Misrepresentation of Preferences

Transcription

1 Misrepresenttion of Preferences Gicomo Bonnno Deprtment of Economics, University of Cliforni, Dvis, USA Socil choice functions Arrow s theorem sys tht it is not possible to extrct from profile of individul preferences preference rnking for society with procedure tht stisfies five desirble properties: Unrestricted Domin, Rtionlity, Unnimity, Nondicttorship nd Independence of Irrelevnt Alterntives. Perhps Arrow s pproch is too demnding, in tht it requires tht rnking of the entire set of lterntives be obtined for society. After ll, if the purpose of voting procedures is to rrive t some choice mong the lterntives, then we cn dispense with complete rnking nd just focus on the finl choice. Thus, we could look for simpler object tht extrcts from profile of individul preferences one lterntive, to be thought of s society s choice. Such n object is clled Socil Choice Function (SCF). Definition.. Let X = {x, x,..., x m } (m ) be finite set of lterntives, N = {,,..., n} (n ) finite set of individuls, R the set of complete nd trnsitive binry reltions on X, R n the crtesin product R R R (thus } {{ } n times n element of R n is list of complete nd trnsitive preference reltions on the set of lterntives X, one for ech individul; we cll n element of R n profile of preferences) nd let S be subset of R n. A socil choice function is function f : S X tht tkes s input profile of preferences for the individuls

2 Misrepresenttion of Preferences (,,..., n ) S nd produces s output n lterntive f (,,..., n ) X to be thought of s society s choice. For exmple, suppose tht there re only two lterntives, nd b (thus X = {, b}), only strict rnkings cn be reported (tht is, S = { b, b } { b, b }), nd two voters (N = {, }). Then, in order to construct SCF we need to replce ech in the following tble with either n or b: Individul s rnking Individul s b rnking b Thus, there re 4 = 6 possible SCFs, which re listed below: (SCF-) b b (SCF-) b b b (SCF-3) b b b (SCF-4) b b b (SCF-5) b b b (SCF-6) b b b b (SCF-7) b b b b (SCF-8) b b b b (SCF-9) b b b b (SCF-0) b b b b

3 G Bonnno 3 (SCF-) b b b b (SCF-) b b b b b (SCF-3) b b b b b (SCF-4) b b b b b (SCF-5) b b b b b (SCF-6) b b b b b b Which of these SCFs should one reject on the bsis of some generl resonble requirements? First requirement: Unnimity. If ll the individuls list lterntive x t the top of their reported rnkings then x should be chosen. In the bove exmple this requirement mounts to insisting tht the min digonl be s follows: b. By ppeling to Unnimity we cn thus reject SCF-, SCF-3, SCF-4, SCF-5, SCF-8, SCF-9, SCF-0, SCF-, SCF-3, SCF-4, SCF-5 nd SCF-6. Thus, we re left with the following four SCFs: (SCF-) b b b (SCF-6) b b b b (SCF-7) b b b b (SCF-) b b b b b Second requirement: Non-dicttorship. There should not be dicttor, tht is, n individul whose top lterntive is lwys chosen. In the bove

4 4 Misrepresenttion of Preferences exmple there should not be n individul who is such tht if he reports b then is chosen nd if he reports b then b is chosen. On the bsis of Non-dicttorship we must thus reject SCF-6 (where Individul is dicttor) nd SCF-7 (where Individul is dicttor). Hence, we re left two SCFs: (SCF-) b b b (SCF-) b b b b b Cn these two remining SCFs be considered resonble or good? Are there ny other requirements tht one should impose? One issue tht we hve not ddressed so fr is the issue of misrepresenttion of preferences. We hve implicitly ssumed up to now tht ech individul, when sked to report her rnking of the lterntives, will do so sincerely, tht is, she will not report rnking tht is different from her true rnking. Is this n issue one should worry bout? In the next section we will go through number of populr SCFs nd show tht they ll provide incentives for individuls to lie in reporting their preferences. Test your understnding of the concepts introduced in this section, by going through the exercises in Section 5. t the end of this chpter. Strtegic voting We shll illustrte the issue of strtegic voting, or misrepresenttion of preferences, in severl populr voting methods which cn be viewed s socil choice functions. Plurlity voting with defult lterntive. We illustrte this procedure for the cse of three lterntives: X = {, b, c} nd three voters: N = {,, 3}. We ssume tht ech voter cn only report strict rnking of the lterntives (tht is, indifference is not llowed). Thus s we sw in the previous chpter there re six possible rnkings tht n individul cn choose from when deciding wht to report: b c, c b, b c, b c, c b, c b. To simplify the nottion, we shll write them s bc, cb, bc, bc, cb, cb, tht is, we red xyz s x y z. We tke to be the designted defult lterntive nd the voting procedure is s follows:

5 G Bonnno 5 If two or more individuls list lterntive b t the top of their rnking, then b is chosen, if two or more individuls list lterntive c t the top of their rnking, then c is chosen, otherwise, the defult lterntive is chosen (thus, is chosen when two or more individuls list it t the top of their rnking or when there is complete disgreement, in the sense tht one individul lists t the top, nother lists b t the top nd the third lists c t the top). How cn we represent this voting procedure or SCF? We need six tbles: ech tble lbeled with one possible reported rnking of Individul 3; ech tble hs six rows: ech row lbeled with one possible reported rnking of Individul, nd six columns: ech column lbeled with one possible reported rnking of Individul. Inside ech cell of ech tble we write the lterntive chosen by the procedure described bove. This is shown in Figure. Let us first check if this SCF stisfies Unnimity nd Non-dicttorship. Unnimity requires tht when n lterntive is listed t the top of ech reported rnking then it should be chosen, tht is, it requires the following, which is highlighted in Figure :. in the two tbles t the top (corresponding to the cses where Voter 3 reports bc or cb) there should be n in the following cells: (row, column ), (row, column ), (row, column ) nd (row, column ) [these re the cses where every voter rnks t the top],. in the two tbles in the middle (corresponding to the cses where Voter 3 reports bc or bc) there should be b in the following cells: (row 3, column 3), (row 3, column 4), (row 4, column 3) nd (row 4, column 4) [these re the cses where every voter rnks b t the top], 3. in the two tbles t the bottom (corresponding to the cses where Voter 3 reports cb or cb) there should be c in the following cells: (row 5, column 5), (row 5, column 6), (row 6, column 5) nd (row 6, column 6) [these re the cses where every voter rnks c t the top]. Thus, Unnimity only restricts the vlues in four cells in ech tble s shown in Figure. Non-dicttorship is lso stisfied, since for ech individul there is t lest one sitution where she lists n lterntive, sy x, t the top nd yet tht lterntive is not chosen becuse the other two individuls list different lterntive, sy y, t the top.

6 6 Misrepresenttion of Preferences s bc cb bc b b bc b b cb c c cb c c bc cb bc bc cb cb s bc cb bc bc cb cb bc cb bc b b bc b b cb c c cb c c 3 reports bc 3 reports cb s bc b b cb b b bc b b b b b b bc b b b b b b cb b b c c cb b b c c bc cb bc bc cb cb s bc cb bc bc cb cb bc b b cb b b bc b b b b b b bc b b b b b b cb b b c c cb b b c c 3 reports bc 3 reports bc ` bc cb bc bc cb cb ` bc cb bc bc cb cb bc c c bc c c cb c c cb c c bc b b c c bc b b c c bc b b c c bc b b c c cb c c c c c c cb c c c c c c cb c c c c c c cb c c c c c c 3 reports cb 3 reports cb Figure : Plurlity voting with s the defult lterntive

7 G Bonnno 7 s bc cb bc b b bc b b cb c c cb c c bc cb bc bc cb cb s bc cb bc bc cb cb bc cb bc b b bc b b cb c c cb c c 3 reports bc 3 reports cb s bc b b cb b b bc b b b b b b bc b b b b b b cb b b c c cb b b c c bc cb bc bc cb cb s bc cb bc bc cb cb bc b b cb b b bc b b b b b b bc b b b b b b cb b b c c cb b b c c 3 reports bc 3 reports bc ` bc cb bc bc cb cb ` bc cb bc bc cb cb bc c c bc c c cb c c cb c c bc b b c c bc b b c c bc b b c c bc b b c c cb c c c c c c cb c c c c c c cb c c c c c c cb c c c c c c 3 reports cb 3 reports cb Figure : The highlights show the restrictions imposed by Unnimity

8 8 Misrepresenttion of Preferences It remins to verify if it is the cse tht no individul cn ever gin by lying bout her preferences, tht is, by reporting rnking tht is not her true rnking. We cll this requirement Non-mnipulbility or Strtegy-proofness. Unfortuntely, this requirement is violted in this voting procedure. To see this, focus on the first tble in Figure, corresponding to the cse where Individul 3 reports the rnking bc. This tble is reproduced in Figure 3. Consider the sixth column, corresponding to the cse where Individul reports the rnking cb. Suppose tht the true rnking of Individul is bc (4 th row); if she reports her preferences truthfully, tht is, if she reports bc (recll tht this mens b c ) then the chosen lterntive is, which is the worst, ccording to her true preferences; if, on the other hnd, she lies nd reports the flse rnking cb then the chosen lterntive is c, which ccording to her true rnking is better thn (in her true rnking, nmely bc, c is the middle-rnked lterntive while is the worst). s bc bc cb bc bc cb cb cb bc b b bc b b cb c c cb c c 3 reports bc Figure 3: The top-left tble in Figure The Condorcet method with defult lterntive. The Condorcet method selects tht lterntive clled the Condorcet winner tht would win mjority vote in ll the pirwise comprisons with ech of the other lterntives; if such n lterntive does not exist, then pre-determined defult lterntive is Becuse there is complete disgreement: Voter lists b t the top, Voter lists c t the top nd Voter 3 lists t the top; hence, the defult lterntive, nmely, is chosen.

9 G Bonnno 9 selected. As we did with plurlity voting, we illustrte this procedure for the cse of three lterntives: X = {, b, c} nd three voters: N = {,, 3}, ssuming tht ech voter cn only report strict rnking of the lterntives. As before, we denote the rnking x y z by xyz. We tke to be the designted defult lterntive. Let us first see wht lterntive the Condorcet method would select in couple of situtions. If the reported rnkings re s follows: Voter Voter Voter 3 best c b b b worst c c then b is the Condorcet winner: mjority (consisting of Voters nd ) prefers b ro nd mjority (consisting of Voters nd 3) prefers b to c. Thus, b is selected. On the other hnd, if the reported rnkings re: Voter Voter Voter 3 best c b b c worst c b then there is no Condorcet winner: bets b but is beten by c, b bets c but is beten by nd c bets but is beten by b (indeed, the mjority rule yields cycle: mjority prefers to b, mjority prefers b to c nd mjority prefers c to ). Thus, since there is no Condorcet winner, the defult lterntive is chosen. As we did with plurlity voting, we cn represent this SCF by mens of six tbles, ech with six rows nd six columns, s shown in Figure 4. The reder might wnt to try to construct the tbles before looking t Figure 4. Note tht this is different SCF from the one shown in Figure. For exmple, the entry in the first tble, row 6 nd column 3 (corresponding to reported rnkings cb for Voter, bc for Voter nd bc for Voter 3) is for plurlity voting (the defult lterntive since no two voters rnk the sme lterntive t the top), but b for the Condorcet method (b is the Condorcet winner). It is strightforwrd to verify tht the SCF shown in Figure 4 stisfies Unnimity nd Non-dicttorship (see Exercise 6). On the other hnd, it fils to stisfy Non-mnipulbility. To see this, suppose tht Voter s true rnking is bc nd consider the first tble, row 5 nd column 4, corresponding to the cse where Voter reports cb, Voter reports bc (thus, truthful report) nd Voter 3 reports bc. Then the chosen lterntive is, which is the worst in Voter s

10 0 Misrepresenttion of Preferences s bc cb bc b b b bc b b b cb c c cb b b c c bc cb bc bc cb cb s bc cb bc bc cb cb bc cb bc b b bc b b c c cb c c c cb c c c 3 reports bc 3 reports cb s bc b b cb cb b cb b b bc b b b b b b bc b b b b b b cb b b c c cb b b b c c bc cb bc bc s 3 reports bc bc cb bc bc cb cb bc b b b cb b b c c bc b b b b b b bc b b b b b b cb c b b c c cb b c b b c c 3 reports bc ` bc cb bc bc cb cb ` bc cb bc bc cb cb bc c c bc b b c c cb c c c cb c c c bc b b c c bc b b b c c bc c b b c c bc b c b b c c cb c c c c c c cb c c c c c c cb c c c c c c cb c c c c c c 3 reports cb 3 reports cb Figure 4: The Condorcet method with s the defult lterntive Pge of 3

11 G Bonnno true rnking. If Voter were to misrepresent his preferences by reporting cb, then the chosen lterntive would be c, which ccording to his true rnking bc is better thn. The Bord count. The Bord count is the following SCF. Ech voter sttes strict rnking (tht is, no indifference is llowed) of the m lterntives. For ech voter s rnking, m points re ssigned to the lterntive rnked first, m points to the lterntive rnked second, nd so on, up to point for the worst lterntive. Then, for ech lterntive, ll the points re dded up nd the lterntive with the lrgest score is chosen. A tie-breking rule must be specified in cse two or more lterntives receive the lrgest score. Like the previous two SCFs, the Bord count stisfies Unnimity nd Nondicttorship but fils to stisfy Non-mnipulbility. For exmple, suppose tht there re five lterntives: X = {, b, c, d, e} nd five voters: N = {,, 3, 4, 5}. Suppose tht Voter s true rnking is: best worst Voter s true rnking c d b e Suppose lso tht Voter expects the other voters to report the following rnkings: Voter Voter 3 Voter 4 Voter 5 best b b c c c d b e e d e e d worst d b c If Voter reports her true rnking, then we get the following profile of rnkings: Voter Voter Voter 3 Voter 4 Voter 5 score best b b c 5 c c c d b 4 d e e 3 b d e e d worst e d b c ()

12 Misrepresenttion of Preferences Applying the Bord count we get the following scores, so tht lterntive c is chosen. : = 7 b : = 7 c : = 8 d : = e : = If, insted of her true rnking (), Voter were to report the following rnking: best e d c worst b then we would get the following profile of rnkings: Voter Voter Voter 3 Voter 4 Voter 5 score best b b c 5 e c c d b 4 d e e 3 c d e e d worst b d b c Applying the Bord count we get the following scores: : = 7 b : = 6 c : = 6 d : = e : = 4 so tht lterntive would be chosen, which ccording to her true rnking () Voter prefers to c. Hence, Voter hs n incentive to misrepresent her preferences. Note tht mnipulbility of SCF does not men tht for every individul there is sitution where tht individul cn bring bout better outcome by misrepresenting her preferences. A SCF is mnipulble s long s there is t lest one individul who cn bring bout better outcome by reporting rnking which is different from her true rnking in t lest one sitution.

13 G Bonnno 3 For exmple, consider the following SCF. There re three lterntives: X = {, b, c} nd three voters: N = {,, 3}. Ech voter reports strict rnking of the lterntives. Voter is given privileged sttus in tht her top-rnked lterntive is ssigned.5 points (nd the other two lterntives 0 points), while for ech of the other two voters his top-rnked lterntive is ssigned point (nd the other two lterntives 0 points). The lterntive with the lrgest number of points is selected. This SCF is shown in Figure 5. The privileged sttus of Voter s bc cb bc b b b b bc b b b b cb c c c c cb c c c c bc cb bc bc cb cb s bc cb bc bc cb cb bc cb bc b b b b bc b b b b cb c c c c cb c c c c 3 reports bc 3 reports cb s bc b b cb cb cb b b bc b b b b b b bc b b b b b b cb c c b b c c cb c c b b c c bc cb bc bc s 3 reports bc bc cb bc bc cb cb bc b b cb b b bc b b b b b b bc b b b b b b cb c c b b c c cb c c b b c c 3 reports bc ` bc cb bc bc cb cb ` bc cb bc bc cb cb bc c c bc c c cb c c cb c c bc b b b b c c bc b b b b c c bc b b b b c c bc b b b b c c cb c c c c c c cb c c c c c c cb c c c c c c cb c c c c c c 3 reports cb 3 reports cb Pge of Figure 5: Mjority voting with slight dvntge given to Voter hs n impct only when there is complete disgreement, s is the cse, for exmple, in the first tble, row 3, column 5, corresponding to the cse where Voter reports bc, Voter reports cb nd Voter 3 reports bc. In this cse b gets.5 points, c gets point nd gets point, so tht b the top lterntive in Voter s reported rnking is selected. In this SCF there is no sitution where Voter cn benefit from misreporting her preferences. To see this, suppose tht the

14 4 Misrepresenttion of Preferences top-rnked lterntive in Voter s true rnking is x. One of two scenrios must occur:. Voters nd 3 report the sme lterntive, cll it y, t the top of their rnking (it might be tht y = x or might it be be tht y x). In this cse lterntive y is chosen nd it will be chosen no mtter wht rnking Voter reports (y lredy gets points nd Voter s report either dds.5 points to y or ssigns.5 points to n lterntive different from y). Thus, in this scenrio, telling the truth nd lying produce the sme outcome; in prticulr, lying cnnot be better thn truthful reporting.. Voters nd 3 report different lterntives t the top of their rnkings. In this cse if Voter reports truthfully, the chosen lterntive will be x (either there is complete disgreement nd x gets.5 points, while the other two lterntives get point ech, or one of Voters nd 3 hs x t the top, in which cse x gets.5 points). If Voter lies then the lterntive t the top of her reported rnking is chosen (sme resoning: either it is chosen becuse there is complete disgreement or it is chosen becuse Voter forms mjority with one of the other two voters). Thus, if the lterntive t the top of her reported rnking is not x, then Voter is worse off by lying. On the other hnd, for both Voter nd Voter 3 there re situtions where they gin by misrepresenting their preferences. We shll show this for Voter nd let the reder show tht Voter 3 cn gin by misrepresenttion (Exercise 7). For Voter, consider the sitution represented by the first tble (Voter 3 reports bc) nd row 6 (Voter reports cb) nd suppose tht Voter s true rnking is bc (column 3). If Voter reports truthfully, then the selected lterntive is c, which is the worst from his point of view; if, on the other hnd, Voter 3 reports bc, then the selected lterntive is, which ccording to Voter s true rnking bc is better thn c. Test your understnding of the concepts introduced in this section, by going through the exercises in Section 5. t the end of this chpter.

15 G Bonnno 5 3 The Gibbrd-Stterthwite theorem The Gibbrd-Stterthwite theorem is result published independently by philosopher Alln Gibbrd in 973 nd economist Mrk Stterthwite in As for the cse of Arrow s theorem, the objective is to determine if there re Socil Choice Functions (see Definition.) tht stisfy some resonble properties, which we will cll xioms, s we did in the previous chpter. We ssume tht the domin of the Socil Choice Function (SCF) is the set of profiles of strict rnkings of the set of lterntives X (tht is, indifference is ruled out). Let P denote the set of strict rnkings of the elements of X. Then the domin of the SCF is tken to be P n = P P. Thus, individuls re not } {{ } n times llowed to report indifference between ny two lterntives, but subject to this restriction ny strict rnking cn be reported. Hence, this is limited form of the property of Unrestricted Domin considered in the previous chpter. The xioms tht we consider re the following: Axiom : Unnimity. If lterntive x is the top-rnked lterntive in the reported rnking of every individul, then it should be chosen by society: if, for every individul i, x i y for every lterntive y x then f (,..., n ) = x. Axiom : Non-dicttorship. There is no individul i whose top lterntive in her reported rnking is lwys chosen. Formlly, this cn be stted s follows: for every individul i N, there is profile of reported preferences (,..., n ) such tht if f (,..., n ) = x X then x is not t the top of i (tht is, there exists y X such tht y x nd y i x). Axiom 3: Non-mnipulbility or Strtegy-proofness. There is no sitution where some individul cn gin by reporting rnking different from her true rnking. Formlly, this cn be stted s follows. Fix n rbitrry individul i N nd n rbitrry profile (,..., i, i, i+,..., n ) P n Alln Gibbrd, Mnipultion of voting schemes: generl result, Econometric, 973, Vol. 4 (4), pges Mrk Stterthwite, Strtegy-proofness nd Arrow s conditions: existence nd correspondence theorems for voting procedures nd socil welfre functions, Journl of Economic Theory, 975, Vol. 0 (), pges 87-7.

16 6 Misrepresenttion of Preferences nd let f (,..., i, i, i+,..., n ) = x X. Then there is no i P such tht f (,..., i, i, i+,..., n ) i x (think of i s the true rnking of individul i nd s possible lie). i The following theorem provides n impossibility result similr to Arrow s impossibility theorem. Theorem. [Gibbrd-Stterthwite theorem] If the set of lterntives X contins t lest three elements, there is no Socil Choice Function f : P n X tht stisfies Unnimity, Non-dicttorship nd Non-mnipulbility. An lterntive wy of stting the bove theorem is s follows: if SCF stisfies Unnimity nd one of the other two xioms then it fils to stisfy the third xiom (for exmple, if SCF stisfies Unnimity nd Non-dicttorship then it violtes Non-mnipulbility). 4 In the next section we illustrte the logic of the proof of Theorem by focusing on the simple cse of three lterntives nd two voters. 5 Test your understnding of the concepts introduced in this section, by going through the exercises in Section 5.3 t the end of this chpter. 4 Illustrtion of the proof of the Gibbrd-Stterthwite theorem In this section we prove the Gibbrd-Stterthwite theorem (Theorem ) for the cse where there re three lterntives, clled x, y nd z, nd two individuls (N = {, }). There re six possible strict rnkings of the set X = {x, y, z} nd thus ny SCF cn be represented s tble with six rows nd six columns. We will show tht ny SCF tht stisfies Unnimity nd Non-mnipulbility must violte Non-dicttorship. Fix SCF tht stisfies Unnimity. Then the blocks on the min digonl must be filled s shown in Figure 6 (Unnimity forces the vlues of out of 36 entries; s usul, xyz mens x y z nd similrly for the other rnkings). 4 Sometimes the Gibbrd-Stterthwite theorem is stted with the premise if the rnge of the SCF contins t lest three lterntives..., but this cluse is implied by the ssumptions tht the set X contins t lest three elements nd tht the SCF stisfies Unnimity. 5 A reltively simple proof for the generl cse cn be found in Jen-Pierre Benoit, The Gibbrd- Stterthwite theorem: simple proof, Economics Letters, Vol. 69, 000, pges See lso the references therein for lterntive proofs.

17 G Bonnno 7 Voter xyz xzy yxz yzx zxy zyx A xyz x x V o t e r B xzy x x C yxz y y D yzx y y E zxy z z F zyx z z Figure 6: The requirement of Unnimity Now consider the highlighted cell A4 in Figure 6. By Non-mnipulbility there cnnot be z there otherwise Voter, with true preferences xyz (row A), would gin by lying nd reporting yxz (row C) when Voter reports yzx (column 4). Thus, in cell A4 there must be either n x or y. The strtegy of the proof is to show tht if there is n x in cell A4 then Voter must be dicttor, while if there is y in cell A4 then Voter must be dicttor. We will only prove the first prt, tht is, tht if there is n x in cell A4 then Voter must be dicttor. Suppose tht there is n x in cell A4. Then there must be n x lso in cell B4 (the cell mrked with in Figure 7) otherwise Voter, with true preferences xzy (row B), would gin by reporting xyz (row A) when Voter reports yzx (column 4). Now, from Voter s point of view, there must be n x in ll the boxes mrked with otherwise Voter, with true preferences yzx (column 4), would gin by moving either left or right to get the non-x which she prefers to x. Thus, the top two rows re entirely mde of x s. Now consider the highlighted cell C6 in Figure 7. There cnnot be z there becuse Voter, with true preferences yxz (row C), would gin by reporting xzy (row B) when Voter reports zyx (column 6); furthermore, there cnnot be n x in cell C6 becuse Voter, with true preferences zyx (column 6), would gin by reporting yzx (column 4) when Voter reports yxz (row C). Thus, there

18 8 Misrepresenttion of Preferences V o t e r Voter xyz xzy yxz yzx zxy zyx A xyz x x (not z) x x x x B xzy x x C yxz y y x D yzx y y E zxy z z x x x F zyx z z Figure 7: Inferences from the presence of x in cell A4

19 G Bonnno 9 must be y in cell C6. It follows tht there must be y lso in cell D6 below, otherwise Voter, with true preferences yzx (row D), would gin by reporting yxz (row C) when Voter reports zyx (column 6). Thus, we hve reched the configurtion shown in Figure 8. Voter xyz xzy yxz yzx zxy zyx A xyz x x x x x x V o t e r B xzy x x x x x x C yxz y y y D yzx y y not x not z E zxy z z y F zyx z z Figure 8: Further inferences Now consider the highlighted cell D5: there cnnot be z there otherwise Voter, with true preferences zyx (column 6), would gin by reporting zxy (column 5) when Voter reports yzx (row D) nd there cnnot be n x, otherwise Voter, with true preferences yzx (row D), would gin by reporting zxy (row E) when Voter reports zxy (column 5). Hence, there must be y in cell D5. Then there must be y lso in cell C5 otherwise Voter with true preferences yxz (row C) would gin by reporting yzx (row D) when Voter reports zxy (column 5). Thus, we hve reched the configurtion shown in Figure 9. Now there must be y in the remining cells of rows C nd D (mrked with in Figure 9) becuse otherwise Voter with true preferences zxy (column 5) would gin by reporting either xyz (column ) or xzy (column ) when Voter reports rnking corresponding to either row C or row D. Thus, we hve shown tht rows C nd D consist entirely of y s. Now consider cell E4 in Figure 0: there cnnot be y there becuse Voter, with true preferences zxy (row E), would gin by reporting xyz (row A) in

20 0 Misrepresenttion of Preferences Voter xyz xzy yxz yzx zxy zyx A xyz x x x x x x V o t e r B xzy x x x x x x C yxz y y y y D yzx y y y y E zxy z z F zyx z z Figure 9: Updted configurtion the sitution represented by column 4 nd there cnnot be n x becuse Voter, with true preferences yzx (column 4), would gin by reporting zxy (column 5) in the sitution represented by row E. Thus, there must be z in cell E4. Then there must be z lso in cell F4 below otherwise Voter, with true preferences zyx (row F), would gin by reporting zxy (row E) in the sitution represented by column 4. Now in the highlighted cells F, F nd F3 there cnnot be n x becuse Voter, with true preferences zyx (row F), would gin by reporting yzx (row D) nd there cnnot be y becuse Voter, with true preferences yzx (column 4), would gin by reporting the rnking corresponding to either column or column or column 3 in the sitution represented by row F. Thus, there must be z in F, F nd F3. This implies tht there must be z in the remining cells too becuse Voter, with true preferences zxy (row E) would gin by reporting zyx (row F). Hence, we hve shown tht the SCF must hve ll x s in rows A nd B, ll y s in rows C nd D nd ll z s in rows E nd F, mking Voter Dicttor (in rows A nd B her reported top lterntive is x nd it is chosen no mtter wht Voter reports, in rows C nd D her reported top lterntive is y nd it is chosen no mtter wht Voter reports nd in rows E nd F her reported top lterntive is z nd it is chosen no mtter wht Voter reports).

21 G Bonnno Voter xyz xzy yxz yzx zxy zyx A xyz x x x x x x V o t e r B xzy x x x x x x C yxz y y y y y y D yzx y y y y y y E F zxy zyx not x not y z z not x not x not x not y not y not y (z) z z Figure 0: The lst steps The proof tht if there hd been y in cell A4, then Voter would hve been Dicttor is similr nd we will omit it. 5 Exercises The solutions to the following exercises re given in Section 6 t the end of this chpter. 5. Exercises for Section : Socil choice functions Exercise. Suppose tht there re three lterntives: X = {, b, c} nd two voters: N = {, } nd consider SCFs tht only llow the reporting of strict rnkings so tht ech individul must report one of the following: b c, c b, b c, b c, c b, c b. To simplify the nottion, write them s bc, cb, bc, bc, cb, cb. In this cse we cn represent SCF by mens of tble with six rows (ech row lbeled with one rnking for Individul ) nd six columns (ech column lbeled with one rnking for Individul ).

22 Misrepresenttion of Preferences () How mny SCFs re there? (b) Fill in the tble s much s you cn by using only the Unnimity principle. (c) How mny SCFs tht stisfy the Unnimity principle re there? (d) Show the SCF tht corresponds to the cse where Individul is dicttor. Exercise. Consider gin the cse where there re three lterntives: X = {, b, c} nd two voters: N = {, } nd only strict rnkings cn be reported. Consider the SCF shown in Figure. () Does this SCF stisfy Unnimity? (b) Show tht this SCF stisfies Non-dicttorship. s rnking s rnking bc cb bc bc cb cb bc b c cb b c bc b b b b c bc b b b c b cb c c b c c cb c b c c c Figure : An SCF when X = {, b, c} nd N = {, }

23 G Bonnno 3 5. Exercises for Section : Strtegic voting Exercise 3. In Section we considered the cse of two lterntives nd two voters, with only strict rnkings being llowed. We sw tht in this cse there re 6 possible SCFs, but by ppeling to Unnimity nd Non-dicttorship, one cn reduce the number to the two SCFs shown below: (SCF-) b b b (SCF-) b b b b b () For SCF- show tht neither individul cn ever gin by misrepresenting his/her preferences. Give enough detils in your rgument. (b) For SCF- show tht neither individul cn ever gin by misrepresenting his/her preferences. Give enough detils in your rgument. Exercise 4. Consider the SCF of Exercise, which is reproduced in Figure. () Show tht there is t lest one sitution where Individul cn gin by misrepresenting her preferences. (b) Show tht there is t lest one sitution where Individul cn gin by misrepresenting his preferences.

24 4 Misrepresenttion of Preferences s rnking s rnking bc cb bc bc cb cb bc b c cb b c bc b b b b c bc b b b c b cb c c b c c cb c b c c c Figure : An SCF when X = {, b, c} nd N = {, } Exercise 5. Consider the Bord count explined in Section, with the following tiebreking rule: if two or more lterntives get the highest score, then the lterntive tht comes first in lphbeticl order is chosen. Suppose tht there re three lterntives: X = {, b, c} nd three voters: N = {,, 3}. Voter s true rnking is: best worst Voter s true rnking b c () Suppose tht Voter expects the other two voters to report the following rnkings: Voter Voter 3 best c c b b worst () Wht lterntive will be chosen if Voter reports her true rnking ()?

25 G Bonnno 5 (b) Show tht, by misrepresenting her preferences, Voter cn obtin better lterntive thn the one found in Prt (). Exercise 6. Consider the SCF shown in Figure 3 (the Condorcet method with defult lterntive, previously shown in Figure 4), where there re three voters nd three lterntives. () Show tht it stisfies the Unnimity principle. (b) Show it stisfies the Non-dicttorship principle. s bc cb bc b b b bc b b b cb c c cb b b c c bc cb bc bc cb cb s bc cb bc bc cb cb bc cb bc b b bc b b c c cb c c c cb c c c 3 reports bc 3 reports cb s bc b b cb cb b cb b b bc b b b b b b bc b b b b b b cb b b c c cb b b b c c bc cb bc bc s bc cb bc bc 3 reports bc cb cb bc b b b cb b b c c bc b b b b b b bc b b b b b b cb c b b c c cb b c b b c c 3 reports bc ` bc cb bc bc cb cb ` bc cb bc bc cb cb bc c c bc b b c c cb c c c cb c c c bc b b c c bc b b b c c bc c b b c c bc b c b b c c cb c c c c c c cb c c c c c c cb c c c c c c cb c c c c c c 3 reports cb 3 reports cb Pge of 3 Figure 3: The Condorcet method with s the defult lterntive Exercise 7. Consider the SCF shown in Figure 4 (mjority voting with slight dvntge given to Voter ), which reproduces Figure 5. Show tht there is t lest one sitution where Voter 3 cn benefit from misrepresenting his preferences. Exercise 8. Consider gin the SCF shown in Figure 4 (which reproduces Figure 5). At the end of Section we showed tht there is no sitution where Voter (the one

26 6 Misrepresenttion of Preferences s bc cb bc b b b b bc b b b b cb c c c c cb c c c c bc cb bc bc cb cb s bc cb bc bc cb cb bc cb bc b b b b bc b b b b cb c c c c cb c c c c 3 reports bc 3 reports cb s bc b b cb cb cb b b bc b b b b b b bc b b b b b b cb c c b b c c cb c c b b c c bc cb bc bc s 3 reports bc bc cb bc bc cb cb bc b b cb b b bc b b b b b b bc b b b b b b cb c c b b c c cb c c b b c c 3 reports bc ` bc cb bc bc cb cb ` bc cb bc bc cb cb bc c c bc c c cb c c cb c c bc b b b b c c bc b b b b c c bc b b b b c c bc b b b b c c cb c c c c c c cb c c c c c c cb c c c c c c cb c c c c c c 3 reports cb 3 reports cb Pge of Figure 4: Mjority voting with slight dvntge given to Voter

27 G Bonnno 7 who hs slight dvntge in tht her top lterntive is ssigned.5 points insted of point) cn gin by misrepresenting her preferences. Suppose now tht (perhps s result of previous discussions) it is common knowledge mong the three voters tht Voter s true rnking is cb (tht is, c b). Hence, it is resonble to ssume tht Voter will report her rnking truthfully (she cnnot gin by lying) nd, indeed, it is common knowledge between Voters nd 3 tht they expect Voter to report cb. By postulting tht Voter reports cb, we cn reduce the SCF of Figure 4 to n SCF with only two voters: Voter nd Voter 3. () Drw tble tht represents the reduced SCF. For exmple, this tble should show tht if Voter reports bc nd Voter 3 reports cb, then the chosen lterntive is (it gets.5 points from Voter s report, while ech of b nd c get only point ech). (b) In the reduced SCF, cn Voter ever gin from misrepresenting his preferences? (c) In the reduced SCF, cn Voter 3 ever gin from misrepresenting his preferences? (d) Suppose tht Voter 3 s true rnking is bc. Cn Voter 3 gin by reporting different rnking? (e) Suppose tht Voter knows tht Voter 3 s true rnking is bc nd expects her to report truthfully. Suppose lso tht Voter s true rnking is cb. Wht rnking should Voter report? 5.3 Exercises for Section 3: The Gibbrd-Stterthwite theorem Exercise 9. Consider the two SCFs of Exercise 3 (SCF- nd SCF-), reproduced below: (SCF-) b b b (SCF-) b b b b b In Section they were shown to stisfy Unnimity nd Non-dicttorship nd in Exercise 3 they were shown to stisfy Non-mnipulbility. Explin why these two SCFs do not constitute counterexmple to the Gibbrd-Stterthwite theorem (Theorem ).

28 8 Misrepresenttion of Preferences Exercise 0. There re five lterntives (X = {, b, c, d, e}) nd fourteen voters (N = {,,..., 4}). Consider the following SCF: ech voter submits strict rnking of the lterntives nd there re no restrictions on wht strict rnking cn be submitted. Then the procedure is s follows:. if Individuls -5 ll rnk the sme lterntive t the top, then tht lterntive is chosen, otherwise. if Individuls 6-0 ll rnk the sme lterntive t the top, then tht lterntive is chosen, otherwise 3. if there is Condorcet winner in the reported rnkings of Individuls -3 (the definition of Condorcet winner ws explined in Section ) then tht lterntive is chosen, otherwise 4. the top-rnked lterntive of individul 4 is chosen. Does this SCF stisfy Non-mnipulbility? 6 Solutions to Exercises Solution to Exercise. () The tble hs 36 cells tht need to be filled, ech with one of, b or c. Thus, there re 3 36 = , tht is, more thn 50,000 trillions (recll tht trillion is 0 or million million) SCFs! (b) The Unnimity principle restricts only the vlues in of the 36 cells, s shown in Figure 5 below. (c) In Figure 5 there re 4 remining cells to be filled in (ech with one of, b or c) nd thus there re 3 4 = (tht is, more thn 8 billions) SCFs tht stisfy the Unnimity principle!

29 G Bonnno 9 (d) The cse where Individul is dicttor corresponds to the SCF shown in Figure 6 below. s bc bc cb bc bc cb cb cb bc b b bc b b cb c c cb c c Figure 5: The restrictions imposed by the Unnimity principle s bc bc cb bc b bc b cb c cb c cb b b c c bc b b c c bc b b c c cb b b c c cb b b c c Solution to Exercise. Figure 6: Individul is dicttor () Yes, this SCF stisfies Unnimity: when both individuls rnk lterntive x t the top, x is chosen by society (the min digonl consists of block of four s, block of four b s nd block of four c s, fulfilling the requirement shown in Figure 5). (b) This SCF lso stisfies Non-dicttorship. In Figure 7 we hve highlighted two cells to show this. For Individul, consider the cell in row 4 nd

30 30 Misrepresenttion of Preferences column : her rnking is bc, thus her top-rnked lterntive is b, nd yet the chosen lterntive (when Individul reports the rnking bc) is, not b. For Individul, consider the cell in row nd column 6: his rnking is cb, thus his top-rnked lterntive is c, nd yet the chosen lterntive (when Individul reports the rnking bc) is, not c. (Of course, other cells could hve been used to mke the sme point.) s rnking s rnking bc cb bc bc cb cb bc b c cb b c bc b b b b c bc b b b c b cb c c b c c cb c b c c c Figure 7: Neither individul is dicttor Solution to Exercise 3. () SCF-. Individul cnnot gin by misrepresenting her preferences: if her true rnking is b then by reporting truthfully she gets her top lterntive nd by misrepresenting she might get or b; if her true rnking is b nd she reports truthfully, then there re two possibilities: () Individul reports b, in which cse the outcome is, nd would still be if Individul lied, nd () Individul reports b, in which cse if Individul reports truthfully then she gets her top lterntive b, while if she lies then she get her worst lterntive, nmely. Individul cnnot gin by misrepresenting his preferences: if his true

31 G Bonnno 3 rnking is b then by reporting truthfully he gets his top lterntive nd by misrepresenting he might get or b; if his true rnking is b nd he reports truthfully, then there re two possibilities: () Individul reports b, in which cse the outcome is, nd would still be if Individul lied, nd () Individul reports b, in which cse if Individul reports truthfully then he gets his top lterntive b, while if he lies then he gets his worst lterntive, nmely. (b) SCF-. Individul cnnot gin by misrepresenting her preferences: if her true rnking is b then by reporting truthfully she gets her top lterntive b nd by misrepresenting she might get or b; if her true rnking is b nd she reports truthfully, then there re two possibilities: () Individul reports b, in which cse the outcome is b, nd would still be b if Individul lied, nd () Individul reports b, in which cse if Individul reports truthfully then she gets her top lterntive, while if she lies she get her worst lterntive, nmely b. Individul cnnot gin by misrepresenting his preferences: if his true rnking is b then by reporting truthfully he gets his top lterntive b nd by misrepresenting he might get or b; if his true rnking is b nd he reports truthfully, then there re two possibilities: () Individul reports b, in which cse the outcome is b, nd would still be b if Individul lied, nd () Individul reports b, in which cse if Individul reports truthfully then he gets his top lterntive, while if he lies he gets his worst lterntive, nmely b. Solution to Exercise 4. () There re severl situtions where Individul cn gin by lying. For exmple, suppose tht her true rnking is bc (row 4) nd Individul reports bc (column ). Then, by reporting truthfully, Individul brings bout outcome, which is her worst outcome, but by lying nd reporting bc she obtins her most preferred outcome, nmely b.

32 3 Misrepresenttion of Preferences (b) There re severl situtions where Individul cn gin by lying. For exmple, suppose tht his true rnking is bc (column ) nd Individul reports bc (row 3). Then, by reporting truthfully, Individul brings bout outcome b, which is his middle-rnked outcome, but by lying nd reporting cb he obtins his most preferred outcome, nmely. Solution to Exercise 5. () If Voter reports her true rnking then we get the following profile: Voter Voter Voter 3 score best c c 3 b b b worst c The scores computed ccording to the Bord rule re: : = 5 b : + + = 6 c : = 7 so tht lterntive c is chosen (Voter s worst). (b) If insted of her true rnking Voter reports the following rnking: best worst b c then the profile of reported rnkings is Voter Voter Voter 3 score best b c c 3 b b worst c The scores computed ccording to the Bord rule re: : + + = 4 b : = 7 c : = 7

33 G Bonnno 33 The lrgest score is 7 nd is shred by both b nd c. According to the tiebreking rule, in cse of ties the lterntive tht comes first in lphbeticl order is chosen. Thus, the chosen lterntive b, which Voter (ccording to her true rnking b c) prefers to c. Thus, Voter gins by lying nd reporting rnking which is different from her true rnking.

34 34 Misrepresenttion of Preferences Solution to Exercise 6. () Unnimity requires the following: () in the two tbles t the top (corresponding to the cses where Voter 3 reports bc or cb) there should be n in the following cells: (row, column ), (row, column ), (row, column ) nd (row, column ) [these re the cses where every voter rnks t the top], () in the two tbles in the middle (corresponding to the cses where Voter 3 reports bc or bc) there should be b in the following cells: (row 3, column 3), (row 3, column 4), (row 4, column 3) nd (row 4, column 4) [these re the cses where every voter rnks b t the top], (3) in the two tbles t the bottom (corresponding to the cses where Voter 3 reports cb or cb) there should be c in the following cells: (row 5, column 5), (row 5, column 6), (row 6, column 5) nd (row 6, column 6) [these re the cses where every voter rnks c t the top]. The SCF shown in Figure 3 indeed stisfies these constrints. (b) To see tht Voter is not dicttor, consider the first tble (Voter 3 reports bc), row 4 (Voter reports bc) nd column (Voter reports bc): b is t the top of Voter s reported rnking nd yet the chosen lterntive is. To see tht Voter is not dicttor, consider the first tble (Voter 3 reports bc), row (Voter reports bc) nd column 4 (Voter reports bc): b is t the top of Voter s reported rnking nd yet the chosen lterntive is. Finlly, to see tht Voter 3 is not dicttor, consider the first tble (Voter 3 reports bc), row 6 (Voter reports cb) nd column 4 (Voter reports bc): is t the top of Voter 3 s reported rnking nd yet the chosen lterntive is b. Solution to Exercise 7. Consider the sitution where Voter reports the rnking cb (row 5 of ny tble) nd Voter reports bc (column 4 of ny tble). Suppose tht Voter 3 s true rnking is bc (first tble). If Voter 3 reports truthfully, then the selected lterntive is c, which is the worst from his point of view; if, on the other hnd, Voter 3 reports bc (the second tble in the first column of tbles), then the selected lterntive is b, which ccording to Voter 3 s true rnking bc is better thn c. Solution to Exercise 8. () The reduced SCF is shown in Figure 8 below.

35 G Bonnno 35 (b) Yes, there re situtions where Voter cn gin by misrepresenting his preferences. For exmple, if his true rnking is cb (row 6) nd he expects Voter 3 to report bc (column 3), then by reporting truthfully he brings bout his worst outcome, nmely, while by lying nd reporting bc (row 4) he brings bout outcome b which he prefers to. (c) Yes, there re situtions where Voter 3 cn gin by misrepresenting her preferences. For exmple, if her true rnking is cb (column 6) nd she expects Voter to report bc (row 3), then by reporting truthfully she brings bout her worst outcome, nmely, while by lying nd reporting bc (column 4) she brings bout outcome b which she prefers to.

36 36 Misrepresenttion of Preferences (d) If Voter 3 s true rnking is bc, then there is no sitution where she cn gin by misrepresenting her preferences: () if Voter reports bc or cb, then the outcome is no mtter wht Voter 3 reports, () if Voter reports bc or bc, then, by reporting truthfully, Voter 3 gets her best outcome, nmely b, (3) if Voter reports cb or cb, then, by reporting truthfully, Voter 3 gets outcome nd by lying she gets either or her worst outcome, nmely c. (e) If Voter s true rnking is cb nd he expects Voter 3 to report bc, then he should lie nd report either bc or bc (nd thus bring bout outcome b which he prefers to, which is the outcome he would get if he reported truthfully). 3 s s bc bc cb bc bc cb cb cb bc b b bc b b cb c c cb c c Assuming tht Voter reports cb Figure 8: The reduced SCF from Figure 5 Solution to Exercise 9. They do not constitute counterexmple to the Gibbrd- Stterthwite theorem becuse the set of lterntives contins only two elements. The Gibbrd-Stterthwite theorem is bsed on the premise tht there re t lest three lterntives. Solution to Exercise 0. This SCF fils to stisfy Non-mnipulbility. One cn try to show this by identifying sitution where some individul cn gin by misrepresenting her preferences, but quicker proof is by invoking the

37 G Bonnno 37 Gibbrd-Stterthwite theorem. All we need to do is show tht this SCF stisfies Unnimity nd Non-dicttorship, so tht by the Gibbrd-Stterthwite theorem it must violte Non-mnipulbility. Tht Unnimity is stisfied is obvious: if ll the individuls list the sme lterntive x t the top, then in prticulr the first five individuls list x t the top nd thus x is chosen. Tht the SCF stisfies Non-dicttorship is lso strightforwrd: () for ny of the first five individuls, if she reports x t the top, but t lest one of the other first five does not nd ll of individuls 6-0 report y x t the top, then y is chosen ; () for ny of Individuls 6-4, if he reports x t the top but the first five individuls ll report y x t the top, then y is chosen.