Severer Difficulties in Social Choice and Welfare Ning Neil Yu1
Abstract This note presents social welfare theorems allowing for indifference. JEL Classification Numbers: D7, D70, D71 Keywords: Arrow’s Impossibility Theorem, The Muller-Satterthwaite Theorem, The Gibbard-Satterthwaite Theorem, Dictatorship, Social Choice and Welfare
1. An Impossibility Result about Social Choice Functions Individuals numbered 1, · · · , N , with N ≥ 2, each have some preferences over M ≥ 3 alternatives A = {a1 , · · · , aM }. Let P be the set of all possible relations on M that are complete and transitive, i.e., weak preferences, and P¯ ⊂ P be asymmetric ones, i.e., strict preferences. The set of preference profiles P N is a collection of functions from {1, · · · , N } to P that specify ~ = (1 preferences of each individuals. An element of P N is an ordered list , · · · , N ) with n ∈ P for every 1 ≤ n ≤ M . If ai n aj and not aj n ai , then ai is strictly preferred to aj or ai n aj . If ai n aj and aj n ai , then ai is indifferent to aj or ai ∼n aj . We can now define social choice functions designed to narrow down a choice from A given each profile, and social welfare functions that map profiles to social preferences. 1
Ning Neil Yu (
[email protected]): Economics Department, Stanford University. The article benefited greatly from comments by Mohammad Akbarpour, Matthew Jackson, and Paul Milgrom. Yiqing Xing clarified relations among several definitions. I am also grateful for generous support from the Koret Foundation Stanford Graduate Fellowship Fund. 2
October 31, 2012
Definition 1 (SCF). A social choice function F : P N → A assigns to each ~ ∈ P N a choice F () ~ ∈ A. Definition 2 (SWF). A social welfare function R : P N → P assigns to each ~ ∈ P N social preferences R() ~ ∈ P. Several definitions are in place to facilitate communication. ~ and ~ 0 share rankings over {ai , aj } if for every n, Definition 3. Profiles ai n aj is necessary and sufficient for ai 0n aj , and aj n ai is necessary and sufficient for aj 0n ai . ~ or A0 is at the top if ai n aj for every n, Definition 4. A0 ⊂ A tops ~ an every ai ∈ A0 , and every aj ∈ A\A0 . If in addition |A0 | = 2, we call 0 3 A -runoff. Definition 5. A topping function Tij : P N → P N brings {ai , aj } to the top keeping other rankings intact.4 Let a topping choice function Fij ≡ F ◦ Tij , an SCF that agrees with F for {ai , aj }-runoffs. Analogously, we have topping welfare function Rij ≡ R ◦ Tij . Obviously, Tij = Tji , Fij = Fji , and Rij = Rji . It is worth emphasizing ~ and {ai , aj }-runoff Tij () ~ share rankings over {ai , aj }, convenient for that (SC-WIIA). Definition 6 (SC-WIIA). An SCF F is weakly independent of irrelevant ~ and {ai , aj }-runoff ~ 0 share rankings over {ai , aj }, alternatives if whenever ~ 0 ) = ai implies F () ~ 6= aj . F ( ~ = aj Otherwise, the designer of F can have difficulty explaining F () to those in favor of ai over aj : ai wins the two-party runoff and people’s preferences over {ai , aj } are intact. (SC-WIIA) precludes these complaints. Another justification is that conditional on relative rankings of {ai , aj }, a runoff offers the highest position of aj in each individual’s preferences. If aj loses the runoff, there is absolutely no reason to choose it at no better positions. 3
If F is a two-round system which automatically singles out two candidates first, it is natural that A0 enter the second round. 4 ~ 0 = Tij () ~ means for every n: first, ai 0n as and aj 0n as for every Mathematically, s 6= i, j; second, ai n aj ⇔ ai 0n aj and aj n ai ⇔ aj 0n ai ; third, as n at ⇔ as 0n at for every s, t 6= i, j.
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A stronger condition is (SC-IIA), which worries about people making a case out of any other profiles, not only salient profiles like runoffs. Definition 7 (SC-IIA). An SCF F is independent of irrelevant alternatives if ~ and ~ 0 share rankings over {ai , aj }, F () ~ = ai implies F ( ~ 0 ) 6= whenever aj . ~ is ai , then aj should not overturn and In words, if the choice for 0 ~ , lest it disgruntle supporters of ai . As will be become the choice for made clear, this condition is also necessary for “monotonicity” used in the Muller-Satterthwaite theorem or “strategy-proofness” used in the GibbardSatterthwaite theorem, justifying its appeal from another perspective. One particular SCF F that satisfies (SC-WIIA) but violates (SC-IIA) honors the proverb “two dogs strive for a bone and the third one runs off with it”. Let M = 3. Whenever ai and aj enter a runoff, a third alternative ak wins, but F always picks a1 otherwise. The premises in (SC-WIIA) is never met. When a1 n a2 n a3 for some n and a1 n0 a3 n0 a2 for the others, the choice is a1 . When a1 0n a2 0n a3 for every n, the choice is a3 . But these two profiles share rankings over {a1 , a3 }, violating (SC-IIA). In another dimension, a clearly desirable property for an SCF to possess is unanimity, that is, if ai is the favorite for all, then the choice should be ai . ~ implies Definition 8 (SC-U). An SCF F is unanimous 5 if {ai } topping ~ F () = ai . (SC-U) makes sure that an SCF is minimally pro-social. Another prosocial condition is runoff unanimity. ~ ∈ {ai , aj } Definition 9 (SC-RU). An SCF F is runoff unanimous if F () ~ for every {ai , aj }-runoff . A runoff unanimous SCF respects a two-party system: if the society agrees that {ai , aj } are most preferred by everyone, then the choice should be between them. Note that (SC-RU) is neither stronger nor weaker than (SC-U). A stronger condition than both of them is weak Pareto efficiency, sufficient ~ implies F () ~ ∈ A0 , while (SC-U) for the statement that A0 6= ∅ topping 0 0 restricts |A | = 1 and (SC-RU) rules for |A | = 2. Definition 10 (SC-WP). An SCF F is weakly Paretian 6 if aj n ai for every ~ 6= ai . n implies F () 5 6
“Pareto efficient” in Reny (2001). Consistent with Mas-Colell, Whinston, and Green (1995, page 808).
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~ have a1 1 a2 1 a3 and a2 2 a1 2 a3 . (SC-U) For example, let ~ = a3 , but (SC-RU) or (SC-WP) does not. Let the profile be permits F () ~ = a2 , but a1 1 a2 1 a3 and a1 2 a3 2 a2 . (SC-RU) permits F () (SC-U) or (SC-WP) does not. Claim 1. (SC-WIIA) and (SC-U) imply (SC-WP). ~ = ai by (SC-U), so The proof is trivial. If ai n aj for every n, Fij () ~ (SC-WIIA) rules out F () = aj . Claim 2. (SC-WIIA) and (SC-RU) imply (SC-IIA). ~ and ~ 0 share rankings over {ai , aj }. If F () ~ = ai and Indeed, let 0 ~ ) = aj , Fij () ~ is not either aj or ai by (SC-WIIA), violating (SC-RU). F ( Since (SC-WP) is stronger than (SC-RU), we now know that (SC-WIIA) and (SC-U) is sufficient for (SC-IIA). A dictator rules the social choice to be among her favorite alternatives, and an SCF is dictatorial if we can identify a dictator. Definition 11 (SC-D). An SCF F is dictatorial if there exists a dictator n ~ ∈ {ai : ai n aj for all j 6= i} for every . ~ such that F () Theorem 1. If an SCF F satisfies (SC-WIIA) and (SC-U), then it is dictatorial. Proof of Theorem 1: Claim 1 and 2 bring in (SC-WP) and (SC-IIA). Consider ~ with {ai } and {ai , aj } at the top. Swap the positions of {ai , aj } an arbitrary sequentially from 1 to N . By (SC-WP), the choice is either ai or aj , starting with ai and ending with aj . The (i, j)-pivotal voter nij is the first whose ~ by (SC-IIA). swap makes a difference. The definition is independent of
~0
~ 00
1 aj ak ai
··· ··· ··· ···
aj ai
··· ··· ··· ···
nij − 1 nij aj ai ak ai aj ak aj aj ai ai ak 4
nij + 1 ai
··· ···
N ai
aj ak
··· ···
aj ak
ai aj
··· ··· ··· ···
ai aj
~ 0 and ~ 00 with the depicted rankings at the top, Consider arbitrary where squares mark possible positions of ak , with indifference between aj ~ 0 ) = ai and Fij ( ~ 00 ) = aj , and ak allowed. The definition of nij requires Fij ( ~ 0 ) 6= aj and F ( ~ 0 ) 6= ai by (SC-IIA). We get F ( ~ 0 ) = ai , for (SCso F ( ~ 0 and ~ 00 share rankings over {ai , ak }, WP) rules out ak and others. As ~ 00 ) 6= ak and we are left with F ( ~ 00 ) = aj . Focusing on aj and ak in ~ 00 , F ( we conclude by (SC-IIA) that ~ 6= ak . aj n ak ⇒ F ()
(∗)
In the swapping process that defines njk , (∗) says that the choice remains aj as long as nij ranks j above k, so njk ≥ nij . For nkj , the choice should become aj no later than nij makes the change, so nkj ≤ nij . We have njk ≥ nij ≥ nkj . As j and k are arbitrary, nkj ≥ njk also holds, implying njk = nkj = nij , which can be easily extended to all nts ’s. But (∗) requires this unique pivotal voter to be a dictator. In summary, (SC-U), (SC-RU), and (SC-WP) are disciplines about “prosocialness”, which rules that when the society have well-formulated opinions, the SCF respects them. (SC-WIIA) and (SC-IIA) belong to the class of “complaint-freeness”, which protect the designer from claims by the unsatisfied based on her own decisions. A pair of the weakest conditions in these two classes generate dictatorship. What a disappointment! 2. The Muller-Satterthwaite and Gibbard-Satterthwaite Theorems We are unenthusiastic about an SCF that rules out alternatives from the ~ = ai for at least some ~ ∈ PN . outset, that is, for every ai , we want F () Definition 12 (SC-O). An SCF F is onto if F (P N ) = A. (SC-O) says that F does consider every available alternative, but it allows for anti-social choices. Even (SC-U) is stronger than (SC-O), so (SC-WIIA) is too weak here. Monotonicity forbids F to discard the current choice ai due to an adjustment that makes it no less favorable. ~ 0 maintains ai ’s positions in ~ if ai n aj implies Definition 13. A profile 0 0 ai n aj and ai n aj implies ai n aj for every n and every j 6= i. ~ = ai and Definition 14 (SC-M). An SCF F is monotonic if whenever F () 0 0 ~ maintains ai ’s positions in , ~ F ( ~ ) = ai . 5
Assuming (SC-O), (SC-M) is sufficient for generating dictatorship, but it is also sufficient to discipline only two special maintenances of positions. (SC-TM) says that if ai is in consideration after all, ai should be the choice when it is everyone’s favorite. (SC-TM) rules that if ai wins in some cases, it wins any runoff as long as its rankings against the opponent remain the same. ~ = ai for Definition 15 (SC-TM). An SCF F is topping monotonic if F () 0 0 ~ implies F ( ~ ) = ai for every ~ with {ai } at the top. some ~ and Definition 16 (SC-RM). An SCF F is runoff monotonic if whenever 0 0 ~ share rankings over {ai , aj }, F () ~ = ai implies F ( ~ ) = ai . {ai , aj }-runoff We can demonstrate that (SC-TM) and (SC-RM) combined are strictly weaker than (SC-M). Let M ≥ 3. The SCF considers only {a1 , a2 }. It picks between them the one that an anti-dictator n dislikes unless the society decide to save her and everyone puts one alternative above the other. Ties are broken in favor of a1 . When a1 or a2 is at the top, it is picked, so (SC-TM) is true. To see (SC-RM), we reason that an alternative a1 is selected either as the societal dominant one or as a worse or tied one for n but undominated. In the first case, entering a corresponding runoff keeps ai the dominant one. In the second case, a runoff against aj ∈ {a1 , a2 } with j 6= i keeps its relative status, but a runoff against ak ∈ / {a1 , a2 } makes it dominant. For the violation of (SC-M), note that simply moving ai up for n could result in its elimination. Claim 3. An SCF F satisfies (SC-U) if and only if it satisfies (SC-O) and (SC-TM). Both directions are trivial. Claim 4. (SC-RM) implies (SC-IIA). ~ and ~ 0 share rankings over {ai , aj }. Suppose F () ~ = ai and Indeed, let 0 ~ ) = aj . (SC-RM) rules Fij () ~ = ai and Fij () ~ = aj , a contradiction. F ( Theorem 2 (Muller-Satterthwaite). If an SCF F satisfies (SC-O) and (SCM), then it is dictatorial. (SC-IIA) is stronger than (SC-WIIA), and (SC-M) is stronger than (SCTM) and (SC-RM) combined. It follows directly from theorem 1, claim 3, claim 4. Pro-socialness of (SC-U) in theorem 1 is dissembled and distributed to (SC-O) and (SC-M) in theorem 2. Although we can replace (SC-M) with a 6
strictly weaker combination of (SC-TM) and (SC-WIIA), the theorem still suggests an illuminating introduction to the Gibbard-Satterthwaite theorem, a pathbreaking development that brought incentives into the field. We write out the well-known connection for completeness. When individual preferences are private information, any implementations of SCFs could only count on reported preferences. The incentive to misrepresent one’s own preferences could be an important concern, urging designers to look for mechanisms that minimize the incentive to manipulate the social choice. The Gibbard-Satterthwaite theorem presents an obstacle that this enterprise can never remove. We can start with some logistics. When we replace the preferences of ~ −n ). Strategy~ with 0n ∈ P, denote the new profile (0n , individual n in proofness ensures truth-telling by always switching to a strictly worse alternative for an individual if only her misreport alters the outcome. ~ −n ) 6= F () ~ Definition 17 (SC-SP). An SCF F is strategy-proof if F (0n , ~ n F (0n , ~ −n ) for every n, every , ~ and every 0n . implies F () Muller and Satterthwaite (1977) shows that strategy-proofness is equivalent to monotonicity, but we only need one direction. Claim 5. (SC-SP) implies (SC-M). ~ = ai and ~ 0 maintains ai ’s positions in . ~ If ~ 00 ≡ (01 , ~ −1 ), Let F () 00 00 ~ = (1 , ~ −1 ). Suppose aj = F ( ~ ) 6= ai . (SC-SP) demands ai 1 aj then 0 ~ lowers ai ’s positions relative to aj at least for individual and aj 01 ai , so ~ 00 ) = ai . Following the 1, contradicting the assumption. As a result, F ( ~ to ~ 0 one by one without altering the choice. same logic, we can update Interestingly, (SC-M) wraps up (SC-TM) and (SC-IIA) in such a simple and informative way that we can reach theorem 3 through the famous onestep proof above. Theorem 3 (Gibbard-Satterthwaite). If an SCF F satisfies (SC-O) and (SC-SP), then it is dictatorial. It is a direct consequence of theorem 2 and claim 5. 3. A Strong Form of Arrow’s Impossibility Theorem To explore the links between SCFs and SWFs, we can imagine a world with M ≥ 3 songs produced per year. A firm collects personal preferences 7
of N ≥ 2 critics so as to publish an “annual ranking” and present a “song of the year” award.7 In devising an award algorithm or SCF, the firm can never succeed in preventing one critic’s opinion from dominating and at the same time meeting the premises of theorem 1, 2, or 3, which all seem reasonable. Arrow (1951) casts doubt on the possibility of a desirable ranking algorithm or SWF. Our proof utilizes the fact that a ranking algorithm contains an award algorithm: the song of the year is often the number one song in ranking. The setup is substantially shorter if we attend only to strict social preferences, but we demonstrate how to deal with the complication of indifference here. Let ˆ ~ )aj denote ai R( ~ )aj and not aj R( ~ )ai , the strict social preferences. ai R( ˜ ~ ~ ~ Let ai R( )aj denote ai R( )aj and aj R( )ai , the indifference. Definition 18. The candidate set function C R : P N → 2A derived from an ~ ∈ P N a subset {ai : ai R()a ~ j for all j 6= i} of A. SWF R assigns to each Definition 19. An SCF F R (·; ∗ ) is derived from an SWF R with a tie~ ∗ ) ∈ C R () ~ and ai ∗ aj for every breaking rule ∗ ∈ P¯ if ai ≡ F R (; ~ with j 6= i. aj ∈ C R () R F (·; ∗ ) is unique, and it is worth noting that RF can not be defined in a natural way unless we assume more structures, as an SCF F does not discipline rankings among losing alternatives. We can thus easily define (SW) conditions for R by requiring F R (·; ∗ ) to satisfy (SC-) conditions for every ∗ ∈ P, but not the other way around. We define some but leave others to the reader. Definition 20 (SW-U). An SWF R is unanimous if F R (·; ∗ ) satisfies (SC~ implies C R () ~ = {ai }. U) for every ∗ ∈ P, i.e., {ai } topping Definition 21 (SW-RU). An SWF R is runoff unanimous if F R (·; ∗ ) sat~ implies C R () ~ ⊂ isfies (SC-RU) for every ∗ ∈ P, i.e., {ai , aj } topping {ai , aj }. Definition 22 (SW-WP). An SWF R is weakly Paretian if F R (·; ∗ ) satisfies ~ (SC-WP) for every ∗ ∈ P, i.e., aj n ai for every n implies ai ∈ / C R (). (SW-WP) still implies (SW-U) and (SW-RU). We attach “Arrow” to concepts undefinable in terms of F R . 7
Arrow (2000) offers interpretations of SWFs in the context of expert systems.
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Definition 23 (SW-AU). An SWF R is Arrow unanimous if ai n aj for ˆ )a ~ j. every n implies ai R( Definition 24 (SW-AIIA). An SWF R satisfies Arrow’s independence of ~ and ~ 0 share rankings over {ai , aj }, irrelevant alternatives if whenever ~ j implies ai R( ~ 0 )aj . ai R()a Definition 25 (SW-AD). An SWF R is Arrow dictatorial if there exists a ˆ )a ~ j for every . ~ dictator n such that ai n aj implies ai R( These three concepts and (SW-AWIIA) to be introduced are stronger than (SW-WP), (SW-IIA), (SW-D), and (SW-WIIA) respectively in that they also regulate losing alternatives. The common form of Arrow’s theorem assumes (SW-AU) and (SW-AIIA) to generate Arrow dictatorship (Yu, 2012, and innumerable works of others). We can replace them with strictly weaker conditions separately. Definition 26 (SW-AWIIA). An SWF R satisfies Arrow’s weak indepen~ and {ai , aj }-runoff ~ 0 share dence of irrelevant alternatives if whenever ˆ )a ~ 0 ) = {ai } implies ai R( ~ j , and C R ( ~ 0) = rankings over {ai , aj }, C R ( ˜ )a ~ j. {ai , aj } implies ai R( Similar to the relation between (SC-IIA) and (SC-WIIA), (SW-AIIA) eliminates complaints about a particular social ranking of {ai , aj } justified by any profile with the same rankings over them, while (SW-AWIIA) deals only with justifications based on runoffs: winning a runoff announces the superiority of an alternative in a salient way, as tying in a runoff reveals social indifference. The “bone assignment” algorithm in section 2 can still support the claim that (SW-AWIIA) is strictly weaker than (SW-AIIA). Claim 6. (SW-U) and (SW-AWIIA) implies (SW-AU). ~ C R designates To see this, let ai n aj for every n. Since {ai } tops Tij (), ˆ )a ~ j. {ai } by (SW-U). (SW-AWIIA) requires ai R( Claim 7. (SW-RU) and (SW-AWIIA) imply (SW-AIIA). ˆ ~ and ~ 0 share rankings over {ai , aj }. Suppose ai R()a ~ j and aj R( ~ 0 )ai . Let ~ (SW-AWIIA) denies {ai }, {aj }, and {ai , aj }, With {ai , aj } topping Tij (), violating (SW-RU). Theorem 4 (Strong Form of Arrow’s). If an SWF R satisfies (SW-U) and (SW-AWIIA), then it is Arrow dictatorial. 9
Indeed, claim 6 and 7 bring in (SW-AU) and (SW-AIIA), requiring F R (·, ∗ ) for any ∗ to satisfy (SC-U) and (SC-WIIA). Theorem 1 says that F R presents a social choice dictator n. She is a social welfare dictator too. If ˆ ij ()a ~ ∗ ) = ai , meaning ai R ~ j by the ai n aj , individual n rules FijR (, ˆ )a ~ j , and we are done. construction of FijR . By (SW-AIIA), ai R( 4. Another result about Social Welfare Functions There also exists an SWF analogue of the Muller-Satterthwaite theorem. ˆ )a ~ j for some Definition 27 (SW-AO). An SWF R is Arrow onto if ai R( ~ . Definition 28 (SW-AM). An SWF R is Arrow monotonic if whenever ˆ )a ˆ ~ j and ~ 0 maintains ai ’s positions in , ~ ai R( ~ 0 )aj . ai R( ˆ )a ~ j Definition 29 (SW-ATM). An SWF R is Arrow topping monotonic if ai R( 0 0 ˆ ~ implies ai R( ~ )aj for every ~ with {ai } at the top. for some Note that (SW-AO) is weaker than (SW-O), but (SW-AM) and (SWATM) are more restrictive than (SW-M) and (SW-TM) respectively. (SWATM) is still part of (SW-AM). The following statement has an interesting counterpart in claim 3. Claim 8. An SWF R satisfies (SW-U) if and only if it satisfies (SW-AO) and (SW-ATM). We skip the proof which merely checks the definitions. Claim 9. (SW-AM) implies (SW-AIIA). ~ and ~ 0 share rankings over {ai , aj }. Suppose ai R()a ~ j and Indeed, let 0 0 ˆ ~ )ai . Tij () ~ maintains {ai , aj }’s positions in ~ and ~ , so CijR () ~ is aj R( not well-defined. Theorem 5. If an SWF R satisfies (SW-AO) and (SW-AM), then it is Arrow dictatorial. This follows directly from theorem 7, claim 8, and claim 9. As in the case of the Muller-Satterthwaite theorem, (SW-AM) is reducible in a strict sense to (SW-ATM) and (SW-AIIA). References 10
Arrow, K. J., 1951. Social choice and individual values. NewYork: Wiley. Arrow, K. J., 2000. Social choice: Many individuals or many criteria. Peking: Capital University of Economics and Business Press. Mas-Colell, A., Whinston, M., Green, J., 1995. Microeconomic Theory. Oxford University Press. Muller, E., Satterthwaite, M. A., 1977. The equivalence of strong positive association and strategy-proofness. Journal of Economic Theory 14 (2), 412–418. Reny, P. J., 2001. Arrow’s theorem and the Gibbard-Satterthwaite theorem: a unified approach. Economics Letters 70 (1), 99–105. Yu, N. N., 2012. A one-shot proof of Arrow’s impossibility theorem. Economic Theory 50 (2), 523–525.
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