Dividing Connected Chores Fairly

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1 Dividig Coected Chores Fairly Sady Heydrich a,b, Rob va Stee c, a Max Plack Istitute for Iformatics, Saarbrücke, Germay b Saarbrücke Graduate School of Computer Sciece, Germay c Uiversity of Leicester, Leicester, UK Abstract I this paper we cosider the fair divisio of chores (tasks that eed to be performed by agets, with egative utility for them), ad study the loss i social welfare due to fairess. Previous work has bee doe o this so-called price of fairess, cocerig fair divisio of cakes ad chores with o-coected pieces ad of cakes with coected pieces. I this paper, we cosider situatios where each player has to receive oe coected piece of the chores. We provide tight or early tight bouds o the price of fairess with respect to the three mai fairess criteria proportioality, evy-freeess ad equitability ad for utilitaria ad egalitaria welfare. We also give the first proof of the existece of equitable divisios for chores with coected pieces. Keywords: divisio of chores, price of fairess, cake cuttig. Itroductio Fair divisio is a topic that has bee worked o for milleia. It is kow that already the aciet Greeks kew the cut ad choose method for achievig a fair divisio of some good betwee two people (see e.g. []). Motivated by the fact that social iteractio ofte requires dividig goods, researchers i ecoomics, law ad computer sciece dealt with this subject sice the 940 s. I fair divisio, oe tries to divide some good betwee a umber of people that all have idividual prefereces ad dislikes, while satisfyig some fairess coditio. We will oly focus o the case where the goods are divisible, i.e. ca be cut i arbitrary pieces; dividig idivisible goods fairly is a much harder problem. For the settig where each of the players wats to get as much of the goods as possible, oe ofte uses the aalogy of cake cuttig [6], meaig that we wat to divide a cake that has various sectios with differet toppigs. While i cake cuttig we wat to maximize the happiess of the players with the fractio of the cake they receive, i the chore divisio [0] or dirty work problem [5, p. 73] oe tries to miimize the discotet of the players whe dividig work or other disliked goods. May algorithms foud for cake cuttig also apply to the divisio of chores, but iterestigly, as we will see i this work their theoretical properties differ i several cases. Of course oe has to decide how to defie fairess, ad the three criteria proportioality, evy-freeess ad equitability cosidered i may earlier papers (e.g. [5,, 6]) will also be cosidered here. Iformal defiitios for these are give i the ext paragraph. Apart from achievig a divisio which is fair i some sese, aother goal is optimizig the welfare of the divisio, i.e. to maximize the utility or miimize the disutility of the players. As those two quality criteria for divisios are somehow orthogoal, the atural questio arises what the trade-off betwee those two goals is. Caragiais et al. [6] ad Auma ad Dombb [] examied this trade-off for the divisio of cakes ad chores; Caragiais et al. foud upper ad lower bouds for this trade-off, called the price of fairess, for both cakes ad chores, but without ay restrictio o the umber of pieces each player receives. This may lead to the udesirable situatio that players receive a huge umber of small pieces, e.g. a buch of crumbs i the cake aalogy. Therefore, Auma ad Dombb [] examied the price of fairess for coected pieces, requirig that every player receive exactly oe coected part of the cake; however, they did ot cosider divisio of chores. To close the gap, i this paper A prelimiary versio of this paper appeared i Proc. 6th Itl. Symp. o Algorithmic Game Theory (SAGT 03), p addresses: heydrich@mpi-if.mpg.de (Sady Heydrich), rvs4@le.ac.uk (Rob va Stee) Work performed while this author was at Max Plack Istitute for Iformatics, Saarbrücke, Germay Preprit submitted to Elsevier October 30, 04

2 we give bouds o the price of fairess with coected pieces i divisio of chores. A aalogy for this could be that a group of gardeers eeds to maitai a garde ad each of them wats to be resposible for oe coected area... Model I our model, the chores are represeted by the real iterval [0,] ad we cosider players. Each player has a disutility fuctio over this iterval that gives his discotet for a particular piece. These fuctios are required to be o-atomic measures, i.e. they are o-egative ad additive ad if a iterval is valued strictly positive, it must have a subiterval that has a strictly less but still strictly positive value; furthermore the fuctios are ormalized, so the disutility for the whole chores is. The disutility of a player i a divisio is the the disutility of this player for the piece he receives. The utilitaria welfare for a divisio is defied as the sum over the disutilities of all players, while egalitaria welfare is the greatest disutility amog all players (i.e. the disutility of the worst-off player). A divisio is called optimal if it miimizes the welfare. We call a divisio proportioal if every player s disutility is at most, we call it evy-free if o player thiks that aother player receives less chores tha him, ad we call it equitable if the disutilities of all players i the divisio are equal. To quatify the loss i welfare due to fairess we use the otio of the price of fairess. We defie the price of proportioality (respectively evy-freeess, equitability) as the ratio betwee the welfare of the best proportioal (respectively evy-free, equitable) ad the welfare of the optimal divisio... Related Work Moder mathematicias started workig o the topic i the 940 s with Baach, Steihaus ad Kaster givig the Last Dimiisher mechaism for proportioal divisios with players [6]. I the followig years, research maily focused o fidig algorithms for achievig fair divisios ([4, 5, 9, 7]), also tryig to boud the umber of cuts required. Furthermore, Dubis ad Spaier as well as Stromquist gave existece theorems for certai fair divisios [9, 7]. O the problem of fair divisio of chores however, much less work has bee doe. The problem was first metioed by Garder [0], Oskui [5, p. 73] gave the first three perso evy-free chore divisio algorithm. Peterso ad Su gave a four-perso evy-free protocol [3] ad a perso protocol [4]. The result of Steihaus [6] ca be applied to chores too, so proportioal divisios always exist. By Su [8], it is prove that evy-free divisios of chores with coected pieces also always exist. For the existece of equitable divisios with coected pieces, as far as we kow o costructive proof was give so far. The problem of the efficiecy of fair divisios was first addressed by Caragiais et al. [6]. Their work cosidered the price of fairess for utilitaria welfare ad the three fairess otios proportioality, evy-freeess ad equitability, ad examied bouds for these for divisible ad idivisible cakes as well as chores. Auma ad Dombb [] gave bouds for the price of fairess for utilitaria ad egalitaria welfare, restricted to the case that oly coected pieces are allowed to be give to the players so they do ot ed up with a coutable uio of crumbs ([7]), but they oly cosidered cake cuttig. I this work we try to fid bouds for the remaiig case of chore divisio with coected pieces. Followig the work of Caragiais et al. ad Auma ad Dombb, Cohler et al. [8] provided a polyomial time approximatio scheme for computig evy-free cake divisios that are optimal w.r.t. utilitaria welfare. Based o this work, Bei et al. [] give a polyomial time approximatio scheme for computig optimal proportioal cake divisios with coected pieces. Brams et al. [3] coected the topic of efficiet fair divisios with the sphere of Pareto-optimal divisios, i.e. divisios i which it is ot possible to give oe player a strictly higher utility while givig o player a lower utility. They examied whether we ca always fid fair divisios maximizig the (utilitaria) social welfare that are also Paretooptimal ad showed that for a special class of evaluatio fuctios, the optimal (w.r.t utilitaria welfare) equitable divisio has ever a higher social welfare tha the optimal evy-free divisio. However, they did ot take coected pieces ito accout..3. Overview of Results We examie the price of fairess for utilitaria ad egalitaria welfare ad the three fairess otios proportioality, evy-freeess ad equitability as a fuctio of the umber of players. We give tight bouds for all cases except

3 Table : Results of this work, compared to [6] ad []. Some results oly hold for 3. See the text for the case =. Chores: coected (this work) o-coected ([6]) utilitaria egalitaria utilitaria lower upper lower upper Proportioality / Evy-Freeess (+) 4 (+) 4 Equitability Cakes: coected ([]) o-coected ([6]) utilitaria egalitaria utilitaria lower upper lower upper Proportioality + o() Ω( ) O( ) Evy-Freeess + o() / Ω( ) / Equitability + / (+) 4 for the utilitaria price of proportioality, where there is still a small gap betwee the lower ad the upper boud. All results are summarized ad compared to the results by Caragiais et al. [6] ad Auma ad Dombb [] i Table. Essetially the same results, but with a tight boud of for the utilitaria price of proportioality, were achieved idepedetly by Hoffma et al. []..3.. The Price of Proportioality For utilitaria welfare, we show that the price of proportioality is liear i the umber of players. To be precise, we give a lower boud of / ad a upper boud of for >. The proof for the upper boud is idetical to the oe give by Caragiais et al. [6]. For = we show a tight boud of. The bouds that were give by Auma ad Dombb [] for cake cuttig with coected pieces are i Θ( ), Caragiais et al. [6] foud bouds i Θ() for chore divisio with o-coected pieces. For egalitaria welfare we have a price of proportioality of for all, meaig that there is o trade-off betwee proportioality ad egalitaria welfare. This is the same result as show by Auma ad Dombb []..3.. The Price of Evy-Freeess While for cake cuttig with coected pieces Auma ad Dombb [] foud almost tight bouds i Θ( ) for utilitaria welfare ad the tight boud / for egalitaria welfare, we ca show that for the divisio of chores with more tha players ad both welfare fuctios, we ca costruct istaces that have a arbitrarily high price of fairess, i.e. the price of evy-freeess is ubouded. So we see that for this fairess otio there is a iheret differece betwee cakes ad chores ad betwee the coected ad the o-coected case. Oly for = we ca derive bouds of (utilitaria welfare) ad (egalitaria welfare) The Price of Equitability As far as we kow, our proof is the first costructive proof for the existece of equitable egalitaria-optimal divisios of chores with coected pieces. I.e. we prove that for all there always exists a equitable divisio that is as good as the optimal divisio i terms of egalitaria welfare, i other words the egalitaria price of equitability is. We show this by costructig a equitable divisio startig with a egalitaria-optimal divisio without icreasig ay player s disutility above the egalitaria welfare. We also give a tight boud of for the utilitaria price of equitability for all, by givig a istace of the chore divisio problem where at least oe player has to get a disutility of ɛ i every divisio. For this price of fairess, Auma ad Dombb [] gave a upper boud of ad a lower boud of +, ad for o-coected chores, Caragiais et al. [6] give also a tight boud of. 3

4 . Defiitios I this sectio we formally defie the chores divisio problem itself, the otios of fairess ad social welfare used i this work ad fially the price of fairess, the measure for the trade-off betwee fairess ad social welfare. The real iterval [0, ] represets the chores we wat to divide. Our players are simply deoted by the umbers,...,, we write [] = {,..., }. Each player i has a certai valuatio fuctio v i ( ), that maps ay possible subset of the chores to a real valuatio betwee 0 ad. This valuatio fuctio eeds to be a o-atomic measure with v i (0, ) =. Defiitio. A divisio x of the chores is a vector x = (x,..., x, π) [0, ] S. The poit x i deotes the positio of the i-th cut, we defie x 0 := 0, x :=, ad the cuts are sorted: x 0 x... x x. π is a permutatio that deotes the assigmet of the pieces to the players: Player i receives the iterval (x π(i), x π(i) ). By X we deote the set of all possible divisios. The uhappiess of the players with a certai divisio is give by the otio of disutility. Defiitio. The disutility of a divisio x for a player i is d i (x) = v i (x π(i), x π(i) ). I this work, three differet otios of fairess are cosidered, give i the followig defiitios. Defiitio 3. A divisio x is proportioal if d i (x) for every player i. Ituitively, a divisio is proportioal if all players get a portio they cosider their fair share of the chores (or less). Defiitio 4. A divisio x is evy-free if v i (x π(i), x π(i) ) v i (x π(j), x π(j) ) for every pair of players i, j. Ituitively, a divisio is evy-free if o player evies ay other player, i the sese that he values the other player s piece less tha his ow piece. Note that every evy-free divisio is also proportioal. Defiitio 5. A divisio x is equitable if d i (x) = d j (x) for every pair of players i, j. So a divisio is equitable if the disutilities of all players are equal (by their ow valuatios). The social welfare of a divisio ca be defied i two ways: Defiitio 6. A divisio x has utilitaria social welfare u(x) = d i (x). i [] Defiitio 7. A divisio x has egalitaria social welfare eg(x) = max i [] d i(x). So i utilitaria welfare, the total disutility of all players is cosidered, whereas egalitaria welfare refers to the disutility of the worst-off player. To quatify the amout of social welfare oe has to sacrifice to achieve fairess, we defie the price of fairess: Defiitio 8. The price of fairess (price of proportioality, respectively evy-freeess, equitability) is the miimal welfare achievable i fair (proportioal, respectively evy-free, equitable) divisios divided by the miimal welfare achievable i arbitrary divisios. For example the price of evy-freeess with egalitaria welfare is mi x X EF eg(x) mi x X eg(x), where X EF deotes the set of all coected evy-free divisios. 3. The Price of Proportioality We start with bouds for the price of proportioality. For utilitaria welfare, the results do ot differ much from the results for o-coected chores by Caragiais et al. [6], although the lower boud is slightly better. Cocerig egalitaria welfare, we ca use a proof aalogous to the proof by Auma ad Dombb for the price of proportioality with coected cakes []. The idea of the lower boud proof for utilitaria welfare is to costruct a istace where oe player, who is idifferet ad dislikes the chores uiformly, receives a piece slightly greater tha i the optimal divisio, ad where it is very costly to give some part of this piece to ay other player. Ituitively oe could say that this is a sceario where oe player sacrifices himself to do more work tha his fair share (i terms of proportioality) would be, as he himself does ot dislike this work as much as the other players. 4

5 3.. Utilitaria Welfare Theorem. The utilitaria price of proportioality for the divisio of chores with coected pieces is lower-bouded by for >. Proof. Cosider the followig istace of the chores divisio problem: Let 0 < ɛ < 4 be arbitrarily small. Players,..., value the piece ( ) as ɛ ad the piece ( ) as + ɛ. Player values the etire chores uiformly. This istace together with its optimal divisio is show i figure. Figure : Example costructio for = 4 players. The two bars at (3/4 3/4) ad ( ) deote the valuatios of players,...,. The last player dislikes the whole chores uiformly, marked by a grey bar. Below the iterval the optimal divisio is show. I a optimal divisio, the piece (0, ɛ) is assiged to the players,..., ad the rest of the chores to player. This gives 0 disutility for the first players ad a disutility of + ɛ for player, so the utilitaria welfare i such a optimal divisio is + ɛ. Assigig oe of the itervals ( ) ad ( ) to oe of the other players would cause a greater disutility, for ɛ < 4 ad > implies + ɛ < ɛ. I a proportioal divisio however, caot be assiged more tha of the chores ad therefore the itervals at ( ) ad ( ) caot completely be give to player. To be more precise, the other players have to receive fractios of the two itervals ( ɛ, ) ad ( ɛ, ) of total legth ɛ. To achieve a optimal proportioal divisio we assig player the iterval ( ) (that is, we give this player the most costly part of the chores w.r.t the other players disutilities), which gives him a disutility of ɛ ad the rest of the chores is divided betwee the other players such that the divisio is proportioal (which is possible as ( ) ɛ for > ). Their total disutility is therefore ɛ. Hece, the utilitaria welfare i this divisio is ɛ + ɛ =. Fially, the utilitaria price of proportioality is therefore lower-bouded by +ɛ, which is arbitrarily close to for sufficietly small ɛ. Note that for this problem istace, it is essetial that players have to receive coected pieces of the chores; otherwise we could give oly the two strogly disliked pieces ( ) ad ( ) to player. For two players, we cosider a problem istace that we will also use i some later proofs: oe player is idifferet o the chores, the other dislikes oly a small part i the middle. Dividig this istace fairly (for all three fairess measures cosidered here) always results i givig the idifferet player slightly more tha oe half of the chores, so that the other player does ot receive ay part of the chores he dislikes. This agai meas that we sacrifice proportioality i order to give the idifferet player more work while pleasig the other player. Theorem. The utilitaria price of proportioality for = players is lower-bouded by. Proof. Cosider the followig istace: Player values the piece ( + ɛ) as ad the rest as 0, while player values the etire iterval uiformly. I a optimal divisio, we give player the piece (0, ɛ) ad player the piece ( ). The utilitaria welfare of such a divisio is + ɛ, the physical size of the piece of player. To achieve a utilitaria-optimal proportioal divisio, we give each player exactly oe half of the chores (ote that this divisio is also evy-free ad equitable), which gives both a disutility of ad a utilitaria welfare of. Hece the price of fairess is arbitrarily close to / = for sufficietly small ɛ. For the upper boud o the utilitaria price of proportioality, we ca cite the proof by Caragiais et al. [6], as it also applies to coected chores. 5

6 Theorem 3. The utilitaria price of proportioality for the divisio of chores with coected pieces is upper-bouded by. Proof. Let a arbitrary istace of the chores divisio problem be give, let x be a optimal ad y a proportioal optimal allocatio for this istace. The either x is already proportioal i this case the price of proportioality is or at least oe player has a disutility >. I this case, also the total (utilitaria) disutility of x is >. I y however, ay player s disutility is at most by defiitio of proportioality. Thus the utilitaria welfare of y is at most, ad the price of proportioality is upper-bouded by / =. 3.. Egalitaria Welfare Completely aalogous to Auma ad Dombb [], we prove that every egalitaria-optimal divisio is also proportioal, hece we do ot have to give up proportioality to achieve egalitaria-optimal divisios i ay istace. Theorem 4. Every egalitaria-optimal divisio of chores with coected pieces is proportioal, ad therefore the egalitaria price of proportioality i this case is. Proof. Let agai a arbitrary istace of the chores divisio problem be give with a optimal divisio x ad a arbitrary proportioal divisio y. By defiitio of proportioality we kow that d i (y) for all players i, hece also eg(y) = max i [] d i (y). By defiitio of egalitaria welfare, for all players i, d i(x) eg(x) ad as x is egalitaria-optimal, we kow d i (x) eg(x) eg(y), i.e. x is proportioal. 4. The Price of Equitability The price of equitability is a more iterestig case tha proportioality, as so far o costructive proof was give for the existece of equitable divisios of chores with coected pieces. The existece proof of this kid of divisio for cakes was give by Auma ad Dombb []. With some mior chages, their proof might actually be carried over to our settig: We start with a egalitaria-optimal divisio with egalitaria welfare OP T. First, we eed to move cuts betwee the first pieces to the right util they all have the value OP T. However, i costrast to the cake cuttig sceario, it might be ot possible to move all these cuts to the right as far as ecessary. If some cut reaches the boudary, we stop this process ad all followig pieces are empty. I the secod step, we eed to move the last cut to the left util the last piece has value OP T ɛ. Auma ad Dombb oly eed to move this oe cut to the right util the last piece has value OP T + ɛ, however, i our settig it might be ecessary to also move other cuts if the rightmost cut touches them, ad therefore pieces of other players might get empty. But the valuatios for all players except the last oe are adjusted i a recursive step, ad additioally we kow by optimality of the origial divisio that at least oe piece will stay at value OP T. Therefore, Auma ad Dombb s further argumetatio carries over. This approach by Auma ad Dombb, however, is strictly speakig ot costructive. They show how to costruct a egalitaria-optimal divisio i which o player s utility deviates from the optimum by more tha ɛ for ay ɛ > 0 (you could say early-equitable divisios ). The existece of a egalitaria-optimal equitable divisio the follows from the compactess of the set of all egalitaria-optimal divisios. Durig the work o this paper, aother existece proof for such cake divisios was give by Cechlárová, Dobos ad Pillárová [7] (they eve prove a stroger claim), but their proof is o-costructive as well. We complete the picture by givig a costructive existece proof for equitable chore divisios with coected pieces that are egalitaria-optimal. I Theorem 5, we show how to trasform ay egalitaria-optimal divisio ito a equitable oe with the same welfare. The costructio relies o the fact that optimality with respect to egalitaria welfare ad the o-atomicity of the evaluatio measures imply that we ca make pieces that are adjacet to a piece with maximal value (amog all pieces) also maximal. Afterwards we give proofs for a tight boud of for utilitaria welfare. This geeralizes the boud for ocoected chores give by Caragiais et al. [6]. 6

7 4.. Egalitaria welfare Theorem 5. For every istace of the chores divisio problem, there exists a equitable divisio with coected pieces. Furthermore, the egalitaria price of equitability for the divisio of chores with coected pieces is. Proof. We eed some more termiology for this proof: The value of a piece is the value that is assiged to this piece by the player who receives it. Let m = mi x X eg(x) be the optimal egalitaria welfare. Pieces that have a value of m are called maximal pieces. We call a sequece of oe or more adjacet maximal pieces p,..., p k a block of maximal pieces, if the left eighbour of p is o-maximal or p is the left-most piece of the divisio, ad the right eighbour of p k is o-maximal or p k is the right-most piece of the divisio We call the left eighbour of p ad the right eighbour of p k the eighbours of this block (if they exist). Cosider a egalitaria-optimal divisio x that has the miimal umber of maximal pieces amog all egalitariaoptimal divisios. We wat to make all pieces i x maximal by movig cuts, ad for this we first show a lemma: Lemma. Cosider a divisio x with a block p,..., p k of k maximal pieces that has a right eighbor p (i.e. the rightmost piece of the block is ot the last piece). Let l be the left border of the block, r the right border ad r the right border of p. The there exists a divisio x which is idetical to x o [0, l] ad [r, ], ad which has oe of the followig two properties. (a) The pieces p,..., p k ad p are all maximal. (b) Noe of the pieces p,..., p k ad p are maximal. Proof. We do a iductio o the umber of pieces i the block k: Iductio base: k =. We move the cut betwee our oly maximal piece p ad its right eighbour p to the left, i.e. we make p physically smaller ad p physically larger. Either at some poit the value of p falls below m, the our whole block is o-maximal, or evetually p becomes maximal. Iductio step: k k. Move r to the left util p is maximal (stop at the first poit where p is maximal). If this is ot possible, we ca make the whole block o-maximal (we move r to the left util it reaches l; the all pieces of the block are empty). So assume, p becomes maximal whe r reaches some poit r > l. If p k is still maximal, we are doe. Otherwise, the pieces to the left of p k are a block of size at most k, hece we ca apply the iductio hypothesis. The either we ca make p k (ad all pieces to the left of p k that became o-maximal) maximal agai by oly movig cuts iside the block; i particular, we do ot chage r ad therefore p is still maximal. Or it is ow possible to make all k pieces o-maximal. The the block icludig p k is o-maximal, ad p is maximal. Say p k s value is m ɛ for some ɛ > 0. As the valuatio fuctios are o-atomic measures, there must exist a δ > 0, such that movig r back to the right by δ icreases p k by less tha ɛ. O the other had, we moved r oly as far to the left as eeded so that p became maximal, hece movig r to the right by δ decreases p s value below m. Therefore, all pieces of the origial block ad p are o-maximal. The same lemma ca also be show for the left eighbour completely symmetrically. Now cosider our optimal divisio x ad look at its left-most block of maximal pieces. By our lemma we ca make its right eighbour maximal, as otherwise the block would become o-maximal ad this would cotradict the assumptio that x is the divisio with miimal umber of maximal pieces. But by applyig the lemma agai ad agai, we ca make all pieces to the right of this (steadily growig) left-most block maximal. Note that every time, if we fid that we ca make the etire block o-maximal, we fid a divisio with less maximal pieces tha x, which is a cotradictio. If we reach a piece that is already maximal durig this process, we just add it to the block without movig cuts of course. We ca the apply the lemma symmetrically for the pieces to the left of this block (which is ow the oly block of maximal pieces i x) ad make them all maximal too. Fially we have a divisio where all pieces are maximal. 7

8 4.. Utilitaria welfare While achievig equitability does ot ifluece the egalitaria optimality, it has a huge impact o the utilitaria welfare, as show i the ext three theorems. The idea of the lower boud proof is to make sure that oe idifferet player has to receive at least a piece of a certai value i both fair ad ufair divisios, which leads to a price of fairess of, as i equitable divisios all player have to receive this certai disutility, while the idifferet player is the oly oe to receive ay disliked piece i the utilitaria-optimal divisio. Theorem 6. The utilitaria price of equitability is lower-bouded by for > players. Proof. We costruct a istace of the chores divisio problem that has a utilitaria price of equitability of at least as follows: Let ɛ > 0 be arbitrarily small. We create ( ) so-called disliked pieces p,..., p ( ), where p i is located i at ( i + ɛ). We divide those pieces ito ( ) blocks of pieces each, ad each block cotais oe piece for every player {,..., }. The first piece of the first block is associated with player, the secod with player ad so o, util the last piece of the first block is associated with player. The pieces of the secod block are the associated with players, 3,...,, (i this order), the pieces of the third block with players 3, 4,...,,, ad so o. Geerally, the pieces of the i-th block are associated with players i, i +,...,,,..., i. Each player values each piece associated with him as ad the rest of the chores as 0, which sums up to a total valuatio of for the etire chores for each of the first players. Fially player values the etire chores uiformly. This costructio is show i figure. Figure : Example costructio for = 4 players. The umbers above the colums deote the player this piece is associated with. Above the iterval the optimal divisio is show. First, we take a look at the optimal divisio for this istace, that is obtaied as follows: we give the piece i ( ) (i+) ( ) (0, ɛ) to player, the piece ( ɛ) to player i for i =,..., ad fially the piece ( ( ) ) to player. We observe the followig: For i =,..., the i-th disliked piece of player i is at ( (i ) ( )+ by costructio, as this piece is the first piece of the i-th block. (i ) ( )+ + ɛ). This follows For i =,..., the i + -st disliked piece of player i is at ( (i+) ( ) (i+) ( ) + ɛ). This also follows by costructio, as this piece is the last piece of the i + -st block. We coclude that the piece that is assiged to player i =,..., as above is betwee the i-th ad i + -st piece of this player. The piece player receives is before his first disliked piece. Hece, the disutility of the players,..., are all 0, as they all do ot receive ay of their disliked pieces. Player s disutility is exactly the physical size of the piece he receives, so the utilitaria welfare i this divisio is u(x) = d (x) = ( ( ) ɛ) = + ɛ. The optimal divisio for the example with = 4 players ca be see i figure. 8

9 Now, we claim that the disutility of player i ay equitable divisio of the chores is at least ɛ. From this it follows that the utilitaria welfare i equitable divisios is at least ( ɛ), as all players must have the same disutility. Hece, the price of equitability is the at least 0. We first show a lemma: ( ) ɛ + ( ) +ɛ +. The boud follows as ɛ approaches Lemma. There does ot exist a divisio i which player receives o piece ad each other player has disutility 0. Proof. Assume we could divide the chores betwee the first players etirely s.t. every player s disutility is 0. Some player must therefore receive a piece startig at 0. But this piece caot cover more tha the first block of disliked pieces, as i this block oe disliked piece of this player is cotaied ad he caot receive that. I particular, his piece must ed somewhere before his disliked piece ad therefore iside the first block. Accordigly, the secod piece must ed somewhere i the secod block ad so o. The, the last piece that is assiged to ay of the players must ed somewhere i the (last) -st block, amely before his last disliked piece. But the the iterval from this last disliked piece to the right border of the chores stays uassiged. Cotradictio. So it remais to show that player s disutility is always at least ɛ. First of all, player caot have a disutility of 0, as the he would t receive aythig of the chores, leavig the whole chores to players,...,. But as show i the lemma above, we caot divide the chores betwee the first players such that each of them gets 0 disutility, so player has to receive some piece of the chores. Assume player s disutility is less tha disliked pieces etirely (as they have a valuatio of > get fractios of their disliked pieces. There are two cases ow: (a) The piece for player is the left-most or right-most piece (b) The piece for player is somewhere i betwee the other players pieces ɛ. That implies that all other players caot get oe of their ɛ for each of those pieces), but they have to If player s piece is the right-most piece, the accordigly to the argumetatio i the proof of Lemma the last piece assiged to ay other player must ed somewhere i the last block (as otherwise some player would get his etire disliked piece), so player s piece must reach from the right border of the iterval ito the last block, therefore his disutility is greater tha ɛ (the size of the slot betwee the last disliked piece ad ). Note that the argumetatio i the proof of Lemma ca also be applied from right to left : Some player receives a piece startig at the right border ad the the i-th piece from the right must ed i block i. Hece the case whe player receives the left-most piece is symmetric to the case that he receives the right-most piece ad we are doe for the first case. Cosider the other case, whe player s piece is i betwee the pieces of the other players. Say there are i players receivig the chores o the left of s piece ad i players receivig the chores o the right. The agai the left border of player s piece is somewhere i the i-th block ad the right border somewhere i the i + -st block, hece player must receive a piece that has a physical size of more tha ɛ, sice this is the size of the slot betwee the last piece of the i-th block ad the first piece of the i + -st block.. For the special case of two players, the lower boud proof uses the same istace we have see before i Theorem Theorem 7. The utilitaria price of equitability for = players is lower-bouded by. Proof. Cosider the same istace as i the proof of Theorem : Player values the piece ( + ɛ) as ad the rest as 0, while player values the etire iterval uiformly. I a optimal divisio the utilitaria welfare is + ɛ, ad i the optimal equitable divisio (the same as i Theorem ), the welfare is. The price of fairess therefore approaches / = for sufficietly small ɛ. For provig a matchig upper boud, we re-use the costructive proof of Theorem 5 to costruct a equitable divisio which is at most -times as bad as the utilitaria-optimal divisio. Theorem 8. The utilitaria price of equitability is upper-bouded by. 9

10 Proof. Cosider a chores divisio istace ad let x be a utilitaria-optimal divisio for it. Of course the egalitaria welfare of x is at most the utilitaria welfare of x, i.e. eg(x) u(x). Accordig to Theorem 5 there exists a equitable divisio y (which we ca costruct with the method described i the proof of Theorem 5) that achieves the same egalitaria welfare as the egalitaria-optimal divisio, hece y has at most the same egalitaria welfare as x (x does ot have to be egalitaria-optimal of course). Obviously, u(y) eg(y). But the, u(y) eg(y) eg(x) u(x), i.e. the utilitaria price of fairess is at most. 5. The Price of Evy-Freeess Fially, we take a look at the price of evy-freeess. For this fairess otio, we get the most iterestig deviatio from former results o coected cakes ad o-coected chores, as we ca prove uboudedess of the price of fairess here (for more tha two players). I cotrast to the previous theorems, the arbitrary high price of fairess ow does ot result from givig a idifferet player more tha his fair share i the optimal divisio, which results i high costs whe assurig fairess, but from the fact that i the optimal divisio for the cocrete istace give below, it is optimal to give the idifferet player o piece of the chores; however ote that this player is ow oly idifferet o several pieces of the chores, ot o the whole chores. But a situatio where oe player does ot receive ay piece i the optimal divisio has a egative effect o the price of evy-freeess, as every other player receivig a disutility > 0 will evy this player. By choosig the prefereces i a certai way, we ca make the price of evy-freeess arbitrarily high, as show below. Theorem 9. The price of evy-freeess for the divisio of chores with coected pieces is ubouded for both utilitaria ad egalitaria welfare for > players. Proof. We use the same costructio we used i the proof of Theorem 6 for the utilitaria price of equitability with other valuatio fuctios, which ca have a arbitrarily large price of evy-freeess. Agai, let 0 < ɛ < ad cosider ( ) disliked pieces arraged i ( ) blocks as before, where piece p i is located at ( ) i ( i + ɛ) for i =,..., ( ). Call the first piece of each block type A piece, the other pieces type B pieces. As before, the pieces of the i-th block are associated with players i, i +,...,,,..., i. Each of them values the oly type A piece that is associated with him as ( )ɛ, the ( ) type B pieces associated with him as ɛ ad the rest of the chores as 0. Player assigs a value of ( ) the rest. A example costructio with 4 players ca be foud i figure 3. to every disliked piece of either type ad 0 to Figure 3: Example costructio for = 4. For better readability, the left image shows the valuatios of players to 3, the right image shows the valuatio of player 4. The umbers above the colums deote the player this piece is associated with. Above the left iterval the optimal divisio is show, above the right iterval the evy-free divisio is show. A optimal divisio is costructed as follows: Similarly to the optimal divisio give i the proof for Theorem i ( ) (i+) ( ) 6, we give the piece (0, ɛ) to player, the piece ( ɛ) to player i for i =,..., 3, but we give the whole remaiig piece ( ( )( ) ) to player. Player does ot receive ay piece of the chores. This optimal divisio is also show i figure 3. We observe the same facts as before: 0

11 For i =,..., the i-th disliked piece of player i is at ( (i ) ( )+ (i ) ( )+ + ɛ). For i =,..., the i + -st disliked piece of player i is at ( (i+) ( ) (i+) ( ) + ɛ). Therefore, the piece player i =,..., 3 receives is betwee the i-th ad i + -st piece of this player. The piece player receives is before his first disliked piece (which starts at ɛ). Hece, all players except player do ot receive ay of their disliked pieces ad therefore have disutility 0. Oly player receives oe type B piece, therefore the maximal disutility, the egalitaria welfare ad the utilitaria welfare of the optimal divisio is ɛ. Note that the optimal divisio from Theorem 6 yields i this case a higher disutility, amely ( ) (which is strictly greater tha ɛ accordig to our assumptios). This optimal divisio however is ot evy-free, as player evies player, for the empty piece is preferred by every player. Aalogous to Lemma, we ca argue that it is impossible to divide the chores betwee the first players etirely without givig oe player a piece he dislikes. We are always left with at least oe disliked piece p that has to be assiged to oe player amog players,..., that values it > 0. Furthermore, this meas we also have to give some piece to player, as otherwise all players receivig a disutility > 0 evy, ad assigig less tha oe half of p makes the player who receives the rest of p evy. Thus, we ca show that we caot do better tha givig oe half of p ad aother player i who also dislikes p the other half. Therefore, to achieve a ( optimal evy-free divisio ) we give players,..., 3, the same pieces as i the optimal divisio, the piece ( )( ) ( ) ( ) to player ad the piece ( ), to player, i.e. we split the type B piece that player received i the optimal divisio betwee players ad. I this divisio x, d (x) = ɛ ad d (x) = ( ). As both get the same amout of player s last disliked piece, does ot evy ay more. Furthermore, player receives half of a disliked piece, whereas every other player receives more tha oe such piece, hece does also ot evy ay other player. Players,..., 3, have 0 disutility ad therefore also evy o other player. Fially, d (x) > d (x) for ɛ < ( ), hece eg(x) = d (x) = ( ), ad u(x) = d (x) + d (x) = ɛ + ( ). So fially we have a utilitaria price of evy-freeess of + ad a egalitaria price of evy-freeess of ( ) ɛ, which both becomes arbitrarily large whe ɛ approaches 0. ( ) ɛ For the case with two players, we ca give tight bouds for both the utilitaria ad egalitaria price of evyfreeess, essetially due to the fact that we ca switch the two players pieces for achievig evy-freeess. Theorem 0. The utilitaria price of evy-freeess for = players is. Proof. First cosider the same istace as i the proof of Theorem : Player values the piece ( + ɛ) as ad the rest as 0, while player values the etire iterval uiformly. I a optimal divisio, the utilitaria welfare is + ɛ, ad i the optimal evy-free divisio (the same as i Theorem ) the welfare is. Hece the price of fairess is arbitrarily close to for sufficietly small ɛ. For the upper boud we cosider two cases: Either the utilitaria-optimal divisio has welfare, which meas that both players must have a disutility of. But the i both players eyes, the piece of the other player has value d (x) respectively d (x), i.e. they both do ot evy the other. Therefore, i this case the price of fairess is. So cosider the case where the utilitaria-optimal divisio has welfare >. Assume, that the utilitaria welfare i the best evy-free divisio was >. That would mea that either d (x) > or d (x) >, but the oe of the players would evy the other oe ad the divisio would ot be evy-free. Therefore, the welfare i utilitaria-optimal evy-free divisios is at most. Thus the price of fairess is at most / =. Theorem. The egalitaria price of evy-freeess for = players is. Proof. If the egalitaria-optimal divisio has a welfare of, we ca agai argue as i the previous theorem that the this egalitaria-optimal divisio is also evy-free. Cosider therefore optimal divisios with welfare >. That meas that oe of the two players has a disutility of >, say it is w.l.o.g. player. But ow we ca shift our oly cut towards the piece of player util his disutility is. Now of course the disutility of player could have become (or still be) >. But i this case, iterchage the pieces the players receive. Player still receives oe half of the chores, player receives ow less tha oe half. Therefore, the welfare i this divisio is, a cotradictio to the assumptio that the best egalitaria welfare i ay achievable divisio is >.

12 6. Coclusio I this work we examied the decrease of social welfare due to fairess whe dividig chores so that every player receives exactly oe coected piece of the chores. We cosidered three importat fairess criteria ad two differet social welfare fuctios ad foud tight bouds for almost all cases. For utilitaria welfare ad proportioality or equitability the bouds are i Θ(), for egalitaria welfare there is o trade-off for these two fairess criteria. For evy-freeess however, o boud exists for both welfare fuctios except for players. Upo fidig that the price of evy-freeess for the divisio of chores is the oly case that is ubouded, oe could ask the questio why there is such a fudametal differece betwee the price of evy-freeess ad the price of proportioality or equitability, ad why this differece does ot appear whe cosiderig cakes (Auma ad Dombb [] gave bouds for this case). To aswer the first questio, oe should ote that there is a iheret differece betwee evy-freeess ad the other two fairess otios, amely that the first relies o the valuatio of a player for pieces other tha his ow, while i the latter case we oly cosider properties of the valuatios each player has for his particular piece. The differece betwee chores ad cakes seems to arise from the differet ature of the two problems. Ifiite evy always results from oe player receivig o piece of the cake/chores - the this player is either the oe who evies other players (cake divisio) or the oe who is evied by other players (divisio of chores). Givig oe player o chores is - at least i some istaces - desirable i chores divisio, as we have see i the proof of Theorem 9. Whe dividig cakes, however, this is either desirable whe cosiderig utilitaria welfare (see Auma ad Dombb [, Theorem ]) or whe cosiderig egalitaria welfare, as here the player receivig the least utility determies the amout of welfare. This differece ca also be see i the results for idivisible cakes ad chores by Caragiais et al. [6], where the price of evy-freeess is bouded for cakes ad ubouded for chores. It should be oted that a differece betwee cake ad chore divisio was also show by Peterso ad Su [4]. They traslated the -perso evy-free cake divisio protocol of Brams ad Taylor [4] to chore divisio ad argued why chore divisio is ot exactly a dual or straightforward extesio of the cake-cuttig problem [4]. Iterestigly, their procedure for chore divisio is more complicated tha the cake cuttig protocol by Brams ad Taylor, but may coverge faster ad require fewer cuts. Some questios still remai ope. For the settig where o-coected pieces are allowed, Caragiais et al. [6] oly cosidered utilitaria welfare. It would be iterestig to give bouds for the egalitaria welfare fuctio for o-coected divisios of cakes ad chores. Furthermore, Caragiais et al. [6] provided a aalysis of the price of fairess for idivisible cakes ad chores, but agai they oly cosidered utilitaria welfare. Further research could examie the impact of fairess o egalitaria welfare for this settig. Refereces [] Yoata Auma ad Yair Dombb. The efficiecy of fair divisio with coected pieces. I Ami Saberi, editor, WINE, volume 6484 of Lecture Notes i Computer Sciece, pages Spriger, 00. [] Xiaohui Bei, Nig Che, Xia Hua, Biaoshuai Tao, ad Edog Yag. Optimal proportioal cake cuttig with coected pieces. I Jörg Hoffma ad Bart Selma, editors, AAAI. AAAI Press, 0. [3] Steve Brams, Michal Feldma, Joh Lai, Jamie Morgester, ad Ariel Procaccia. O maxsum fair cake divisios. 0. [4] Steve J. Brams ad Ala D. Taylor. A evy-free cake divisio protocol. America Mathematical Mothly, 0():9 8, Jauary 995. [5] Steve J. Brams, Ala D. Taylor, ad William Zwicker. A movig-kife solutio to the four-perso evy-free cake divisio. I Proceedigs of the America Mathematical Society, volume 5, pages , 997. [6] Ioais Caragiais, Christos Kaklamais, Paagiotis Kaellopoulos, ad Maria Kyropoulou. The efficiecy of fair divisio. I Stefao Leoardi, editor, WINE, volume 599 of Lecture Notes i Computer Sciece, pages Spriger, 009.

13 [7] Kataría Cechlárová, Jozef Dobos, ad Eva Pillárová. O the existece of equitable cake divisios. If. Sci., 8:39 45, 03. [8] Yuga J. Cohler, Joh K. Lai, David C. Parkes, ad Ariel D. Procaccia. Optimal evy-free cake cuttig. I Proceedigs of the Twety-Fifth AAAI Coferece o Artificial Itelligece, 0. [9] L. E. Dubis ad E. H. Spaier. How to cut a cake fairly. America Mathematical Mothly, 68(): 7, 96. [0] Marti Garder. Aha! Isight. W. F. Freema ad Co., 978. [] Hesiod. Theogoy. [] Yosef Hoffma, Yoata Auma, ad Yair Dombb. Private commuicatio. [3] Elisha Peterso ad Fracis E. Su. Four-perso evy-free chore divisio. Mathematics Magazie, 75():7, 00. [4] Elisha Peterso ad Fracis E. Su. N-perso evy-free chore divisio. arxiv.org: , 009. [5] Jack M. Robertso ad William A. Webb. Cake-cuttig algorithms - be fair if you ca. A K Peters, 998. [6] Hugo Steihaus. The problem of fair divisio. Ecoometrica, 6:0 04, 948. [7] Walter Stromquist. How to cut a cake fairly. America Mathematical Mothly, 87(8): , 980. [8] Fracis E. Su. Retal harmoy: Sperer s lemma i fair divisio. America Mathematical Mothly, 06(0):930 94,