1 Manipulation for binary voters

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1 STAT 206A: Soil Choie nd Networks Fll 2010 Mnipultion nd GS Theorem Otoer 21 Leturer: Elhnn Mossel Srie: Kristen Woyh In this leture we over mnipultion y single voter: whether single voter n lie out his preferene nd produe n outome tht is etter for him. We look t mnipultion in the se of inry voting nd rnking multiple preferenes. We will then over the Gird- Stterthwite (GS) Theorem, whih sys tht under ertin onditions, if the ggregtion funtion is not mnipulle then it must e the dittor funtion. 1 Mnipultion for inry voters Consider the following senrio: n voters eh hoose from { 1, 1} x i is the vote of the ith voter. outome funtion f : { 1, 1} n { 1, 1} We will define mnipultion y single voter in the following wy: Definition 1.1. f is mnipulle y voter 1 if there exists x 2... x n suh tht: f(1, x 2,..., x n ) = 1 nd f( 1, x 2,..., x n ) = 1 If voter 1 hs true preferene of 1, then he must vote -1 in order to get his preferred outome. So, voter 1 hs inentive to lie out his true preferene. Notie tht the mjority funtion is n exmple of funtion tht is not mnipulle: ny voter voting ginst their true preferene n only mke the outome less likely to e their preferene. This exmple n e generlized to sy something out the reltionship etween mnipulility nd monotoniity: Clim 1.2. f is mnipulle if nd only if f is not monotone Proof: if f is monotone, then it is not mnipulle: Assume y ontrdition tht f is mnipulle. Then whih ontrdits monotoniity. 1 = f(1, x 2,..., x n ) 1 f( 1, x 2,..., x n ) = 1 if f is not mnipulle, then it is monotone: If f is non-mnipulle then it is monotone with respet to flip of ny signle oordinte. We know tht this ondition implies monotoniity of f. Mnipultion nd GS Theorem-1

2 STAT 206A Mnipultion nd GS Theorem Otoer 21 Fll Mnipultion for voters with 3 lterntives When there re more thn 2 lterntives, the sitution is more omplited. Consider the following exmple, reminisent of the 2000 presidentil eletion: There re three possile rnkings of the options {,, } with the proility of desiring tht rnking shown elow: 45% 40% 15% (This orresponds roughly to the sitution in 2000 with = Bush, = Gore, nd = Nder). Assume we use plurlity funtion to determine the winner. If everyone votes ording to their true preferenes, then Bush will win. However, if the Nder supporters relize tht they n never win, they my hoose to lie out their preferenes nd vote for Gore insted. Now Gore will win, whih is etter lterntive for the Nder supporters thn Bush. We will ssume tht we wnt voters to vote ording to their tul preferenes (truthful voting), nd we do not wnt voters to try to gme the eletion y voting retively. So, we wnt to understnd under wht ggregtion funtions no voter will e le to mnipulte the eletion. To do this, we need numer of definitions: Definition 2.1. A soil hoie funtion (SCF) is funtion F : S(k) n [k] whih tkes in ll rnkings from the voters nd omputes winner. Definition 2.2. F is mnipulle y voter 1 if (σ 1, σ 1 ), (σ 1, σ 1 ) : σ 1 (F (σ 1, σ 1 )) > σ 1 (F (σ 1, σ 1 )) where σ 1 is the rnking of voter 1, σ 1 is ll of the rnkings of ll voters exept voter 1 (σ 1 = σ 2... σ n ) nd σ 1 (F ) is the rnk of the outome of F in the preferene of the first voter. In other words, there must e rnking tht is different from voter 1 s true rnking tht would mke the SCF return vlue tht is more preferle to voter 1. In the 2000 eletion exmple, Gore is etter option thn Bush for the Nder voters, so it is in their interest to rnk Gore ove Nder despite tully preferring Nder. Notie tht the inequlity in this definition is strit: we do not re out ses when swithing the rnking will result in n equivlently good outome. Definition 2.3. F stisfies unnimity If Then i, is the top lterntive of σ i F (σ) = F (σ 1,..., σ n ) = Mnipultion nd GS Theorem-2

3 STAT 206A Mnipultion nd GS Theorem Otoer 21 Fll 2010 Definition 2.4. F is neutrl if the funtion is fir mong ll lterntives: σ S(,,..., k) nd σ S(,,..., k) : F (σ σ) = σ F (σ) Definition 2.5. F is strtegy-proof if F is not mnipulle Definition 2.6. The ith dittor funtion is the funtion desried y F (σ) = top lterntive(σ i ) A remrk ws mde tht we might e le fix the mnipulility of the plurlity funtion in the lst exmple y implementing run-off eletion. This mens tht the first eletion hooses the two lterntives with the highest numer of votes nd then there is seond eletion etween these two. The winner is the mjority winner of the seond eletion. However, this exmple too is mnipulle. Consider the following se:, go to run-off 31% 29% 40% 60% 40% In the first eletion, nd reeive the most votes, so they re prt of the run-off. In the runoff, holds the most votes, so it is delred the winner. However, if 3% of the (,, ) preferene voters hoose to lie in the initil eletion nd vote for (,, ) insted, then return to their true vote in the run-off, we hve the following sitution:, go to run-off 31% 32% 37% 29% 71% Now, nd re in the run-off, where wins y lndslide. Therefore, there is inentive for numer of voters to not e truthful in the first round. With these definitions in hnd, the rest of the leture is devoted to stting nd proving the GS theorem. 3 Gird-Stterthwite Theorem Theorem 3.1 (Gird-Stterthwite (GS) Theorem). If F : S(k) n nd strtegy-proof with k 3, then F is dittor funtion. [k] is onto A funtion is onto if ll lterntives n e the winner. This theorem is true in prtiulr if F is neutrl. The originl proof ws derivtive of the proof of Arrow s theorem [1, 2]. The proof we will follow here omes from Svensson [3]. We will need ouple of lemms in order to omplete this proof: Mnipultion nd GS Theorem-3

4 STAT 206A Mnipultion nd GS Theorem Otoer 21 Fll 2010 Lemm 3.2 (Monotoniity). If F is strtegy-proof nd F (σ) =, nd τ stisfies for ll lterntive x nd ll voters i tht Then F (τ) = Note tht σ = (σ 1... σ n ) nd τ = (τ 1... τ 1 ). σ i () σ i (x) τ i () τ i (x) Proof: It suffies to prove this lemm ssuming tht σ i = τ i for ll i > 1 euse we n hnge voters one y one to get ny desired onfigurtion. Assume y ontrdition tht F (τ) = Then σ 1 () σ 1 (). (otherwise if σ 1 () > σ 1 () then voter 1 would wnt to het nd vote τ.) Then τ 1 () τ 1 () y the onditions of the lemm. Whih implies tht F (τ 1 ) = (otherwise, voter with preferene τ 1 will vote σ 1 to get e the winner) Lemm 3.3 (Preto). If for ll i, σ i () > σ i () nd F is strtegy-proof nd onto, Then F (σ) Proof: Assume y ontrdition tht F (σ) = Sine F is onto, there exists τ suh tht F (τ) = Let σ i = ordered like σ i By Lemm 3.2 pplied to the preferenes, F (σ ) = F (σ) = But y Lemm 3.2 pplied to the preferenes, F (σ ) = F (τ) = s well. This is ontrdition, so F (σ) = With these two lemms, we will first prove the GS theorem for two voters nd then we will prove it for the generl se. Proof (GS theorem with 2 voters): Let there e two voters, with option rnking u nd v s follows: u = others, v = others From Lemm 3.3, we hve F (u, v) {, }. Without loss of generlity, we ssume F (u, v) =. First we prove tht for ll v with t the top of the ordering, F (u, v ) =. This is true euse if there exists n ordering ( v = others ) Mnipultion nd GS Theorem-4

5 STAT 206A Mnipultion nd GS Theorem Otoer 21 Fll 2010 suh tht F (u, v ) =, then the seond voter will lie out his true ordering nd vote v, nd therefore F would not e strtegy-proof. In prtiulr this is true for v whih rnks t the ottom. Now, y Lemm 3.2, for ll u with t the top of the rnking, nd ll v, F (u, v ) =. To finish, we need to show tht it does not mtter tht it is option tht is on top of the rnking: Let A 1 = set of lterntives x suh tht if x is t the top of voter 1 s rnking, the outome is x. Let A 2 = set of lterntives y suh tht if y is t the top of voter 2 s rnking, the outome is y. Notie tht A 1 A 2 = [k] euse ny of the options n e the outome. We now rgue tht A 2 is empty. Indeed, let then if is t the top of voter 2 nd is t the top of voter 1 then the outome is. Therefore A 2 nnot ontin ny element different thn. Fixing the sme rgument implies tht nnot elong to A 2 so so A 1 = [k] nd A 2 =. Therefore F must e dittor. Now we n uild on this to get the generl se. We will ll the 2-voter se of the GS theorem Lemm 3.4. Now we prove the 3 voter se: Proof: We will do this proof y indution on the numer of voters n. Assume tht F is strtegy-proof nd onto. Let g(u, v) = F (u, v, v,..., v). We need to prove tht g is onto nd strtegy-proof. g is onto: Lemm 3.3 implies tht tht F (u, u,..., u) = top lterntive of u, so g(u, u) = top lterntive of u nd so g must lso e onto. g is strtegy-proof: Voter 1 nnot mnipulte g y definition Voter 2 nnot mnipulte g euse otherwise there must exist u, v, v suh tht v(g(u, v )) > v(g(u, v)). Define u k = (u, k v, (n k 1) v). For voter k to mnipulte g, there must exist k where v(g(u k+1 )) > v(g(u k )) whih is ontrdition to the ft tht F is strtegy proof. Therefore, g is strtegy proof. With g strtegy-proof nd onto, y Lemm 3.4 g must e dittor. Now we prove tht F is lso dittor. if g is dittor on voter 1, then y the monotoniity lemm, F is lso dittor on voter 1. Mnipultion nd GS Theorem-5

6 STAT 206A Mnipultion nd GS Theorem Otoer 21 Fll 2010 Assume g is dittor on voter 2. Fix u nd look t h(v 2,..., v n ) = F (u, v 2,..., v n ). h is onto nd strtegy-proof, so it is dittoril. Without loss of generlity, ssume 2 is the dittor nd fix v 3,..., v n. Then z(u, v) = F (u, v, v 3,..., v n ) is onto nd strtegy-proof nd voter 1 nnot e the dittor. So, z is dittor on voter 2 implies tht F is dittor on voter 2. Referenes [1] Gird, 1973 A. Gird, Mnipultion of voting shemes: generl result, Eonometri 41 (1973), pp [2] Stterthwite, 1975 M.A. Stterthwite, Strtegy-proofness nd Arrows onditions: existene nd orrespondene theorems for voting proedures nd soil welfre funtions, Journl of Eonomi Theory 10 (1975), pp [3] Lrs-Gunnr Svensson, The Proof of the Gird-Stterthwite Theorem Revisited, preprint. Mnipultion nd GS Theorem-6