- Email: [email protected]
Social choice and telling the truth
Journal
of Public
Economics
SOCIAL
21 (1983) 359-375.
CHOICE
AND
North-Holland
TELLING
THE TRUTH
John CRAVEN* Department
of Economics,
Received
This paper choice rule impossibility the relation intended to
January
University
of Guelph, Guelph, Ontario,
1982, revised version
received
presents a straightforward proof of Gibbard’s with at least three outcomes is dictatorial. The theorem, but does not rely on that theorem. between Gibbard’s and Arrow’s theorems. The make this important area of social choice theory
November
Canada
1982
theorem that any non-manipulable proof is similar to proofs of Arrow’s The last part of the paper discusses paper contains no new result, and is accessible to a wider audience.
1. Introduction Two of the most important results in the theory of social choice are those of Arrow (1963) and Gibbard (1973). Straightforward proofs of Arrow’s result are widely known [mainly stemming from Vickrey (1960)], and most proofs of Gibbard’s result, including Gibbard’s own, and that of Sugden (1981) rely on Arrow’s result, but, as Gibbard notes (p. 588) ‘not in a simple way’. Satterthwaite’s (1975) discussion of the connection between Arrow’s result and Gibbard’s, and other results reported by, for example, Pattanaik (1978) are, in the same sense, not simple. The purpose of this paper is to provide an introduction to, and straightforward proof of, Gibbard’s result without relying on Arrow’s result, and to discuss the connections between Arrow’s and Gibbard’s results in a similarly straightforward way. In section 2 we introduce a convenient notation for individual preferences, the idea of a social choice rule and various possible properties of such rules. Gibbard’s result concerns the possibility that an individual may have an incentive to state untrue preferences in order to influence some social choice (such as an election, or the quantity of a public good to be supplied and paid for), So in section 3 we define truth-inducing (or non-manipulable or strategy-proof as they are known elsewhere) choice rules, and see some implications of that definition. We prove Gibbard’s result in section 4. In section 5 we relate Gibbard’s result to the earlier work of Arrow.
*I am very grateful to Cohn comments and suggestions. 0047-2727/83/$3.00
0
Cannon,
Alan
Carruth
1983, Elsevier Science Publishers
and
two anonymous
B.V. (North-Holland)
referees
for their
360
J. Craven, Social choice and telling
the truth
2. Preferences and social choice rules In the context of Gibbard’s result, a social choice rule is a mechanism for choosing a ‘socially best’ alternative (which we refer to as the winner) from a finite set, given individuals’ stated preferences. The alternatives might be economic policies, or candidates in an election, and we assume that, for each pair of alternatives x and y, each individual can state whether he prefers x to y, y to x or that he is indifferent between them. No individual can say that he does not know his preference when faced with a pair of alternatives. Each individual’s preferences must be consistent (for example, x preferred to y, and y preferred to z imply x preferred to z) and Gibbard requires that the social choice rule determines a winner given any of the possible combinations of consistent individuals’ preferences. It is convenient to assume that the alternatives are named, so that we can place them in alphabetical order.’ It should be emphasised that a social choice rule need not choose the winner on alphabetical criteria, but it is convenient to be able to refer to the alphabetical order of the alternatives at various points in our discussion. Gibbard allows that the choice rule may imply that alternatives do not win whatever individual preferences may be. For example, if women are excluded from Parliament, a female candidate cannot win even if every voter prefers that woman to every male candidate. An alternative that wins for some combination of individual preferences is referred to as a potential winner. A simple way to make social choices is to name one individual as dictator, so that the chosen winner is the potential winner that the dictator ranks highest. If the dictator is indifferent between several potential winners (with no potential winner preferred to these), the rule can choose amongst them on the basis of alphabetical order. People do not vote for the choice of dictator, because such voting merely redefines the available alternatives to be the potential dictators. If the choice rule is dictatorial, the name of the dictator is known before the process of social choice begins. Dictatorial rules make social choice easy, but only one person’s preferences are taken into account, and so such rules do not provide a semblance of a democratic judgement when peoples’ views differ. However unsatisfactory they may be, we shall see that dictatorial rules are important in this area of social choice theory. The important novelty in Gibbard’s result (compared to Arrow’s) lies in his discussion of manipulation of choice rules. The theory of public goods gives an example of the problem of manipulation. A government decides to ask individuals to state how many hours of public broadcasting they want each week. It specifies that the amount supplied will be the average’ of these ‘Any order based on characteristics of the alternatives (but not on individual preferences between alternatives) could be used instead. ‘Technically, the number of alternatives needs to be finite, so this average needs to be rounded to, say, the nearest integer, and some finite upper limit placed on total supply.
J. Craven, Social choice and telling the truth
361
individual statements, and that each individual will be charged a fraction of the cost, where the fraction is proportional to his stated demand. An individual has an incentive to say that he wants zero hours, thereby reducing his own share of the cost to zero, whilst only slightly reducing the average compared to the ‘true’ average. Of course, if many people do this (because all have the same incentive), supply may be far below the ‘true’ demand. Another familiar example of manipulation is that of the first-past-the-post electoral system, where each person has a single vote. In an election where two candidates are expected to have similar numbers of votes, supporters of other candidates have an incentive not to vote for their own favoured candidate, but to vote for which ever of the two front-runners they prefer. This may affect the outcome of the election, because the winner may not be the candidate with the most ‘true’ support. It should be remembered that a choice rule is manipulable if there is some combination of individual preferences that gives some individual an incentive to state a false preference. All combinations of consistent individual preferences are possible, and the rule is manipulable even if only one combination gives an incentive to lie. This implies that, in our proof of Gibbard’s result, we can choose any combination of individual preferences and examine the consequences of requiring that there is no incentive for any individual to state a false preference, given that all others tell the truth. This free choice of preferences simplifies the proof considerably.
3. Consequences
of truth-inducing
In order to prove Gibbard’s choice rule is truth-inducing. purposes:
choice rules
theorem, we must formalise the idea that the The following definition is sufficient for our
A choice rule is truth-inducing if there is no individual i and no two alternatives x and y such that the winner is y if preferences are as in table 1, and x if preferences are as in table 2: Table
1
;
Table 2
362
J. Craven, Social choice and telling the truth
[Note: Throughout the paper we represent a set of preferences for all individuals in tables of this form. Each table defines a single set of preferences that must obey the restrictions specified. Where a set of people is specified in the left-hand .column they may all have different preferences, subject -to the ,restrictions listed in the right-hand column. xPy and xly are shorthand for x preferred to y and x indifferent to y, respectively.] Our formal definition is consistent with an intuitive view of truth-inducing choice rules. If the winner is y in table 1 and x in table 2, person i has an incentive to state his table 2 preferences when he truly holds the table 1 preferences, in order to improve (in his view) the choice of winner. The definition involves a variation in one person’s preferences. It is convenient to consider also how the winner changes when several people change their preferences. If we have two sets of preferences (tables 3 and 4) Table 3
we can consider the winners in a sequence of tables of preferences leading from table 3 to table 4. The first of the sequence is table 3; the second has person i with his table 4 preferences, and everyone else with their table 3 preferences; the third has i and j with their table 4 preferences, and everyone The sequence continues, with people else with their table 3 preferences. altering their preferences in the order specified in the left-hand column, until the final table in the sequence, which is table 4. The main result for such sequences is: Result I. If the choice rule is truth-inducing the winner does not change directly from x to y during the sequence from table 5 to table 6:
Table 5
p!qF,
J. Craven, Social choice and telling the truth
363
Table 6
1z;”1~~~}frorn Some or all different table 5
Pro05 Suppose that the winner changes from x to y when j changes his preferences from table 5 to table 6. If j is in A, he has an incentive to state his table 5 preferences when he truly holds table 6. If j is not in A he has an incentive to state table 6 when he truly holds table 5. This contradicts the hypothesis that the rule is truth-inducing. Result 1 does not imply that the winner cannot change indirectly from x to y, as long as some other state z wins during the sequence, and the winner changes from x to y via z, and possibly other states also. We have also: Result 2. If the choice rule is truth-inducing then y does not win in table 7:
and x is a potential
Proof: The proof falls into two parts: we consider table 8 to table 9, and then from table 9 to table 7:
first the sequence
winner,
from
Table 8
1
All
1 P re ferences implying that x wins
I
Table 9 XP all others; All
yPz if ylz in table 7 (for any z other than x); all other preferences
same as in table 7
If the winner changes from x when j changes his preferences in the sequence from table 8 to table 9, then j has an incentive to state table 8 preferences when he truly holds table 9. So x wins in table 9. Result 1 implies that the winner does not change directly from x to y in the sequence from table 9 to table 7 because everyone prefers x to y in both tables. So, if y wins in table 7,
J. Craven,
364
Social choice and telling
the truth
the winner must change directly from some other alternative w to y when some individual k changes his preferences in the sequence from table 9 to table 7. However, if k prefers w to y in table 7, he prefers w to y in table 9, and so has an incentive to state table 9 preferences when he truly holds table 7. If k prefers y to w, or is indifferent between them, in table 7, he prefers y to w in table 9, and has an incentive to state table 7 preferences when he truly holds table 9. Neither of these cases is possible if the rule is truthinducing, and so the winner can never change to y during the sequence from table 9 to table 7. So the winner in table 7 is not y, proving result 2. This result shows that a truth-inducing rule cannot thwart a unanimous preference for x over y, unless x is not a potential winner. So truth-inducing rules are ‘Paretian’ with regard to alternatives that are potential winners. The last result on sequences of preferences is: Result 3. If the choice rule is truth-inducing wins in table 11:
and x wins in table
10, then x
Table 10
pqr
Table 11 r
Result 2 implies that either x or y wins that the winner does not change directly from from table 10 to table 11. If the winner changes his preferences from table 10 to table 11, j has preferences when he truly holds table 11 because Proof.
in table 11. Result 1 implies x to y during the sequence from x to z when j changes an incentive to state table 10 he prefers x to z in table 11.
4. A proof of Gibbard’s result Gibbard’s
result is
there is no choice rule with at least three potential inducing and non-dictatorial.
winners that is both truth-
If there are only two potential winners, majority voting can be used to choose between them (with ties broken by choosing the first in alphabetical order). The choice rule is not dictatorial, and it is not manipulable because
J. Craven, Social choice and telling the truth
365
there is no advantage to be gained from pretending to prefer b to a if one really prefers a to b because this pretence increases the chance that b wins. If there are three or more potential winners, the votes must be taken in a prescribed order to avoid the well-known voting paradox. Then, however, there can be an incentive to state an untrue preference, as the following example shows. Suppose that person I prefers a to b to c, person II prefers b to c to a, and person III prefers c to a to b. The choice rule establishes the winner as follows: (i) majority voting between a and b; (ii) majority voting between c and the alternative chosen in (i). If the three people state true preferences, c wins. However, I has an incentive to state that he prefers b to a to c so that the chosen winner is b, which I prefers to c. So this method is manipulable. Gibbard’s result tells us that no other nondictatorial rule can be used either, unless we accept that people may have incentives to state untrue preferences. A set D of individuals is dominant with respect to alternative x if x wins whenever preferences are consistent with the restrictions of table 12: Table
12
D
xP all other potential
Rest
No restriction
winners
D is dominant if it is dominant with respect to every potential winner. So a unanimous preference by members of a dominant set for a potential winner over all other potential winners determines the social choice. Throughout the proof we assume that the rule is truth-inducing and so result 2 implies that the whole society is dominant. We shall show that there is a smaller dominant set of individuals within any dominant set, so that we can progressively reduce the size of the dominant set from the whole society down to a dominant set consisting of only one individual. That individual is a dictator, proving Gibbard’s result. Consider a dominant set D, containing at least two people, divided into two parts A and B. Suppose that a, b, and c are all potential winners, and consider the sequence from table 13 to table 14:
366
J. Craven, Social choice and telling the truth Table
14
_._I
Result 2 implies that no alternative other than a, b, and c can win during the sequence. Everyone has the same preferences between a and c and between b and c in both tables and so result 1 implies that there is no point during the sequence from table 13 to table 14 when the winner changes directly from c to a or c to b. D is dominant so a wins in table 14 (where all members of D want a to be the winner). Hence, c cannot win in table 13, and therefore either a or b wins in table 13. If tables 13’ and 13” are two sets of preferences consistent with the restrictions of table 13, then every table in the sequence from 13’ to 13” is also consistent with the restrictions of table 13, and so, throughout the sequence, the winner is either a or b. But, by result 1, there can be no direct change of winner from a to b during the sequence, so if a wins for some set of preferences consistent with the restrictions of table 13, then a wins for every set of preferences consistent with the restrictions of table 13. Similarly, if b wins for some set of preferences consistent with the restrictions of table 13, then b wins for any such set. Result 3 implies that, if a wins in table 13, then a wins in table 15: Table
15
I:,,,]
Now consider any two potential may be b), and table 16:
winners
Table
x and y (where x may be a or y
16
A
xPa (or x = a); aPb; bPy or (yz b); yP all others
B + rest
bPy (or y= b); yPx; xPa (or x = a); aP all others
Result 2 implies that, if x is not a, a does not win in table everyone prefers x to a) and that, if y is not b, y does not win (because everyone prefers b to y). Result 3 implies that if b wins then b wins in table 15. So, if a wins in table 15, b does not win
16 in in in
(because table 16 table 16, table 16.
J. Craven, Social choice and telling the truth
Hence, x must win in table 16 if a wins in table then result 3 implies that x wins in table 17:
Table
15. If x wins in table
367
16,
17
/I
We have shown that x wins in table 17 if a wins in table 15, where x and y are any two potential winners, except that x may not be b and y may not be a. We need to demonstrate this case also, ‘and we use the requirement that there are at least three potential winners, which are a, b, and c. In table 17 set x=c, y= b, so that c wins in table 17 if a wins in table 15. We then use the following sequence of tables: Table
18
A
cPb; bPa; aP all others
B + rest
bPa; aPc; cP all others
Table
19
A
cPa; aP all others
B + rest
aPc; cP all others
Table 20 A
bPc; cPa; aP all others
B+rest
aPb; bPc; cP all others
Table 21 A
bPa; aP all others
Bfrest
aPb; bP all others
If c wins in table 17 (with x =c, y= b) then result 3 implies that b cannot win in table 18. Result 2 implies that a cannot win in table 18. So c wins in table 18, and result 3 implies that c wins in table 19. Results 2 and 3 then imply that c and a cannot win in table 20, and so b wins in table 20. So result 3 implies that b wins in table 21. We have shown that, for any two potential winners x and y, x wins in table 17 if a wins in table 15. Now suppose that y wins in table 22 so that, by result 3, y wins in table 23:
368
J. Craven, Social choice and telling the truth Table 22
/:,,,I1
Table 23
xPy and yP all others
if XP or Iy in table 22
Consider the sequence from table 17 to table 23. Result 2 implies that only x and y can win during this sequence. Suppose that the winner changes from x to y when j changes his preference from table 17 to table 23: (i) if j is in A, j has an incentive to state table 17 when he truly holds table 23; (ii) if j is in B+rest, and prefers y to x in table 23, he has an incentive to state table 23 when he truly holds table 17; (iii) if j is in Btrest, and prefers x to y in table 23, he has an incentive to state table 17 when he truly holds table 23. So there can be no change of winner in the sequence from table 17 to table 23. So, if x wins in table 17, x wins in table 23, so that y cannot win in table 22. This conclusion holds for any potential winner y (other than x), and so x must win in table 22 if x wins in table 17. So set A is dominant for every potential winner x if x wins in table 17. However, x wins in table 17 if a wins in table 15, and a wins in table 15 if a wins in table 13. So we must conclude that, if a wins in table 13, A (which is a part of D) is a dominant set. The only alternative winner in table 13 is b. If b wins in table 13, we can construct a similar argument to show that B is a dominant set. To do this, we replace tables 15-23 by tables 15*-23* where
Table 15*
Table 16* B
xPh (or x = b); bPc; cPy (or y = c); yP all others
A + rest
cPy (or y = c); yPx; xPh (or x = h); UP all others
J. Craven, Social choice and telling the truth
through
369
to Table 22*
I,,1
etc. We can then follow a similar argument through to show that if b wins in table 13, b wins in table 15*. Then x wins in tables 16* and 17*, etc. and so x wins in table 22*, so that B is dominant. We have shown that, within any dominant set, there is a smaller dominant set. The whole society is dominant, and so we can find a succession of dominant sets D,, D,, . . ., D, where D, is the whole of society, and each D, is a part of Dk 1. If we continue this succession far enough, D, has only one member (person j) who determines the social choice when he prefers a potential winner to all other alternatives. To demonstrate that j is a dictator, we must show that one of his highest ranked potential winners is the social winner (even if he prefers some other alternative). Suppose that this is not so, so that b wins in table 24 (a and b are both potential winners): Table 24 j
aPb; al’ or I all other potential
Rest
No restriction
winners
We know that a wins in table 25:
Table 25 j
aP all other alternatives
Rest
Same preferences
as table 24
and so j has an incentive to state his table 25 preferences when he truly holds table 24. So, if the choice rule is truth-inducing, a, or a potential winner that j holds indifferent to a, must win in table 24. Therefore j is a dictator. We have therefore shown that, if the choice rule is truth-inducing, if there are at least three potential winners, and if the rule must choose a winner given any consistent individual preferences, then there must be a dictator. This is Gibbard’s result.
J. Craven, Social choice and telling the truth
370
5. Gibbard
and Arrow compared
Gibbard’s result had been earlier conjectured by Vickrey (1960) because of its many links with Arrow’s result. 3 We now examine the links between Gibbard’s and Arrow’s works. Gibbard shows that if a choice rule satisfies conditions Gl-G3, then it cannot satisfy G4: Gl. the choice rule determines a single winner for any set of consistent individual preferences; G2. there are at least three potential winners; G3. the choice rule is truth-inducing; G4. the choice rule is not dictatorial. Arrow is not concerned with choice rules that give a single winner, but with social welfare functions that provide a complete social ordering of the alternatives. So, given any two alternatives x and y the social welfare function must determine whether x is socially better than y, y is socially better than x, or x and y are socially indifferent. These social decisions must be consistent, in the same sense that individual preferences are consistent. Unlike Gibbard’s choice rules, a social welfare function need not determine a single winner, because several alternatives may be socially indifferent and socially better than all others. However, a welfare function provides a social decision between alternatives that do not win, whereas Gibbard’s choice rules do not. So, in one way Arrow requires less than Gibbard and in the other he requires more. Arrow shows that if a welfare function satisfies conditions Al-A5, then it cannot satisfy A6: Al. The welfare function determines a consistent social ordering of all alternatives for any set of consistent individual preferences. A2. There are at least three alternatives. A3. No social judgement is imposed: that is, for any two alternatives x and y there is some set of individual preferences for which x is socially better than y, and some set for which y is socially better than x. A4. The welfare function is independent or irrelevant alternatives (independent, for short): that is, for any two alternatives x and y, and any division of society into sets A, B and C (some of which may have no member), if x is socially better than y for some set of preferences consistent with the restrictions of table 26, then x is socially better than y for all sets of preferences consistent with those restrictions. A5. The welfare function does not relate social judgements negatively to individual preferences (non-negative association): that is, if x is socially better than y in table 26, x is socially better than y in table 27. 3Arrow’s theorem and modifications.
was originally
stated
in 1951. The 1963 version
includes
some corrections
371
J. Craven, Social choice and telling the truth
Table 26
II_ A
XPY
B
XIY
C
YPX
Table 27
Also, the condition requires that, if x is socially indifferent to y in table 26, y is not socially better than x in table 27. A6. The welfare function is not dictatorial: that is, there is no individual j such that for all alternatives x and y, x is socially better than y if j prefers x to y. The condition of independence implies that the social judgement between x and y depends only on individual preferences between x and y, and not on preferences between other pairs of alternatives. The condition of non-negative association implies that if some people who were indifferent between x and y come to prefer x to y, or some who preferred y to x become indifferent between y and x, or come to prefer x to y, the social judgement cannot change in the opposite direction. If x is socially better than a in table 28 Table 28 All
aP all others
then independence (A4) implies that x is socially better than a whenever everyone prefers a to x. Non-negative association (A5) then implies that x is socially better than a for any set of individual preferences between a and x, so that the social judgement between a and x is imposed. So, if no judgement is imposed Al, A4, and A5 imply that a must win in table 28, and so every Given that there are at least three alternative is a potential winner. alternatives (A2), Al-A5 imply that there are at least three potential winners. Independence (A4) implies that, if y wins in table 1, so that y is socially better than x, then, in table 29, y is socially better than x so that x does not win. Table 29 i
xPy; different
Rest
Same preferences
preferences as table
from table
1
1 I
372
J. Craven, Social choice and telling
Non-negative association not win in table 30:
(A5), implies
the truth
that if y wins in table
1, then x does
Table 30 i
yP or Ix
Rest
Same preferences
as table 1
These conclusions together imply that x does not win in table 2 if y wins in table 1. So, when a welfare function chooses a single winner and satisfies AlA5, no individual has an incentive to state an untrue preference to influence the choice of winner. If a welfare function W makes several alternatives indifferent for some sets of preferences, so that there can be joint winners, we can define an extended welfare function W* as follows. For any alternatives x and y: (i) W* implies that x is socially better than y whenever W implies that x is socially better than y; (ii) W* implies that x is socially better than y whenever W implies that x and y are socially indifferent and x is before y in alphabetical order. The function W* satisfies Al-A5 if W satisfies these conditions, and so the choice rule that chooses the winner defined by the function W* must satisfy Gl-G3. Hence, the chosen winner is always one of the winners preferred by a dictator. So the extended function W* implies that x is socially better than y if the dictator prefers x to y. However, W* replicates all the strict judgements (i.e. judgements other than indifference) of the function w so the welfare function W is dictatorial if it satisfies Al-A5. We have therefore proved Arrow’s theorem from Gibbard’s theorem, by deriving a choice rule satisfying Gl-G3 from a welfare function satisfying Al-A5. The choice rule must be dictatorial by Gibbard’s theorem, and we have shown that this implies that the welfare function is dictatorial. To investigate whether we can prove Gibbard’s theorem from Arrow’s theorem, we must derive a welfare function from a choice rule. Given some a choice rule determines a winner that is set of individual preferences, socially preferred to all other alternatives, but the choice rule does not provide a ranking of the alternatives that do not win. If a wins in table 31: Table 31 All
No restriction
we can establish the social preference over x and y for preferences of table 31 by finding the winner in table 32:
the
individual
J. Craven, Social choice and telling the truth
313
Table 32 xPy if xPy in table 31 xly if xly in table 31 All
yPx if yPx in table 31 x and yP all others
Result 2 implies that either x or y wins in table 32. If x wins in table 32, we can say that x is socially preferred to y in table 31. Arrow’s independence condition then requires that x wins in any table consistent with the restrictions of table 32. However, the condition that the choice rule is truth-inducing is not sufficient to ensure that if x wins in some table consistent with the restrictions of table 32, then y wins in no table consistent with the restrictions table 32. If tables 32’ and 32” represent preferences consistent with the restrictions of table 32, the winner defined by a truth-inducing choice rule can change from x to y during the sequence from table 32’ to table 32” when individual j changes his preference if and only if j is indifferent between x and y. Such an individual does not mind which of x or y wins, and so he has no incentive to state an untrue preference to alter the choice of winner. Hence, a truth-inducing rule need not imply a welfare function that satisfies the independence condition. This problem can be avoided in two ways. We can restrict the allowed sets of individual preferences to those that are both consistent and strict4 (i.e. no indifference), so that, by result 3, the winner cannot change from x to y during the sequence from table 32’ to table 32”. Alternatively, we can restrict the choice rules that we consider to those that are regular [as used by Satterthwaite (1975)]. A regular choice rule has the following property? for any z, if z wins in table 33R, then z wins in table 33 (where we refer to table 33R as the regularisation of table 33): Table 33 All
No restriction Table 33R
For all x and y, xPy if xPy in table 33 All
or if xly in table 33 and x is alphabetically
prior to y
%gden (1981) avoids the problem in this way. Arrow’s result is valid whether or not people are allowed to be indifferent. ‘Remember that the alphabetical ordering is only one possible ordering based on the characteristics of alternatives (see section 2).
314
J. Craven, Social choice and telling the truth
There is no indifference in a regularisation, and so the winner cannot change from x to y during the sequence from table 32’R (the regularisation of table 32’) to table 32”R. So truth-inducing, regular choice rules cannot imply that x wins in table 32’ and y wins in table 32”, and so the independence condition holds for the derived welfare function. A second difficulty that arises in deriving a welfare function that satisfies Al-A5 from a choice rule that satisfies Gl-G3 is that Gibbard allows that some alternatives never win. A potential winner is always socially preferred over an alternative that is not a potential winner, and so some social judgements may be imposed. This difficulty is avoided if all alternatives are potential winners. It is straightforward to show that a choice rule satisfying Gl and G3 together with: G2’. there are at least three alternatives; G5. all alternatives are potential winners; G6. the choice rule is regular; does lead to a welfare function satisfying Al-A5. Arrow’s result implies that this function is dictatorial, and so the chosen winner is one of the winners chosen by the dictator. Arrow’s result therefore tells us that a choice rule satisfying Gl, G2’, G3, G5 and G6 must violate G4. The need to adapt and supplement Gibbard’s original conditions in order to use Arrow’s result to prove Gibbard’s helps explain why Gibbard’s proof (using Gl-G4) relies on Arrow’s result ‘not in a simple way’.
6. Conclusions We have not proved any new results in this paper. Instead we have provided a framework which makes the implications for social choice of the work of Gibbard (sections 3 and 4) and Satterthwaite (section 5) simpler to understand. Gibbard’s result is plainly important when any social decisions are made (except in a world of honest people) because it implies either that there must be a dictatorship or that people have incentives to tell untruths. An important area that we have not discussed in this paper is the possibility of a truth-inducing non-dictatorial choice rule when it is known that individuals’ preferences have some similarities [such as single-peakedness, discussed in this context by Pattanaik (1978)]. We also have not discussed the possibility of a truth-inducing non-dictatorial rule that may choose several joint winners, which is also discussed by Pattanaik. It is possible that our simplified framework would allow a straightforward discussion of these issues also.
.I. Craven, Social choice and telling the truth
315
References Arrow, K.J., 1963, Social choice and individual values, 2nd edn. (Wiley, New Haven). Gibbard, A., 1973, The manipulation of voting schemes, Econometrica 41, 587-602. Pattanaik, P.K., 1978, Strategy and group choice, (North-Holland, Amsterdam). Satterthwaite, M.A., 1975. Strategy-proofness and Arrow’s conditions, Journal of Economic Theory 10, 1877217. Sugden, R., 1981, The political economy of public choice (Martin Robertson, Oxford). Vickrey, W., 1960, Utility, strategy and social decision rules, Quarterly Journal of Economics 74, 507-535.
of Public
Economics
SOCIAL
21 (1983) 359-375.
CHOICE
AND
North-Holland
TELLING
THE TRUTH
John CRAVEN* Department
of Economics,
Received
This paper choice rule impossibility the relation intended to
January
University
of Guelph, Guelph, Ontario,
1982, revised version
received
presents a straightforward proof of Gibbard’s with at least three outcomes is dictatorial. The theorem, but does not rely on that theorem. between Gibbard’s and Arrow’s theorems. The make this important area of social choice theory
November
Canada
1982
theorem that any non-manipulable proof is similar to proofs of Arrow’s The last part of the paper discusses paper contains no new result, and is accessible to a wider audience.
1. Introduction Two of the most important results in the theory of social choice are those of Arrow (1963) and Gibbard (1973). Straightforward proofs of Arrow’s result are widely known [mainly stemming from Vickrey (1960)], and most proofs of Gibbard’s result, including Gibbard’s own, and that of Sugden (1981) rely on Arrow’s result, but, as Gibbard notes (p. 588) ‘not in a simple way’. Satterthwaite’s (1975) discussion of the connection between Arrow’s result and Gibbard’s, and other results reported by, for example, Pattanaik (1978) are, in the same sense, not simple. The purpose of this paper is to provide an introduction to, and straightforward proof of, Gibbard’s result without relying on Arrow’s result, and to discuss the connections between Arrow’s and Gibbard’s results in a similarly straightforward way. In section 2 we introduce a convenient notation for individual preferences, the idea of a social choice rule and various possible properties of such rules. Gibbard’s result concerns the possibility that an individual may have an incentive to state untrue preferences in order to influence some social choice (such as an election, or the quantity of a public good to be supplied and paid for), So in section 3 we define truth-inducing (or non-manipulable or strategy-proof as they are known elsewhere) choice rules, and see some implications of that definition. We prove Gibbard’s result in section 4. In section 5 we relate Gibbard’s result to the earlier work of Arrow.
*I am very grateful to Cohn comments and suggestions. 0047-2727/83/$3.00
0
Cannon,
Alan
Carruth
1983, Elsevier Science Publishers
and
two anonymous
B.V. (North-Holland)
referees
for their
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2. Preferences and social choice rules In the context of Gibbard’s result, a social choice rule is a mechanism for choosing a ‘socially best’ alternative (which we refer to as the winner) from a finite set, given individuals’ stated preferences. The alternatives might be economic policies, or candidates in an election, and we assume that, for each pair of alternatives x and y, each individual can state whether he prefers x to y, y to x or that he is indifferent between them. No individual can say that he does not know his preference when faced with a pair of alternatives. Each individual’s preferences must be consistent (for example, x preferred to y, and y preferred to z imply x preferred to z) and Gibbard requires that the social choice rule determines a winner given any of the possible combinations of consistent individuals’ preferences. It is convenient to assume that the alternatives are named, so that we can place them in alphabetical order.’ It should be emphasised that a social choice rule need not choose the winner on alphabetical criteria, but it is convenient to be able to refer to the alphabetical order of the alternatives at various points in our discussion. Gibbard allows that the choice rule may imply that alternatives do not win whatever individual preferences may be. For example, if women are excluded from Parliament, a female candidate cannot win even if every voter prefers that woman to every male candidate. An alternative that wins for some combination of individual preferences is referred to as a potential winner. A simple way to make social choices is to name one individual as dictator, so that the chosen winner is the potential winner that the dictator ranks highest. If the dictator is indifferent between several potential winners (with no potential winner preferred to these), the rule can choose amongst them on the basis of alphabetical order. People do not vote for the choice of dictator, because such voting merely redefines the available alternatives to be the potential dictators. If the choice rule is dictatorial, the name of the dictator is known before the process of social choice begins. Dictatorial rules make social choice easy, but only one person’s preferences are taken into account, and so such rules do not provide a semblance of a democratic judgement when peoples’ views differ. However unsatisfactory they may be, we shall see that dictatorial rules are important in this area of social choice theory. The important novelty in Gibbard’s result (compared to Arrow’s) lies in his discussion of manipulation of choice rules. The theory of public goods gives an example of the problem of manipulation. A government decides to ask individuals to state how many hours of public broadcasting they want each week. It specifies that the amount supplied will be the average’ of these ‘Any order based on characteristics of the alternatives (but not on individual preferences between alternatives) could be used instead. ‘Technically, the number of alternatives needs to be finite, so this average needs to be rounded to, say, the nearest integer, and some finite upper limit placed on total supply.
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individual statements, and that each individual will be charged a fraction of the cost, where the fraction is proportional to his stated demand. An individual has an incentive to say that he wants zero hours, thereby reducing his own share of the cost to zero, whilst only slightly reducing the average compared to the ‘true’ average. Of course, if many people do this (because all have the same incentive), supply may be far below the ‘true’ demand. Another familiar example of manipulation is that of the first-past-the-post electoral system, where each person has a single vote. In an election where two candidates are expected to have similar numbers of votes, supporters of other candidates have an incentive not to vote for their own favoured candidate, but to vote for which ever of the two front-runners they prefer. This may affect the outcome of the election, because the winner may not be the candidate with the most ‘true’ support. It should be remembered that a choice rule is manipulable if there is some combination of individual preferences that gives some individual an incentive to state a false preference. All combinations of consistent individual preferences are possible, and the rule is manipulable even if only one combination gives an incentive to lie. This implies that, in our proof of Gibbard’s result, we can choose any combination of individual preferences and examine the consequences of requiring that there is no incentive for any individual to state a false preference, given that all others tell the truth. This free choice of preferences simplifies the proof considerably.
3. Consequences
of truth-inducing
In order to prove Gibbard’s choice rule is truth-inducing. purposes:
choice rules
theorem, we must formalise the idea that the The following definition is sufficient for our
A choice rule is truth-inducing if there is no individual i and no two alternatives x and y such that the winner is y if preferences are as in table 1, and x if preferences are as in table 2: Table
1
;
Table 2
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J. Craven, Social choice and telling the truth
[Note: Throughout the paper we represent a set of preferences for all individuals in tables of this form. Each table defines a single set of preferences that must obey the restrictions specified. Where a set of people is specified in the left-hand .column they may all have different preferences, subject -to the ,restrictions listed in the right-hand column. xPy and xly are shorthand for x preferred to y and x indifferent to y, respectively.] Our formal definition is consistent with an intuitive view of truth-inducing choice rules. If the winner is y in table 1 and x in table 2, person i has an incentive to state his table 2 preferences when he truly holds the table 1 preferences, in order to improve (in his view) the choice of winner. The definition involves a variation in one person’s preferences. It is convenient to consider also how the winner changes when several people change their preferences. If we have two sets of preferences (tables 3 and 4) Table 3
we can consider the winners in a sequence of tables of preferences leading from table 3 to table 4. The first of the sequence is table 3; the second has person i with his table 4 preferences, and everyone else with their table 3 preferences; the third has i and j with their table 4 preferences, and everyone The sequence continues, with people else with their table 3 preferences. altering their preferences in the order specified in the left-hand column, until the final table in the sequence, which is table 4. The main result for such sequences is: Result I. If the choice rule is truth-inducing the winner does not change directly from x to y during the sequence from table 5 to table 6:
Table 5
p!qF,
J. Craven, Social choice and telling the truth
363
Table 6
1z;”1~~~}frorn Some or all different table 5
Pro05 Suppose that the winner changes from x to y when j changes his preferences from table 5 to table 6. If j is in A, he has an incentive to state his table 5 preferences when he truly holds table 6. If j is not in A he has an incentive to state table 6 when he truly holds table 5. This contradicts the hypothesis that the rule is truth-inducing. Result 1 does not imply that the winner cannot change indirectly from x to y, as long as some other state z wins during the sequence, and the winner changes from x to y via z, and possibly other states also. We have also: Result 2. If the choice rule is truth-inducing then y does not win in table 7:
and x is a potential
Proof: The proof falls into two parts: we consider table 8 to table 9, and then from table 9 to table 7:
first the sequence
winner,
from
Table 8
1
All
1 P re ferences implying that x wins
I
Table 9 XP all others; All
yPz if ylz in table 7 (for any z other than x); all other preferences
same as in table 7
If the winner changes from x when j changes his preferences in the sequence from table 8 to table 9, then j has an incentive to state table 8 preferences when he truly holds table 9. So x wins in table 9. Result 1 implies that the winner does not change directly from x to y in the sequence from table 9 to table 7 because everyone prefers x to y in both tables. So, if y wins in table 7,
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the winner must change directly from some other alternative w to y when some individual k changes his preferences in the sequence from table 9 to table 7. However, if k prefers w to y in table 7, he prefers w to y in table 9, and so has an incentive to state table 9 preferences when he truly holds table 7. If k prefers y to w, or is indifferent between them, in table 7, he prefers y to w in table 9, and has an incentive to state table 7 preferences when he truly holds table 9. Neither of these cases is possible if the rule is truthinducing, and so the winner can never change to y during the sequence from table 9 to table 7. So the winner in table 7 is not y, proving result 2. This result shows that a truth-inducing rule cannot thwart a unanimous preference for x over y, unless x is not a potential winner. So truth-inducing rules are ‘Paretian’ with regard to alternatives that are potential winners. The last result on sequences of preferences is: Result 3. If the choice rule is truth-inducing wins in table 11:
and x wins in table
10, then x
Table 10
pqr
Table 11 r
Result 2 implies that either x or y wins that the winner does not change directly from from table 10 to table 11. If the winner changes his preferences from table 10 to table 11, j has preferences when he truly holds table 11 because Proof.
in table 11. Result 1 implies x to y during the sequence from x to z when j changes an incentive to state table 10 he prefers x to z in table 11.
4. A proof of Gibbard’s result Gibbard’s
result is
there is no choice rule with at least three potential inducing and non-dictatorial.
winners that is both truth-
If there are only two potential winners, majority voting can be used to choose between them (with ties broken by choosing the first in alphabetical order). The choice rule is not dictatorial, and it is not manipulable because
J. Craven, Social choice and telling the truth
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there is no advantage to be gained from pretending to prefer b to a if one really prefers a to b because this pretence increases the chance that b wins. If there are three or more potential winners, the votes must be taken in a prescribed order to avoid the well-known voting paradox. Then, however, there can be an incentive to state an untrue preference, as the following example shows. Suppose that person I prefers a to b to c, person II prefers b to c to a, and person III prefers c to a to b. The choice rule establishes the winner as follows: (i) majority voting between a and b; (ii) majority voting between c and the alternative chosen in (i). If the three people state true preferences, c wins. However, I has an incentive to state that he prefers b to a to c so that the chosen winner is b, which I prefers to c. So this method is manipulable. Gibbard’s result tells us that no other nondictatorial rule can be used either, unless we accept that people may have incentives to state untrue preferences. A set D of individuals is dominant with respect to alternative x if x wins whenever preferences are consistent with the restrictions of table 12: Table
12
D
xP all other potential
Rest
No restriction
winners
D is dominant if it is dominant with respect to every potential winner. So a unanimous preference by members of a dominant set for a potential winner over all other potential winners determines the social choice. Throughout the proof we assume that the rule is truth-inducing and so result 2 implies that the whole society is dominant. We shall show that there is a smaller dominant set of individuals within any dominant set, so that we can progressively reduce the size of the dominant set from the whole society down to a dominant set consisting of only one individual. That individual is a dictator, proving Gibbard’s result. Consider a dominant set D, containing at least two people, divided into two parts A and B. Suppose that a, b, and c are all potential winners, and consider the sequence from table 13 to table 14:
366
J. Craven, Social choice and telling the truth Table
14
_._I
Result 2 implies that no alternative other than a, b, and c can win during the sequence. Everyone has the same preferences between a and c and between b and c in both tables and so result 1 implies that there is no point during the sequence from table 13 to table 14 when the winner changes directly from c to a or c to b. D is dominant so a wins in table 14 (where all members of D want a to be the winner). Hence, c cannot win in table 13, and therefore either a or b wins in table 13. If tables 13’ and 13” are two sets of preferences consistent with the restrictions of table 13, then every table in the sequence from 13’ to 13” is also consistent with the restrictions of table 13, and so, throughout the sequence, the winner is either a or b. But, by result 1, there can be no direct change of winner from a to b during the sequence, so if a wins for some set of preferences consistent with the restrictions of table 13, then a wins for every set of preferences consistent with the restrictions of table 13. Similarly, if b wins for some set of preferences consistent with the restrictions of table 13, then b wins for any such set. Result 3 implies that, if a wins in table 13, then a wins in table 15: Table
15
I:,,,]
Now consider any two potential may be b), and table 16:
winners
Table
x and y (where x may be a or y
16
A
xPa (or x = a); aPb; bPy or (yz b); yP all others
B + rest
bPy (or y= b); yPx; xPa (or x = a); aP all others
Result 2 implies that, if x is not a, a does not win in table everyone prefers x to a) and that, if y is not b, y does not win (because everyone prefers b to y). Result 3 implies that if b wins then b wins in table 15. So, if a wins in table 15, b does not win
16 in in in
(because table 16 table 16, table 16.
J. Craven, Social choice and telling the truth
Hence, x must win in table 16 if a wins in table then result 3 implies that x wins in table 17:
Table
15. If x wins in table
367
16,
17
/I
We have shown that x wins in table 17 if a wins in table 15, where x and y are any two potential winners, except that x may not be b and y may not be a. We need to demonstrate this case also, ‘and we use the requirement that there are at least three potential winners, which are a, b, and c. In table 17 set x=c, y= b, so that c wins in table 17 if a wins in table 15. We then use the following sequence of tables: Table
18
A
cPb; bPa; aP all others
B + rest
bPa; aPc; cP all others
Table
19
A
cPa; aP all others
B + rest
aPc; cP all others
Table 20 A
bPc; cPa; aP all others
B+rest
aPb; bPc; cP all others
Table 21 A
bPa; aP all others
Bfrest
aPb; bP all others
If c wins in table 17 (with x =c, y= b) then result 3 implies that b cannot win in table 18. Result 2 implies that a cannot win in table 18. So c wins in table 18, and result 3 implies that c wins in table 19. Results 2 and 3 then imply that c and a cannot win in table 20, and so b wins in table 20. So result 3 implies that b wins in table 21. We have shown that, for any two potential winners x and y, x wins in table 17 if a wins in table 15. Now suppose that y wins in table 22 so that, by result 3, y wins in table 23:
368
J. Craven, Social choice and telling the truth Table 22
/:,,,I1
Table 23
xPy and yP all others
if XP or Iy in table 22
Consider the sequence from table 17 to table 23. Result 2 implies that only x and y can win during this sequence. Suppose that the winner changes from x to y when j changes his preference from table 17 to table 23: (i) if j is in A, j has an incentive to state table 17 when he truly holds table 23; (ii) if j is in B+rest, and prefers y to x in table 23, he has an incentive to state table 23 when he truly holds table 17; (iii) if j is in Btrest, and prefers x to y in table 23, he has an incentive to state table 17 when he truly holds table 23. So there can be no change of winner in the sequence from table 17 to table 23. So, if x wins in table 17, x wins in table 23, so that y cannot win in table 22. This conclusion holds for any potential winner y (other than x), and so x must win in table 22 if x wins in table 17. So set A is dominant for every potential winner x if x wins in table 17. However, x wins in table 17 if a wins in table 15, and a wins in table 15 if a wins in table 13. So we must conclude that, if a wins in table 13, A (which is a part of D) is a dominant set. The only alternative winner in table 13 is b. If b wins in table 13, we can construct a similar argument to show that B is a dominant set. To do this, we replace tables 15-23 by tables 15*-23* where
Table 15*
Table 16* B
xPh (or x = b); bPc; cPy (or y = c); yP all others
A + rest
cPy (or y = c); yPx; xPh (or x = h); UP all others
J. Craven, Social choice and telling the truth
through
369
to Table 22*
I,,1
etc. We can then follow a similar argument through to show that if b wins in table 13, b wins in table 15*. Then x wins in tables 16* and 17*, etc. and so x wins in table 22*, so that B is dominant. We have shown that, within any dominant set, there is a smaller dominant set. The whole society is dominant, and so we can find a succession of dominant sets D,, D,, . . ., D, where D, is the whole of society, and each D, is a part of Dk 1. If we continue this succession far enough, D, has only one member (person j) who determines the social choice when he prefers a potential winner to all other alternatives. To demonstrate that j is a dictator, we must show that one of his highest ranked potential winners is the social winner (even if he prefers some other alternative). Suppose that this is not so, so that b wins in table 24 (a and b are both potential winners): Table 24 j
aPb; al’ or I all other potential
Rest
No restriction
winners
We know that a wins in table 25:
Table 25 j
aP all other alternatives
Rest
Same preferences
as table 24
and so j has an incentive to state his table 25 preferences when he truly holds table 24. So, if the choice rule is truth-inducing, a, or a potential winner that j holds indifferent to a, must win in table 24. Therefore j is a dictator. We have therefore shown that, if the choice rule is truth-inducing, if there are at least three potential winners, and if the rule must choose a winner given any consistent individual preferences, then there must be a dictator. This is Gibbard’s result.
J. Craven, Social choice and telling the truth
370
5. Gibbard
and Arrow compared
Gibbard’s result had been earlier conjectured by Vickrey (1960) because of its many links with Arrow’s result. 3 We now examine the links between Gibbard’s and Arrow’s works. Gibbard shows that if a choice rule satisfies conditions Gl-G3, then it cannot satisfy G4: Gl. the choice rule determines a single winner for any set of consistent individual preferences; G2. there are at least three potential winners; G3. the choice rule is truth-inducing; G4. the choice rule is not dictatorial. Arrow is not concerned with choice rules that give a single winner, but with social welfare functions that provide a complete social ordering of the alternatives. So, given any two alternatives x and y the social welfare function must determine whether x is socially better than y, y is socially better than x, or x and y are socially indifferent. These social decisions must be consistent, in the same sense that individual preferences are consistent. Unlike Gibbard’s choice rules, a social welfare function need not determine a single winner, because several alternatives may be socially indifferent and socially better than all others. However, a welfare function provides a social decision between alternatives that do not win, whereas Gibbard’s choice rules do not. So, in one way Arrow requires less than Gibbard and in the other he requires more. Arrow shows that if a welfare function satisfies conditions Al-A5, then it cannot satisfy A6: Al. The welfare function determines a consistent social ordering of all alternatives for any set of consistent individual preferences. A2. There are at least three alternatives. A3. No social judgement is imposed: that is, for any two alternatives x and y there is some set of individual preferences for which x is socially better than y, and some set for which y is socially better than x. A4. The welfare function is independent or irrelevant alternatives (independent, for short): that is, for any two alternatives x and y, and any division of society into sets A, B and C (some of which may have no member), if x is socially better than y for some set of preferences consistent with the restrictions of table 26, then x is socially better than y for all sets of preferences consistent with those restrictions. A5. The welfare function does not relate social judgements negatively to individual preferences (non-negative association): that is, if x is socially better than y in table 26, x is socially better than y in table 27. 3Arrow’s theorem and modifications.
was originally
stated
in 1951. The 1963 version
includes
some corrections
371
J. Craven, Social choice and telling the truth
Table 26
II_ A
XPY
B
XIY
C
YPX
Table 27
Also, the condition requires that, if x is socially indifferent to y in table 26, y is not socially better than x in table 27. A6. The welfare function is not dictatorial: that is, there is no individual j such that for all alternatives x and y, x is socially better than y if j prefers x to y. The condition of independence implies that the social judgement between x and y depends only on individual preferences between x and y, and not on preferences between other pairs of alternatives. The condition of non-negative association implies that if some people who were indifferent between x and y come to prefer x to y, or some who preferred y to x become indifferent between y and x, or come to prefer x to y, the social judgement cannot change in the opposite direction. If x is socially better than a in table 28 Table 28 All
aP all others
then independence (A4) implies that x is socially better than a whenever everyone prefers a to x. Non-negative association (A5) then implies that x is socially better than a for any set of individual preferences between a and x, so that the social judgement between a and x is imposed. So, if no judgement is imposed Al, A4, and A5 imply that a must win in table 28, and so every Given that there are at least three alternative is a potential winner. alternatives (A2), Al-A5 imply that there are at least three potential winners. Independence (A4) implies that, if y wins in table 1, so that y is socially better than x, then, in table 29, y is socially better than x so that x does not win. Table 29 i
xPy; different
Rest
Same preferences
preferences as table
from table
1
1 I
372
J. Craven, Social choice and telling
Non-negative association not win in table 30:
(A5), implies
the truth
that if y wins in table
1, then x does
Table 30 i
yP or Ix
Rest
Same preferences
as table 1
These conclusions together imply that x does not win in table 2 if y wins in table 1. So, when a welfare function chooses a single winner and satisfies AlA5, no individual has an incentive to state an untrue preference to influence the choice of winner. If a welfare function W makes several alternatives indifferent for some sets of preferences, so that there can be joint winners, we can define an extended welfare function W* as follows. For any alternatives x and y: (i) W* implies that x is socially better than y whenever W implies that x is socially better than y; (ii) W* implies that x is socially better than y whenever W implies that x and y are socially indifferent and x is before y in alphabetical order. The function W* satisfies Al-A5 if W satisfies these conditions, and so the choice rule that chooses the winner defined by the function W* must satisfy Gl-G3. Hence, the chosen winner is always one of the winners preferred by a dictator. So the extended function W* implies that x is socially better than y if the dictator prefers x to y. However, W* replicates all the strict judgements (i.e. judgements other than indifference) of the function w so the welfare function W is dictatorial if it satisfies Al-A5. We have therefore proved Arrow’s theorem from Gibbard’s theorem, by deriving a choice rule satisfying Gl-G3 from a welfare function satisfying Al-A5. The choice rule must be dictatorial by Gibbard’s theorem, and we have shown that this implies that the welfare function is dictatorial. To investigate whether we can prove Gibbard’s theorem from Arrow’s theorem, we must derive a welfare function from a choice rule. Given some a choice rule determines a winner that is set of individual preferences, socially preferred to all other alternatives, but the choice rule does not provide a ranking of the alternatives that do not win. If a wins in table 31: Table 31 All
No restriction
we can establish the social preference over x and y for preferences of table 31 by finding the winner in table 32:
the
individual
J. Craven, Social choice and telling the truth
313
Table 32 xPy if xPy in table 31 xly if xly in table 31 All
yPx if yPx in table 31 x and yP all others
Result 2 implies that either x or y wins in table 32. If x wins in table 32, we can say that x is socially preferred to y in table 31. Arrow’s independence condition then requires that x wins in any table consistent with the restrictions of table 32. However, the condition that the choice rule is truth-inducing is not sufficient to ensure that if x wins in some table consistent with the restrictions of table 32, then y wins in no table consistent with the restrictions table 32. If tables 32’ and 32” represent preferences consistent with the restrictions of table 32, the winner defined by a truth-inducing choice rule can change from x to y during the sequence from table 32’ to table 32” when individual j changes his preference if and only if j is indifferent between x and y. Such an individual does not mind which of x or y wins, and so he has no incentive to state an untrue preference to alter the choice of winner. Hence, a truth-inducing rule need not imply a welfare function that satisfies the independence condition. This problem can be avoided in two ways. We can restrict the allowed sets of individual preferences to those that are both consistent and strict4 (i.e. no indifference), so that, by result 3, the winner cannot change from x to y during the sequence from table 32’ to table 32”. Alternatively, we can restrict the choice rules that we consider to those that are regular [as used by Satterthwaite (1975)]. A regular choice rule has the following property? for any z, if z wins in table 33R, then z wins in table 33 (where we refer to table 33R as the regularisation of table 33): Table 33 All
No restriction Table 33R
For all x and y, xPy if xPy in table 33 All
or if xly in table 33 and x is alphabetically
prior to y
%gden (1981) avoids the problem in this way. Arrow’s result is valid whether or not people are allowed to be indifferent. ‘Remember that the alphabetical ordering is only one possible ordering based on the characteristics of alternatives (see section 2).
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J. Craven, Social choice and telling the truth
There is no indifference in a regularisation, and so the winner cannot change from x to y during the sequence from table 32’R (the regularisation of table 32’) to table 32”R. So truth-inducing, regular choice rules cannot imply that x wins in table 32’ and y wins in table 32”, and so the independence condition holds for the derived welfare function. A second difficulty that arises in deriving a welfare function that satisfies Al-A5 from a choice rule that satisfies Gl-G3 is that Gibbard allows that some alternatives never win. A potential winner is always socially preferred over an alternative that is not a potential winner, and so some social judgements may be imposed. This difficulty is avoided if all alternatives are potential winners. It is straightforward to show that a choice rule satisfying Gl and G3 together with: G2’. there are at least three alternatives; G5. all alternatives are potential winners; G6. the choice rule is regular; does lead to a welfare function satisfying Al-A5. Arrow’s result implies that this function is dictatorial, and so the chosen winner is one of the winners chosen by the dictator. Arrow’s result therefore tells us that a choice rule satisfying Gl, G2’, G3, G5 and G6 must violate G4. The need to adapt and supplement Gibbard’s original conditions in order to use Arrow’s result to prove Gibbard’s helps explain why Gibbard’s proof (using Gl-G4) relies on Arrow’s result ‘not in a simple way’.
6. Conclusions We have not proved any new results in this paper. Instead we have provided a framework which makes the implications for social choice of the work of Gibbard (sections 3 and 4) and Satterthwaite (section 5) simpler to understand. Gibbard’s result is plainly important when any social decisions are made (except in a world of honest people) because it implies either that there must be a dictatorship or that people have incentives to tell untruths. An important area that we have not discussed in this paper is the possibility of a truth-inducing non-dictatorial choice rule when it is known that individuals’ preferences have some similarities [such as single-peakedness, discussed in this context by Pattanaik (1978)]. We also have not discussed the possibility of a truth-inducing non-dictatorial rule that may choose several joint winners, which is also discussed by Pattanaik. It is possible that our simplified framework would allow a straightforward discussion of these issues also.
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References Arrow, K.J., 1963, Social choice and individual values, 2nd edn. (Wiley, New Haven). Gibbard, A., 1973, The manipulation of voting schemes, Econometrica 41, 587-602. Pattanaik, P.K., 1978, Strategy and group choice, (North-Holland, Amsterdam). Satterthwaite, M.A., 1975. Strategy-proofness and Arrow’s conditions, Journal of Economic Theory 10, 1877217. Sugden, R., 1981, The political economy of public choice (Martin Robertson, Oxford). Vickrey, W., 1960, Utility, strategy and social decision rules, Quarterly Journal of Economics 74, 507-535.