A $O(log m)$, deterministic, polynomial-time computable approximation of Lewis Carrolls scoring rule

Ef ficient, determ inistic v oting rules that approxim ate Dodgson and Y oung scores Jason A. Cov ey and Christopher M. Homan August 18, 2021 Abstract W e provid e deterministic, polynomial-time computable vo ting rules that approximate Dodgson’ s and (the “minimization version” of) Y oun g’ s scoring rules to within a logarithmic factor . Our approximation of Dodgson’ s rule is tight up to a constant factor , as Dodgson’ s rule is N P -hard to approximate to within some logarithmic factor . The “maximization version” o f Y oun g’ s rule is kno wn to be N P -hard to approximate by an y constant factor . Both approximations are simple, and natural as rules in their own right: G iv en a candidate we wish to score, we can regard eithe r its Dod gson or Y oung score as the edit dis- tance between a given set of voter preferences and one in which t he candidate to be scored is the Condorcet winner . (The difference be- tween the two scoring rules is the type of edits allowed.) W e regard the marginal cost of a sequence of edits to be the number of edits di vided by the number of reductions (in the candidate’ s deficit against any of its oppon ents in the pairwise race against that opponent) that t he edits yield. Over a series of rounds, our scoring rules greedily choose a se- quence of edits that modify e xactly one voter’ s preferen ces and whose marginal cost is no greater than any other such single-vote-modifyin g sequence. 1 Introd uction A v oting rule tak es a collectio n of v oter preferences (over some fixed set of can didates, or a lternatives) an d ag gegrates them into a single rankin g, id eally in a way th at is as “fair” as po ssible to the voters. Arrow’ s famous impo ssibility theorem [Ar r50] states that n o such ru le over th ree o r more candidates me ets all r easonable fairness cr iteria. So when co nsidering su ch ru les it may b e im portant to know wh ich cr iteria they do and do not meet. One such criterion, which actually pr edates tho se studied explicitly b y Arr ow , is credited to the Ma rquis de Condorcet [Con8 5]. 1 A cand idate that, ag ainst any op posing candidate, is preferred by a majority of voters is c alled the Condo rcet win ner . Note 1 In fact , centurie s earli er Lull considered essentially the same criteri on [HP01]. 1 that different majorities m ay prefer the Condorcet winner to different oppo nents. Note also that su ch a win ner may not exist, but if it d oes then it is unique. The Cond orcet criterion states that, when ev er a Condo rcet win ner does exist, it must be declared the winner . See [Y ou 77] for a nice discussion of its virtues. Unfortu nately , ma ny wid ely used rules, such as p lurality , instant-runo ff, and Borda count do not have this very natura l p roperty . Many that do bring with them u ndesirable features, for in stance Copeland elections [Cop5 1] ten d to frequently result in ties. Oth - ers, such as th ose due to Do dgson [Dod7 6 ], Kemeny [Kem59], and Y ou ng 2 [Y ou7 7 ] are N P -har d to com pute [BTT89, RSV03]. In fact, they are complete with respect to parallel access to N P [HHR97, HSV05, RSV03], whic h me ans that even if the p rob- lem of determining the winner accord ing to one these r ules is “m erely” in N P , the polyno mial hierarchy would collapse (to N P ). W e can view Dodg son’ s and Y ou ng’ s r ules as variations on a theme: Given a list, or profile, of the v oters’ preferences (h ere as is standard in the theor y of voting we tak e each voter’ s prefere nces to be a total ran king over all the cand idates) and a candida te we wish to scor e, either rule takes as the can didate’ s score the ed it distance [CLRS01] between the g iv en prefer ence p rofile and o ne that makes the candidate a Condorcet winner . In o ther words, it is the num ber of edits (exactly wh at an edit is depen ds on the particu lar scoring rule) n eeded to red uce to z ero the vote deficit b etween the given candidate and each of its riv a ls. Can didates are then ranked in increasing o rder by their scores. In Y oung’ s rule, an e dit simply de letes one voter from the list. For Do dgson, an edit takes on e voter’ s rankin g and replaces it with on e just like it, excep t that in the new o ne the p ositions of o ne pa ir o f can didates ra nked adjac ently in the origin al list are swapped. Clearly , bo th rules satisfy the Con dorcet criterio n, as a ny Condorc et winner has a score of zero. A simple examp le illu strates how scoring work s. Let a , b , c , d , and e b e five candidates and let a > 1 b > 1 c > 1 d > 1 e a > 2 b > 2 c > 2 d > 2 e d > 3 a > 3 e > 3 c > 3 b d > 4 a > 4 e > 4 c > 4 b c > 5 e > 5 b > 5 d > 5 a be a preference p rofile ha ving fi ve v oters. In this example, n o candidate is a C on dorcet winner . Note that c is prefe rred over b , d , and e b y majorities of voters an d is losing to a b y fou r votes. T o make c the Con dorcet win ner, we cou ld swap c with b and then with a in voter one and two’ s rankings. It turn s ou t there is no shorter sequen ce of swaps that makes c th e Condor cet winner , so the Dodgson score of c is fou r . Note that in this ca se, the swaps between c a nd b d o n ot actu ally r educe c ’ s vote d eficit, since c is already beating b . 2 Ke meny’ s voting rule is sometimes called the Kemeny -Y oung rule, as Y oung studied it and made some important breakt hroughs [YL78, Y ou88] , e.g., he s ho wed tha t it satisfies the Condorcet c riterion. The Y oung- only rule to which we refer is distinct from the K emeny-Y oung rule , which to av oid confusion we will call simply “K emeny . ” 2 Candidate d is losing to c and b by one v ote each. T o make d the C on dorcet winner, we cou ld r emove v oters one a nd two. Thus d ’ s Y ou ng scor e is two. In this case, both removals yield two deficit reduc tions, but in gen eral the numbe r of deficit reductio ns that each removal yields will vary . As Pr ocaccia et al. observe [PFR07], McCab e-Dansted effectively pr oves that it is hard to Ω(log m ) -appro ximate Dodgson elections, where m is the number o f candi- dates [M D06]. In the same paper, Procaccia et al. show that it is hard to appr oximate the “m aximization version” of Y o ung’ s score—i.e., where the Y oun g score is taken to be the largest subset o f voters that makes a gi ven candidate the Condorc et winner 3 —by any constant factor [PFR07]. In this paper we pr esent a f ramework for efficient, ed it-based scorin g r ules. From this framew ork , we obtain O (log m ) ap proxim ations o f the scoring rules due to Do dg- son and Y oung. The basic idea is very simple: Given a pro file of voter pref erences and a candidate we wish to score, let the margin al cost of a sequence of edits be the number o f ed its d i vid ed by th e n umber of times that, as the e dits are applied , the vote deficit against t he candidate we wish to sco re is red uced. Now , p roceed o ver a series of round s to edit the pro file until the chosen candidate becomes the Condorcet winner . I n each round, greedily choose a v oter and a sequence of edits on that v oter ’ s preferences that, over all such v oters and sequ ences, has the minimum marginal cost. It turns out that, when we re strict th e edits the algorith m makes to those allowed by Dod gson’ s (re spectiv ely , Y oung ’ s) scoring rule, the result is a poly nomial-time O (log m ) -ap proxim ation, where m is th e number of cand idates. Thus, in the case of Dodgson elections, the approx imation is tig ht up to a constant factor . Why care abo ut appro ximations to voting ru les in the first plac e? One reason is that they are themselves voting ru les, ones that in som e way relate to the ru les they approx imate. W e fe el that our fr amew ork supplies appr oximation s th at are simple an d natural enough to function as voting rules in their own right. For instance, supp ose a group of voters ag rees to only a ccept a Condor cet winner . If their stated individual preferen ces fail to yield one , then the election controller holds an auction, to entice some of the voters to chang e their minds. T aking on e cand idate at a time, th e controller o ffers to p ay ea ch voter f or eac h reduction in th e candidate’ s vote d eficit it ca n deliver by changin g its stated prefere nces. The cost to the voter is the nu mber of ed its it needs to make. If the pr ice offered is less tha n the cost to the voter , th e voter will not accept. If not enoug h voters accept, the co ntroller increa ses the am ount offered and the process rep eats un til the ca ndidate becomes th e Condor cet winner . The scor e of the candida te is then the total am ount o f money offered to the voters and the candidate having the lowest score is th e winner . (No pay offs occu r until after all cand idates are scored, and on ly those d eals m ade during the winning can didate’ s scoring roun d are actually hono red, so in e ffect the voters “choose” a Condorce t winner .) The auction thus encou rages voters to r ev eal the true value of their edits, as those 3 This is the actual definition due to Y oung [Y ou77]. Our formulation in terms of del etions is used else- where (see, e.g., [RSV03 , BGN07, Fis77]), and is in many respects equiv alent to the original definition (though certa inly not with respect to optimiza tion and approximat ion results, at least not direct ly). More- ov er , the deletion- based version we use allo ws us to more naturally build Dodgson’ s and Y oung’ s rules into a single frame work. 3 who are willin g to take the least amou nt o f mon ey p er d eficit reduction de li vered are rew arde d first, while th ose holding out for more may get nothin g. Assuming that all voters unif ormly value their edits at some commo n unit price, the score the auction provides ( and the o rder in which it s elects the swaps to make) coincid es with our rules. Related work The study of the appro ximibility o f voting rules is rather new . Ailo n et al. [ACN05], Coppersmith et al. [ CFR06], an d Kenyon-Mathieu an d Schudy [KMS07] stud y appro x- imation algorithms on K emeny elections. As n oted above, McCabe-Dansted [MD0 6] (respe ctiv ely , Procaccia [ PFR07]) pr o- vides lower ( respectively , u pper) b ound s o n ap proxim ating Do dgson ( respectively , Y oung) scores. Ad ditionally , Procaccia et al. p rovide a polynom ial-time, rand omized algo- rithm that with pro bability at least 1/2 O (log m ) -app roximate s the Dodg son score [ PFR07]. They use a linear prog ram whose o ptimal solutio n may assign fra ctional values to counts of the swaps made. They then use randomn ess to help assign integer values to the swap cou nts, in a way that y ields a feasible, integer-valued so lution. Our re- sults im prove on this approac h in th at ou r algo rithm is completely deterministic an d, we feel, mo re straightf ow ard and natural. Additiona lly , we provide a polyn omial-time approx imation of Y oung scores. Sev eral re searchers pr ovide algorithms th at ru n in po lynomial time o n key subsets of the problem domain. Barthold i et al., in the same seminal paper that established N P -h ardness results for Dodg son and Kemeny election s [BTT89], show that Do dgson elections can be scored in poly nomial time when either th e n umber of candidates or the number of voters is fixed. Ou r algorithm run s in polyno mial time o n a ll inputs, ho wever it is do es not gu arantee to provide a cor rect answer . Rathe r , it g uarantees upper b ounds on the degree of error . Homan a nd Hema spaandra [HH07] an d McCabe- Dansted et al. [MPS0 7] use a common insight to p rovide po lynom ial-time, deterministic heuristics that, in cases where the voters greatly ou tnumber the cand idates, c ompute with h igh pro bability the exact Dodg son sco re on a candid ate and pref erence pro file chosen unifor mly at random from all p rofiles of some fixed size. Our Dodgso n-score-ap proximating algo rithm is a generalizatio n of sorts of their app roach. Tho ugh we do no t analyze the p robability of exactness our alg orithm h as, we n ote h ere that wh enever the Homan an d He maspaan- dra appr oach correctly compu tes th e Dodg son sco re, so do es ours. Howe ver , when their alg orithm is no t exact, it retur ns a score that is less t han the true edit distance. Our algorithm never returns a score that is less than the edit d istance. Mo reover , our alg o- rithm always builds as a side effect an actu al sequence of edits lead ing to a Condor cet winner . Finally , Rothe et al. (in the same paper where they e stablish optimal bounds on the complexity of Y oung electio ns) give a polynom ial-time algorithm for computin g th e “homog eneous” version s (see [Fis77]) o f Do dgson’ s and Y oun g’ s voting ru les [RSV03]. (A voting r ule is homogen ous if cloning each v oter’ s preferences some fixed number of times do es no t affect the score). They do n ot d iscuss th e degre e to wh ich these scores approx imate Dodgson a nd Y oun g rules. 4 2 Definitions 2.1 Elections Let V = { 1 , . . . , n } be a set of voters and C be a set { 1 , . . . , m } of candida tes. A ranking of the can didates is a total orde ring over C , i.e., h c m > c m − 1 > · · · > c 1 i , where { c 1 , . . . , c m } = C . W e den ote th e set of a ll such r ankings L ( C ) . The voters’ preferen ce profile is an n - tuple in L ( C ) n . For a given preference pr ofile h > 1 , . . . , > n i ∈ L ( C ) n , i ∈ V , and c ∈ C , let c ( > i ) denote ||{ d ∈ C | c > i d }|| . For every pair of distinct cand idates c, d ∈ C and ev ery p referen ce pr ofile P = h > 1 , . . . , > n i , c ’ s v ote deficit in P with d is Deficit ( P , c, d ) = min { 0 , ||{ i ∈ V | d > i c }|| − ||{ i ∈ V | c > i d }||} . The total deficit of c is Deficit ( P , c ) = X d ∈ C − { c } Deficit ( P , c, d ) . Thus c is a Condo r cet winner if an d o nly if Deficit ( P , c ) = 0 . Deficit ( P , c ) is some - times known as the Tideman scor e [Tid87 ], which form s th e basis of the Tideman (a.k.a., ranked pairs) v oting rule. 2.2 Edit-based scoring rules The building blocks o f this paper are edits an d deficit reductio ns. It will be usef ul to view them as objects we can label. W e now sho w how to d o this. An e dit is a mappin g e : S ∞ i =0 L ( C ) i → S ∞ i =0 L ( C ) i . Let P ◦ e d enote th e application of e to some p referenc e profile P . A sequen ce of edits h e 1 , . . . , e p i is called a Condorcet sequence if Deficit ( P ◦ e 1 ◦ · · · ◦ e p , c ) = 0 A sw ap is an edit, designated b y an ordered pair ( i, j ) ∈ N 2 , that takes a prefer ence profile P = h > 1 , . . . , > n i and outpu ts h > ′ 1 , . . . , > ′ n i , which is just like P except that, if 1 ≤ i ≤ n an d 0 < j < m , th en f or c, d ∈ C satisfying d ( > i ) = j = c ( > i ) + 1 it holds that c ( > ′ i ) = j = d ( > ′ i ) + 1 , i.e., c, d ar e adjacent in bo th ranking s, d > i c , and c > ′ i d . Candidates c an d d are said to be in volved in the swap. A deletion is an edit, designated by some i ∈ N , that takes a pr eference p rofile P = h > 1 , . . . , > n i in N 2 and outpu ts h > 1 , . . . , > i − 1 , > i +1 , . . . , > n i . A deficit r eduction is a 4 -tuple ( P , c, e, d ) where P is a pre ference pro file, c an d d are ca ndidates, and e is an ed it such that Deficit ( P, c, d ) > Deficit ( P ◦ e, c, d ) . The full sequen ce of d eficit reductio ns with respect to ca ndidate c over a seque nce of edits h e 1 , . . . , e p i on a preferen ce profile P , denoted D ( P , c, h e 1 , . . . , e p i ) , is the nonr epeat- ing seq uence of deficit reductions h ( P 1 , e i 1 , c, d 1 ) , . . . , ( P q , e i q , c, d q ) i of maxim um length such tha t, f or all k ∈ { 1 , . . . , q } , P k = P ◦ e i i ◦ · · · ◦ e i k − 1 , D eficit ( P k , c, d k ) > Deficit ( P k ◦ e i k , c, d k ) , and for all j ∈ { 1 , . . . , k − 1 } , i j ≤ i k . W e now d efine, using the terms g iv en ab ove, Dodg son a nd Y o ung’ s scorin g r ules. Let S be the collection of all sequences of swaps. The Dodgson s cor e of cand idate c in profile P is the smallest p ∈ N such that ( ∃h e 1 , . . . , e p i ∈ S )[ Deficit ( P ◦ e 1 ◦ · · · ◦ e p , c ) = 0] . 5 Let D be the c ollection of all seq uences of deletions. The Y o ung scor e of can didate c in profile P is the smallest p ∈ N such that ( ∃h e 1 , . . . , e p i ∈ D )[ Deficit ( P ◦ e 1 ◦ · · · ◦ e p , c ) = 0] . 2.3 The generic framework Below is a gen eric algor ithm for the voting rules w e study and app roximate. Here, E is a collection of “le gal” sequences of edits, whose e xact makeup depen ds on the particular scoring rule in q uestion. The variable E is imp lemented as a p riority queu e, wher e priority is given to sequences of edits S ′ that, when app lied to th e preference pro file P , have the fewest edits per deficit redu ction, i.e., th at minimize | S ′ | / | D ( P , c, S ′ ) | . W e call this quan tity the marginal c ost of S ′ . W e define | S ′ | / | D ( P , c, S ′ ) | = ∞ whenever | D ( P, c, S ′ ) | = 0 . S is a list of edits made. In order to em phasize the key com ponen ts of th is a lgorithm, we h av e om itted im- portant but mu ndane steps. For instan ce, the algorithm needs to compute Deficit ( P , c ) . W e will discuss such details when we d iscuss the actual Dod gson—and Y oung— approx imation rules. Input: A preference profile P and a candidate c . 1. let S = hi 2. while Deficit ( P, c ) > 0 3. let S ′ = argmin S ′′ ∈E | S ′′ | / | D ( P , c, S ′′ ) | 4. let h e 1 , . . . , e p i = S ′ 5. let P = P ◦ e 1 ◦ · · · ◦ e p 6. concatenate ( S, h e 1 , . . . , e p i ) 7. output | S | 3 A ppr oximating Dodgson’ s scoring rule For any cand idate c , we say that a sequence of swaps s 1 , . . . , s p is c -no rmal on P if, for each k ∈ { 1 , . . . , p } , c is inv olved in swap s k = ( i, j ) on P ◦ s 1 ◦ · · · ◦ s k − 1 and c ( < i ) = j − 1 . Let P be a p referen ce profile and let E ′ be the collectio n of all c -norm al swap sequences wh ere, for each seq uence, th ere is a sing le voter’ s pre ference list to which all swaps in the seq uence apply . No te then th at ev ery such sequ ence has a distinct la st element, so we can represen t each seque nce in E ′ by storin g its last elemen t on ly . Let 6 us call the voting rule based on the gen eric algorith m with E = E ′ “Marginal-Cost- Greedy-Do dgson. ” Theorem 1. The run ning time of Mar ginal-Cost-Greedy-Dodgson, when E = E ′ , is O ( N 2 log N ) , wher e N is the leng th of the input. Pr oo f. Let ( P , c ) be the inp ut to the algorithm , where E = E ′ and P ha s m candidates and n voters. W e first need to in itialize the data stru ctures used. It takes linea r time to calculate Deficit ( P, c, d ) on a ll d ∈ C −{ d } (note th at we can compu te Deficit ( P, c ) at the same time). Next we need to initialize E ′ . Th ere are at most n ( m − 1 ) seq uences S ′ in E ′ , and there are at most m ( m − 1) / 2 d istinct v alues for | S ′ | / | D ( S ′ ) | that any such sequence can t ake. So (regard ing E a s a p riority queue) it takes O (log m ) comparisons to add any such sequence (which w e recall is r epresented b y the last element o f the sequence) to E . Note that we can calcu late | S ′ | / | D ( S ′ ) | for every sequence S ′ in E in a single pass through P . Th e worst case is when n is as small a s possible, so the w orst case running time for initialization is O ( N log N ) After initialization, th e algorithm per forms swaps on P until c is the Co ndorcet winner . Note that a ny given swap is per formed at mo st on ce. For each swap applied, the alg orithm must re move the corresp onding swap fro m the q ueue (since whe never a swap is ap plied it follows that the swap sequ ence ending with that swap has a lso been applied), and it must u pdate the margina l cost of each swap sequen ce rem aining in E that app lies to the cu rrent voter’ s preferen ces. Thus, every swap may r equire O ( m ) updates to E . Assuming tha t all swaps in E sharing a commo n voter are conne cted via a linked list, ea ch upd ate can happen in co nstant time. As du ring initialization, the worst case f or these p rocedu res occ urs wh en n is as small a s p ossible, so the run ning time for this part of the algorithm is O ( N 2 ) Finally , every time a swap ca uses the deficit aga inst some o pponen t to go from positive to zero the en tire q ueue needs to be reprio ritized, whic h m eans we mu st pass th rough all swap sequen ces and r ecalculate This can hap pen at most ( m − 1) times. A gain, th e worst-case runnin g time is when n is a s small as p ossible, so it is O ( N 2 log N ) . W e turn now to the app roximatio n b ound. Our pr oof assum es there is a c -no rmal Condorce t sequen ce o f swaps witnessing th e Dodg son score of c . The f ollowing pro po- sition shows th at our assumption is valid. Proposition 2. F or every pr efer ence pr o file P and candida te c there is a c -n ormal Condorcet swap sequen ce of length equal to the Dodgson scor e of c . Pr oo f. Let p be th e Dodg son scor e of c and h s 1 , . . . , s p i be a Condo rcet swap se- quence with respect to cand idate c on prefere nce profile P = h > 1 , . . . > n i . Let h > ′ 1 , . . . , > ′ n i = P ◦ s 1 ◦ · · · ◦ s p . Choo se i ∈ V and let h s ′ 1 , . . . , s ′ q i be the sub- sequence of h s 1 , . . . , s p i consisting o f all swaps on voter i ’ s pr eference s. L et d ′ = argmax d ∈ C : c> ′ i d ( d ( > i ) − c ( > i )) . Since it req uires a t least d ′ ( > i ) − c ( > i ) swaps in order fo r c > ′ i d ′ to ho ld, it mu st be th e case that |h s ′ 1 , . . . , s ′ q i| ≥ d ′ ( > i ) − c ( > i ) . So, r emoving from S each swap in h s ′ 1 , . . . , s ′ q i and append ing the seque nce h ( i, c ( > i 7 ) + 1) , . . . , ( i, d ′ ( > i )) i yields a Con dorcet sequenc e that has no more swaps than S originally had. Theorem 3. Marginal-Cost-Gr eedy-Do dgson is an (ln m + 1) -appr oxima tion of Dodg- son scor e, wher e m is the numb er of candidates in the input election. Pr oo f. Let P be a pr eference p rofile o ver m candidates and n voters and let c b e a can- didate in { 1 , . . . , m } . Let x be the Dod gson score o f c o n P and let S ∗ be a c -norm al Condorce t seq uence of P . Let y = Deficit ( P , c ) and let h ( P ∗ 1 , c, s ∗ 1 , d ∗ 1 ) , . . . , ( P ∗ y , c, s ∗ y , d ∗ y ) i = D ( P, c, S ∗ ) . L et S be the sam e as in the algorith m o n inp ut ( P , c ) at the time lin e 7 is re ached (i.e. , it is the sequ ence of all swaps the algorithm a pplies to P ), and let h ( P 1 , c, s 1 , d 1 ) , . . . , ( P y , c, s y , d y ) i = D ( P, c, S ) . The basic idea b ehind our p roof is that the num ber of deficit reductio ns in a se- quence th at witnesses the Dodgson score of c , such as S ∗ , is eq ual to th e number of deficit red uctions in the sequence S th at the algorithm prod uces. So to compare | S | to | S ∗ | we partition the swaps in S (respectively , S ∗ ) amon g the deficit reductions and then match the deficit re ductions in S to th ose in S ∗ . The partition ing is ea sy: For S it is ju st the m arginal cost associated with ea ch deficit red uction. For S ∗ we fu dge the marginal cost in a straig htforward way . The match ing and the or der in which match ed elements are compare d are the trickiest parts of the proof . For every k ∈ { 1 , . . . , y } , let r ( s k ) d enote the margina l cost th e algorithm a s- sociates with s k (i.e., | S ′ | / | D ( P , c, S ′ ) | , where S ′ and P are as in lin e 3 du ring the iteration when the algorithm chooses s k to be in S ′ ). Clearly , | S | = y X k =1 r ( s k ) . Let σ denote a permu tation o ver { 1 , . . . , y } that satisfies the following constraints. 1. For every j ∈ { 1 , . . . , y } , d ∗ j = d σ ( j ) . 2. For every j, k ∈ { 1 , . . . , y } , if s ∗ k = s j then k = σ ( j ) . Clearly , such a mapping exists. For each i ∈ { 1 , . . . , n } , let S ∗ i (respectively , D ∗ i ) be the subseq uence of all swaps in S ∗ (respectively , h s ∗ 1 , . . . , s ∗ y i ) that apply to voter i only (i.e., all swaps that for some j are of the form ( i, j ) ). Let p = | D ∗ i | an d let D i = h s k 1 , . . . , s k p i be the subsequence of all swaps in h s 1 , . . . , s y i that σ maps to some e lement in D ∗ i . In p articular, this subsequen ce p reserves the order in which the algorithm applies the sw aps. W e claim, for every q ∈ { 1 , . . . , p } , that r ( s k q ) ≤ | S ∗ i | / ( | D ∗ i | + 1 − q ) . This is because, by our constru ction of σ , at the time the algor ithm is abo ut to choo se s k q it has not cho sen s ∗ σ ( k q ) nor any of the other swaps in S ∗ i that come af ter it (in fact, the algorithm may not have chosen a single swap in S ∗ i ). Because the subse- quence h s k 1 , . . . , s k p i preserves the o rder in which th e swaps were m ade, the algo- rithm sti ll needs at this poin t to close deficits against the candid ates d k q , d k q +1 . . . , d k p (= d ∗ σ ( k q ) , d ∗ σ ( k q +1) , . . . , d ∗ σ ( k p ) ) . So at the time the algor ithm ch ooses swap s k q , it could instead take the lo ngest subsequen ce of S ∗ i that remains unchosen. O bviously , this subsequence is at most | S ∗ i | 8 swaps lon g an d, as discussed above, it yields at least | D ∗ i | + 1 − q deficit reduction s. Since s k q was chosen as part of a sequ ence S ′ for which | S ′ | / | D ( P , S ′ , c ) | ( = r ( s k q ) , where P he re is taken to be in th e same state as when S ′ was chosen) was as small as possible, our claim holds. But then | S | = y X k = i r ( s k ) ≤ n X i =1 | D ∗ i | X q =1 | S ∗ i | / ( | D ∗ i | + 1 − q ) ≤ n X i =1 m X q =1 | S ∗ i | / ( m + 1 − q ) ≤ | S ∗ | ln m + 1 4 A ppr oximating Y oung’ s scoring rule For a given prefe rence profile P , let E ′′ be the collection of all single-elem ent se- quences o f deletion s on P . Let u s call the voting rule based on the g eneric alg orithm with E = E ′′ “Marginal-Cost-Greed y-Y ou ng. ” Theorem 4. Marginal-Cost-Gr eedy-Y oung runs in time O ( N 2 log N ) . Theorem 5. Marginal-Cost-Gr eedy-Y oung is a O (log m ) appr oximation of the Y o ung scor e, wher e m is the numb er of candidates in a given input pr eference pr o file. The pro ofs of the ab ove theorems are essentially ana logous to tho se of theo rems 1 and 3. 5 Conclusion W e provide scoring r ules that approximate Dodg son’ s and Y o ung’ s r ules to within lo g- arithmic f actor s. Assuming P 6 = N P , the bound on Dodgson’ s scorin g rule is within a constant factor o f the optimal polyno mial-time appr oximation . Many n atural q uestions arise fr om this work. What ar e the actual optim al po lynomial- time approxim ations to Dodgson and Y oun g scores, assuming P 6 = N P ? How frequ ently do th e fin al can- didate r anking s according to ou r scoring rules eq ual th ose giv en b y Dod gson’ s and Y oung ’ s rules on the same input? Our paper gives a general framework for edit-based sco ring rules, in which different types of edits could be combined to produce an endless stream of distinct voting rules. The b asic p roblem of comparing, in such a b roaden ed setting , edit distanc es ag ainst the edit sequ ences pr oduced by th e kind algorithms presented h ere seem s worthy of further research. 9 Finally , in the intr oduction we explained ou r voting ru les in ter ms of an a uction-like mechanism, where we assumed that all voters value all edits equa lly . 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