation, then the Gibbard-Satterthwaite theorem shows that either a function is dictatorial or excludes most alternatives from ever being chosen (which are non-democratic properties), or the function is manipulable. If a function is democratic, in the sense of being anonymous and neutral, and decisive, e.g., in the sense that it satisfies the Condorcet criterion, then Theorem 3 shows that the function is manipulable. Further support for the general conclusion can be obtained from the fact that the examples of democratic and stable social choice functions we have been able to construct are all very undecisive.

REFERENCES

1. K. J. ARROW, “Social Choice and Individual Values,” 2nd ed., Wiley, New York, 1963.

2. P. C. FISHBURN, “The Theory of Social Choice,” Princeton Univ. Press, Princeton, 1973.

3. P. G.XRDENFORS, Positionalist voting functions, Theory and Decision 4 (1973), l-24. 4. R. GARDNER, Some implications of the Gibbard-Satterthwaite theorem, mimeo-

graphed, 1974. 5. A. GIBBARD, Manipulation of voting schemes: A general result, Econometrica 41

(1973), 587-601. 6. P. K. PATTANAIK, On the stability of sincere voting situations, J. Bon. Theory 6

(1973), 558-574. 7. P. K. PATTANAIK, Stability of sincere voting under some classes of non-binary group

decision procedures, J. Econ. Theory 8 (1974), 206-224.

228 PETER GiiRDENFORS

8. M. SATTERTHWAITE, Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions, J. Econ. Theory 10 (1975), 187-217.

9. J. H. SMITH, Aggregation of preferences with variable electorate, Econometrica 41 (1973), pp. 1027-1041.

10. H. P. YOUNG, Social choice scoring functions, mimeographed, 1973.