The Efficiency of Fair Division with Connected Pieces

Transcription

1 The Efficiecy of Fair Divisio with Coected Pieces Yoata Auma ad Yair Dombb Abstract We cosider the issue of fair divisio of goods, usig the cake cuttig abstractio, ad aim to boud the possible degradatio i social welfare due to the fairess requiremets. Previous work has cosidered this problem for the settig where the divisio may allocate each player ay umber of ucoected pieces. Here, we cosider the settig where each player must receive a sigle coected piece. For this settig, we provide tight bouds o the maximum possible degradatio to both utilitaria ad egalitaria welfare due to three fairess criteria proportioality, evy-freeess ad equitability. 1 Itroductio Cake Cuttig. The problem of fair divisio of goods is the subject of extesive literature i the social scieces, law, ecoomics, game theory ad more. The famous cake cuttig problem abstracts the fair divisio problem i the followig way. There are players wishig to divide betwee themselves a sigle cake. The differet players may value differetly the various sectios of the cake, e.g. oe player may prefer the marzipa, aother the cherries, ad a third player may be idifferet betwee the two. The goal is to obtai a fair divisio of the cake amogst the players. There are several possible defiitios to what costitutes a fair divisio, with proportioality, evy-freeess ad equitability beig the major fairess criteria cosidered (these otios will be defied i detail later). May previous works cosidered the problem of obtaiig a fair devisio uder these (ad other) criteria. Social Welfare. While fairess is clearly a major cosideratio i the divisio of goods, aother importat cosideratio is the social welfare resultig from the divisio. Clearly, a divisio may be evy-free but very iefficiet, e.g. i the total welfare it provides to the players. Accordigly, the questio arises what, if ay, is the tradeoff betwee these two desiderata? How much social welfare does oe have to sacrifice i order to achieve fairess? The aswer to this questio may, of course, deped o the exact defiitio of fairess, o the oe had, ad the social welfare of iterest, o the other. The first aalysis of such questios was provided i [CKKK09], where Caragiais et al. cosider the three leadig fairess criteria proportioality, evy-freeess ad equitability ad quatify the possible loss i utilitaria social welfare due to such fairess requiremets. Here we cotiue this lie of research, extedig the results i two ways. Firstly, the [CKKK09] aalysis allows dividig the cake ito ay umber of pieces, possibly eve ifiite. Thus, each player may get a collectio of pieces, rather tha a sigle oe. While this may be acceptable i some cases, it may ot be so i others, or at least highly udesirable, e.g. i the divisio of real estate, where players aturally prefer gettig a coected plot. Similarly, i the cake sceario itself, allowig ucoected pieces may lead to a situatio where, i Stromquist s words [Str80], a player who hopes oly for a modest iterval of the cake may be preseted istead with a coutable uio of crumbs. Accordigly, i this work, we focus o divisios i which each player gets a sigle coected piece of the cake. I additio, we cosider both the utilitaria ad the egalitaria social welfare fuctios, whereas Caragiais et al. cosidered oly utilitaria welfare. For each

2 of these welfare fuctios, we give tight bouds o the possible loss i welfare due to the three fairess criteria. 1.1 Defiitios ad Notatios We cosider a rectagular cake that ca be divided by makig parallel cuts. The cake ca thus be represeted by the iterval [0, 1], where each cut is some poit p [0, 1]. The cake eeds to be divided to players (we use the otatio [] for the set {1,..., }), each of which has a valuatio fuctio v i ( ) assigig a o-egative value to every possible iterval of the cake. As customary, we require that for all i, v i ( ) is a oatomic measure o [0, 1] havig v i (0, 1) = 1. Every set of valuatio fuctios {v i ( )} i=1 defies a istace of the cake cuttig problem. Sice we cosider oly divisios i which every player gets a sigle coected iterval, a divisio of the cake to players ca be represeted by a vector x = (x 1,..., x 1, π) [0, 1] 1 S with 0 x 1 x x 1 1. Here, x i determies the positio of the i-th cut, ad π is a permutatio that determies which piece is give to which player. For coveiece, we deote x 0 = 0 ad x = 1, so we ca write that player i [] receives the iterval (x π(i) 1, x π(i) ). We use the otatio u i (x) for the utility that player i gets i the divisio x, i.e. u i (x) = v i (x π(i) 1, x π(i) ). We deote by X the set of all possible divisio vectors, ad ote that X is a compact set. Fairess Criteria. We say that a divisio x X is: Proportioal if every player gets at least 1 of the cake (by her ow valuatio). Formally, x is a proportioal divisio if for all i [], u i (x) 1. Evy-Free if o player prefers gettig the piece alloted to ay of the other players. Formally, x is a evy-free divisio if for all i j [], u i (x) = v i (x π(i) 1, x π(i) ) v i (x π(j) 1, x π(j) ). Equitable if all the players get the exact same utility i x (by their ow valuatios). Formally, x is a equitable divisio if for all i, j [], u i (x) = u j (x). Stromquist [Str80], showed that for every istace of the cake cuttig problem there exists a evy-free divisio with coected pieces. Sice oe ca easily observe that every evyfree divisio is i particular proportioal, this implies that such proportioal divisios also always exist. I this paper we show (Theorem 6) that equitable divisios also always exist for coected pieces (for the case where players eed ot get a sigle iterval, this is well kow). Social Welfare Fuctios. welfare of x, i.e. For a divisio x X, we deote by u(x) the utilitaria social u(x) = u i (x). i [] Likewise, we deote by eg(x) the egalitaria social welfare of x, which is eg(x) = mi i [] u i(x). Note that both these social welfare fuctios are cotiuous ad thus have maxima i X.

3 The Price of Fairess. As described above, we aim to quatify the degradatio i social welfare due to the differet fairess requiremets. This is captured by the otio of Price of Fairess, i its three forms Price of Proportioality, Price of Evy-freeess ad Price of Equitability, defied as follows. The Price of Proportioality (resp. Evy- Freeess, Equitability) of a cake-cuttig istace I, with respect to some predefied social welfare fuctio, is defied as the ratio betwee the maximum possible social welfare for the istace, take over all possible divisios, ad the maximum social welfare attaiable whe divisios must be proportioal (resp. evy-free, resp. equitable). Whe cosiderig divisios with coected pieces, this restrictio is applied to both maximizatios. For example, if X EF X is the set of all (coected) evy-free divisios of a istace, the egalitaria Price of Evy-Freeess for this istace is max x X eg(x) max y XEF eg(y). I this work we show bouds o the maximum utilitaria ad egalitaria Price of Proportioality, Evy-Freeess ad Equitability of ay istace. 1. Results We aalyze the utilitaria ad egalitaria Price of Proportioality, Evy-Freeess ad Equitability for divisios with coected pieces. We provide tight bouds (i some cases, up to a additive costat factor) for all six resultig cases. The results are summarized i Table 1; the last row presets the relevat previous results by Caragaiis et al. i [CKKK09], for compariso. The meaig of the upper bouds is that the respective price of fairess of ay possible istace is ever greater tha the boud. The meaig of the lower boud is that there exists a istace that exhibits at least this price of fairess (for the respective class). Price of: Proportioality Evy-Freeess Equitability Utilitaria Egalitaria (tight) Utilitaria UB: LB: + 1 o(1) UB: coected LB: pieces 1 1 (this work) UB: 1 UB: 1 UB: o-coected LB: LB: LB: Table 1: All results (+1) 4 pieces [CKKK09] Utilitaria Welfare. For the utilitaria social welfare, we show a upper boud of + 1 o(1) o the price of evy-freeess, for ay possible istace. This, we believe, is the first o-trivial upper boud o the Price of Evy-Freeess. It seems that such bouds are hard to obtai sice o the oe had we eed to cosider the best possible evy-free divisio, while o the other had o efficiet method for explicitly costructig ay evyfree divisios is kow. We show that the same upper boud also applies to the Price of Proportioality. For the Price of Equitability, we show that it is always bouded by (though simple, this does require a proof sice a equitable divisio eed ot eve give each player 1/). We also provide a almost matchig lower boud, showig that for ay there exists a istace with utilitaria Price of Equitability arbitrarily close to

4 Egalitaria Welfare. Whe cosiderig the egalitaria social welfare, we show that there is o price for either proportioality or equitability. That is, for ay istace there exist both proportioal ad equitable divisios for which the miimum amout ay player gets is o less tha if there were o fairess requiremets. While perhaps ot surprisig, the proof for the Price of Equitability is somewhat ivolved, especially sice we require that the divisios be with coected pieces. We ote that we are ot aware of ay previous proof that altogether establishes the existece of a equitable divisio with coected pieces. For the Price of Evy-Freeess, we show that it is bouded by /, ad provide a matchig family of istaces that exhibits this price, for ay. Paper Orgaizatio. I Sectio, we preset bouds o the Price of Proportioality ad the Price of Evy-Freeess. We begi i.1 by presetig the upper boud o the utilitaria Price of Evy-Freeess, ad complemet it by a example already give i Caragiais et al. [CKKK09], which is tight up to a small additive factor. Both these upper ad lower bouds apply also to the utilitaria Price of Proportioality. I. we show a simple upper boud of for the egalitaria Price of Evy-Freeess, together with a matchig (tight) lower boud. We also show that the egalitaria Price of Proportioality is trivially 1. I Sectio 3 we preset bouds o the Price of Equitability. I additio to the (metioed above) proof that the egalitaria price is 1, we provide a simple upper boud of o the utilitaria Price of Equitability, together with a lower boud of I Sectio 4 we cosider the reverse questio to that of the Price of Fairess amely, how much fairess may oe have to give up to achieve social optimality. Fially, we coclude this work ad preset some ope questios i Sectio Related Work The problem of fair divisio dates back to the aciet times, ad takes may forms. The piece of property to be divided may be divisible or idivisible: Divisible goods ca be cut ito pieces of ay size without destroyig their value (like a cake, a piece of lad, or a ivestmet accout), while idivisible goods must be give i whole to oe perso (e.g. a car, a house, or a atique vase). Sice such items caot be divided, the problem is usually to divide a set of such goods betwee a umber of players. Fair divisio may also relate to the allocatio of chores (of which every party likes to get as little as possible); this problem is of a somewhat differet flavor from goods allocatio, ad also has the divisible ad idivisible variats. Moder mathematical treatmet of fair divisio started at the 1940s [Ste49], ad was iitially cocered maily with fidig methods for allocatio of divisible goods. Differet algorithms both discrete ad cotiuous ( movig kife algorithms ) were preseted (e.g. [Str80, EP84] ad [BT95], which also surveys older algorithms), as well as o-costructive existece theorems [DS61, Str80]. I the past fiftee years, several books appeared o the subject [BT96, RW98, Mou04]. Followig the evaluatio ad cut queries model suggested by Robertso ad Webb [RW98], much attetio was give to the questio of lower bouds o the umber of steps or cuts required for such divisios i this ad other models [MIBK03, EP06, SW03, Str08, Pro09]. I particular, Stromquist [Str08] proves that o fiite protocol (eve ubouded) ca be devised for a evy-free divisio of a cake amog three or more people i which each player receives a coected piece. However, we ote that this result applies oly to the model preseted i that work (which resembles the oe suggested by Robertso ad Webb), ad ot for cases where, for example, some mediator has full iformatio of the players valuatio fuctios ad proposes a divisio based o this iformatio.

5 Ulike most of the work o cake cuttig, the differet otios of the price of fairess are ot cocered with procedures for obtaiig divisios, but rather with the existece of divisios with differet properties (relatig to social optimality ad fairess). These otios, amely the Price of Proportioality, the Price of Evy-Freeess ad the Price of Equitability, were first preseted i a recet paper by Caragiais et al. [CKKK09]. This lie of work has some resemblace to the lie of work o the Price of Stability [ADK + 04], which attracted much attetio i the past decade. The work i [CKKK09] aalyzes the price of fairess (via the above three measures) with the utilitaria welfare fuctio for divisible ad idivisible goods ad chores, givig tight bouds (up to a costat multiplicative factor) i most cases. However, ulike i this work, o special attetio was give to the case of coected pieces i divisible goods. The results of [CKKK09] for divisible goods are summarized i the last row of Table 1. The Price of Evy-Freeess ad Proportioality.1 Utilitaria Welfare Theorem 1. For every cake-cuttig istace with players, the utilitaria Price of Evy- Freeess with coected pieces is bouded from above by + 1 o(1). I fact, we prove a eve stroger claim: The above boud applies ot oly to the distace of the best evy-free divisio from utilitaria optimality, but also to the distace from (utilitaria) optimality of ay evy-free divisio. Proof. Let x be a evy-free divisio of the cake, ad u(x) = i [] u i(x) its utilitaria ( social welfare. We show that ay other divisio to coected pieces y has u(y) ) u(x). Our proof is based o the followig key observatio: Assume that for some i [], u i (y) α u i (x). Sice i values ay other piece i the divisio x at most as much as her ow, it has to be that i y, i gets a iterval that itersects pieces that beloged to at least α differet players (possibly icludig i herself). We will say that i the divisio y, player i gets the j-th cut of x if i y, i is give a piece startig at a poit p < x j ad edig at the poit p > x j. A more formal statemet of our observatio is therefore that if i y, i gets at most α cuts of x, it holds that u i (y) (α + 1) u i (x). We ca thus boud the ratio u(y) u(x) by the solutio to the followig optimizatio problem, which aims to fid values {u i (x)} i=1 ad {α i} i=1 (the umber of cuts of x each player gets) that maximize this ratio. i=1 maximize (α i + 1)u i (x) i=1 u (1) i(x) subject to α i = 1 i=1 u i (x) 1 1 i () (α i + 1)u i (x) 1 1 i (3) α i {0,..., 1} 1 i () is a ecessary coditio for the evy-freeess of x that provides a lower boud for the deomiator, ad (3) is equivalet to u i (y) 1.

6 We therefore cocetrate o boudig the solutio to the above optimizatio problem. To this ed, the followig observatios are useful: 1. For ay choice of values {u i (x)} i=1, the optimal assigmet for the α i variables is greedy, i.e. givig each player i, i o-icreasig order of u i (x) the maximum possible value for α i that does ot violate ay of the costraits. (This holds sice otherwise there are players i, j with u i (x) > u j (x) ad α j 1 such that icreasig α i by oe at the expese of α j is feasible ad yields a icrease of u i (x) u j (x) > 0 i the umerator of (1), without affectig the deomiator.) We thus ca divide the players ito two groups: Those with high u i (x) values, who receive strictly positive α i values, ad those with low u i (x) values, for which α i = 0.. Sice the players with low u i (x) values add the same amout to both the umerator ad the deomiator i the objective fuctio, maximum is obtaied whe these values are miimized; i.e. i the optimal solutio u i (x) = 1 for all these players. 3. The solutio to the problem above is clearly bouded from above by the solutio to the same problem where the α i variables eed ot have itegral values. Clearly, i the optimal solutio to such a problem, all the players with α i > 0 have (α i + 1)u i (x) = 1. We ca thus boud the solutio to our optimizatio problem by the solutio to the followig problem. Let K be a variable that deotes the umber of players that will have α i > 0; by observatio (3) above, for every such player, (α i + 1)u i (x) = 1, ad thus their total cotributio to the umerator is K. We therefore seek a solutio for: maximize subject to K + ( K) 1 K i=1 u i(x) + ( K) 1 K i=1 ( ) 1 u i (x) 1 (4) = 1 (5) K It ca be verified (e.g. usig Lagrage multipliers) that for ay value of K this is maximized whe u i (x) = u j (x) for all i, j [K], i.e. whe u i (x) = K K+1 for all i [K]. We thus coclude that the maximum solutio to the above problem maximizes the ratio K K + ( K) 1 K +K 1 + ( K) 1 by elemetary calculus this is maximized at K =, where the value is as stated. ( + )( + 1) + ( )( + 1) ; = ( + 1 ) + ( + ) 1 + = = + 1 o(1), Sice every evy-free divisio is i particular proportioal, we immediately get that the boud o the utilitaria Price of Evy-Freeess also applies to the Price of Proportioality: Corollary. For every cake-cuttig istace with players, the utilitaria Price of Proportioality i coected pieces is bouded from above by + 1 o(1).

7 We coclude by showig that these bouds are essetially tight (up to a small additive factor). The costructio we show is idetical to the oe i [CKKK09], ad we provide it here agai for completeess. Propositio 3. The utilitaria Price of Proportioality (ad thus also the utilitaria Price of Evy-Freeess) i coected pieces is larger tha. Proof. For some iteger m, cosider = m players with the followig valuatio fuctios. For i = 1,...,, player i assigs a value of 1 to the piece ( i 1 i, ) ad 0 to the rest of the cake (we call these players the focused players ). All other players (players i = ( + 1),...,, the idifferet players ) assig a uiform value to the etire cake. I ay proportioal divisio, the idifferet players must get a total of at least of the physical 1 cake, ad their total utility is less tha 1. This leaves the focused players with at most of the physical cake, ad so they obtai (together) a total utility of at most 1; the utilitaria value of a proportioal divisio is therefore less tha. O the other had, the divisio givig each of the focused players the etire iterval they desire (ad leavig othig to the idifferet players) has a utilitaria social welfare of. The Price of Proportioality for this case is therefore larger tha, as stated.. Egalitaria Welfare Propositio 4. For every cake-cuttig istace, the egalitaria Price of Proportioality is 1. Proof. Let x be a proportioal divisio, ad y the egalitaria optimal divisio. By proportioality, every player i has u i (x) 1, ad thus eg(x) 1. Sice y is the egalitaria optimal divisio, we have that for every i [], u i (y) eg(y) eg(x) 1 ; this implies that y is proportioal as well. Theorem 5. The egalitaria Price of Evy-Freeess for cake-cuttig istaces with players ad coected pieces is. I particular, this is also a upper boud o the egalitaria Price of Evy-Freeess for players ad o-coected pieces. Proof. First, ote that if the egalitaria optimal divisio is itself evy-free, the Price of Evy-Freeess is 1, ad that every divisio with egalitaria welfare of 1 is evy-free. We therefore assume that this is ot the case, ad that i the egalitaria optimal y divisio some player i has u i (y) < 1. Let x be some evy-free divisio, the x is i particular proportioal ad thus has u i (x) 1 ; the upper boud follows. It remais to show a lower boud for the coected case. Let ɛ > 0 be a arbitrarily small costat, ad cosider players with the followig valuatio fuctios. For i = 1,..., ( 1), player i assigs a value of 1 + ɛ to the piece (i ɛ, i + ɛ) (her favorite piece ), a value of 1 i+1 i+1 ɛ to the piece (1 ɛ, 1 + ɛ) (her secod-favorite piece ), ad value of 0 to the rest of the cake. Fially, player assigs a uiform value to the etire cake. I order for player to get utility of α, this player eeds to receive a α fractio of the cake (i physical size). However, every coected piece of physical size at least 1 + ɛ ecessarily cotais some other player s favorite piece, ad it is immediate that if a sigle player receives the etire favorite piece of aother player, there is evy. Thus, i every evy-free divisio of the cake, player gets utility of less tha 1 + ɛ. However, there exists a divisio i which every player gets utility of at least 1 ɛ. Such a divisio is achieved by givig players i = their favorite pieces, players i = ( + 1)... ( 1) their secod-favorite pieces, ad player the iterval ( 1 + ɛ, 1) (the remaiig parts of the cake ca be give to ay of the players closest to them). The stated boud follows as ɛ approaches zero.

8 3 The Price of Equitability I order to talk about the Price of Equitability, we first have to make sure that the cocept is well-defied. Whe o-coected pieces are cocered, it is kow that every cake cuttig istace has a equitable divisio [DS61]. However, the proof of Dubis ad Spaier allows a piece of the cake to be ay member of the σ-algebra of subsets, which is quite far from our restricted case of pieces that are all sigle itervals. Aother result by Alo [Alo87] establishes the existece of a equitable divisio givig every player exactly 1 by each measure; however, such a divisio may require up to 1 cuts. The questio thus arises whether equitable divisios with coected pieces always exist; to the best of our kowledge, this questio has ot bee addressed before, ad we aswer it here to the affirmative. Furthermore, we show that such a divisio requires o sacrifice of egalitaria welfare. Theorem 6. For every cake-cuttig istace there exists a equitable divisio of the cake with coected pieces. Furthermore, there always exists such a divisio i which the egalitaria social welfare is as high as possible i ay divisio with coected pieces. This holds eve for cake cuttig istaces that do ot have v i (0, 1) = 1 for all i (i.e. eve if some players valuatio of the etire cake is ot 1). Proof. Recall that the egalitaria welfare is a cotiuous fuctio ad X is compact, ad thus eg( ) has a maximum i X; we deote OP T = max x X eg(x). We also deote by Y X the set of divisios with egalitaria value OP T, i.e. Y = {y = (y 1,..., y 1, π) X } eg(y) = OP T. We ote that Y is a compact set; this follows from the fact that it is a closed subset of X (which is compact itself). To show that Y is closed, we show that Y = X \ Y is ope. Let z Y be some divisio ot i Y ; the the divisio z must have egalitaria value smaller tha OP T ad i particular there must exist a player i ad ɛ > 0 such that u i (z) OP T ɛ. Sice player i s valuatio of the cake is a oatomic measure, there must exist δ L, δ R > 0 such that extedig i s piece to the iterval (z π(i) 1 δ L, z π(i) + δ R ) icreases i s utility (compared to the origial divisio z) by less tha ɛ. Therefore, i the ball of radius δ = mi{δ L, δ R } aroud z (e.g. i L ), every divisio still gives i utility smaller tha OP T, ad thus this ball does ot itersect Y. It thus follows that Y is a ope set, ad so Y is closed ad compact. Recall that our aim is to show that Y cotais a equitable divisio; to that ed, we defie a fuctio : Y R by settig { (y) = max ui (y) u j (y) } { = max ui (y) OP T }. i,j [] i [] We complete the proof by showig that for ay ɛ, there exists a devisio y (ɛ) Y, such that (y (ɛ) ) ɛ. Sice Y is a compact set ad ( ) is cotiuous, the image of Y is also compact. We therefore coclude that there must be some y Y with (y ) = 0 (sice the image of Y is i particular a closed subset of R cotaiig a poit p < ɛ for every ɛ > 0); such y is clearly equitable. It remais to prove that for ay ɛ, y (ɛ) exists. We prove this by iductio o the umber of players. For = 1 there is oly oe possible divisio, which obtais exactly OP T for the sigle player. Assume for 1, we prove for. Let y be ay divisio i Y (assumig w.l.o.g. that y uses the idetity permutatio). We first costruct a divisio y such that for i = 1,..., 1, u i (y ) = OP T, by sequetially movig the border y i (betwee players i ad i + 1) to the left as far as possible while keepig that u i (y ) OP T. This is possible sice i y, u i (y) OP T ad the borders oly eed to move to the left. Cosider the resultig

9 y. If u (y ) OP T + ɛ we are fiished; otherwise, let y be the divisio obtaied from y by movig the border y 1 (betwee players 1 ad ) as far right as ecessary so that u (y ) = OP T + ɛ. Now, omit the rightmost piece (that of player ), ad cosider the ( 1)-player cake cuttig problem, o the remaiig cake. (Note that the players valuatio of the etire ew cake eed ot be idetical to their valuatio of the origial cake, ad that the ew cake has a differet set Y of egalitaria-optimal divisios.) Now, it caot be the case that for this ew problem the egalitaria maximum is more tha OP T, as that would iduce a egalitaria maximum greater tha OP T for the etire problem. O the other had, egalitaria value of OP T is clearly attaiable, as it is obtaied by y (reduced to the first 1 players). Hece, OP T is also the egalitaria maximum for the ew ( 1)-player problem. Thus, by the iductive hypothesis, there exists a divisio for this problem that obtais egalitaria welfare OP T ad such that o player gets more tha OP T +ɛ. Combiig this solutio with the piece (y 1, 1) give to player, we obtai y (ɛ) Y, such that o player gets more tha OP T + ɛ. Theorem 7. The utilitaria Price of Equitability i coected pieces is upper-bouded by, ad for ay there is a example i which it is arbitrarily close to Proof. We begi by showig a upper boud o the utilitaria Price of Equilibility. From Theorem 6 we have that there always exists a equitable egalitaria-optimal divisio with coected pieces. Sice there also always exists a proportioal divisio (whose egalitaria social welfare is at least 1 ), the egalitaria-optimal divisio must have a egalitaria social welfare of at least 1 ad thus a utilitaria social welfare of at least 1. Clearly, the maximum utilitaria social welfare attaiable i ay o-equitable divisio is less tha, ad thus the utilitaria Price of Equitability is also less tha. For the lower boud, fix some small ɛ > 0 ad cosider players with the followig valuatio fuctios. For i = 1,..., ( 1), player i assigs value of 1 to the iterval ( i ɛ, i + ɛ) ad 0 to the rest of the cake. Fially, player assigs uiform value to the etire cake. Sice ay coected piece of (physical) size 1 + ɛ ecessarily cotais the etire desired piece of at least oe player i [ 1], the utility of player i ay equitable divisio must be strictly smaller tha 1 + ɛ; the utilitaria welfare of such a divisio is therefore smaller tha 1+ɛ. Now, cosider the followig (o-equitable) divisio: give player 1 the iterval (0, 1 i 1 + ɛ), players i =,..., ( 1) the iterval ( + ɛ, i + ɛ), ad player the iterval ( 1 + ɛ, 1). The utilitaria welfare of this divisio is ɛ. By appropriately choosig ɛ, the Price of Equitability ca be arbitrarily close to Tradig Fairess for Efficiecy The work o the Price of Fairess is cocered with the trade-off betwee two goals of cake divisio: Fairess, ad efficiecy (i terms of social welfare). However, the results we preseted so far, as well as the results i [CKKK09], cocetrate o oe directio of this trade-off, amely how much efficiecy may have to be sacrificed to achieve fairess. We ow tur to look at the aalogue questio of how much fairess may have to be give up to achieve social optimality; sadly, it seems that at least for the coected-pieces case, the results are somewhat pessimistic, except for equitability ad proportioality with the egalitaria welfare. I order to aswer such questios, oe first has to quatify ufairess. The followig defiitios seem atural: We say that a divisio x: is α-uproportioal if some player i [] has u i (x) 1 α.

10 has evy of α if there exist players i, j [] for which v i (x π(j) 1, x π(j) ) α v i (x π(i) 1, x π(i) ) = α u i (x), i.e. if some i feels that j i received a piece worth α-times more tha the oe she got. is α-iequitable if there are players i, j [] with u i (x) α u j (x). Usig these ufairess otios, we ca obtai the followig simple results: Propositio 8. There are cake-cuttig istaces where a utilitaria-optimal divisio is ecessarily ifiitely ufair, by all three measures above. Proof. Cosider the cake cuttig istace from the proof of Propositio 3. I this istace, the uique utilitaria-optimal divisio gives o cake at all to the idifferet players ; it follows that this divisio is ifiitely uproportioal ad iequitable, ad has iifiite evy. We already kow (Propositio 4 ad Theorem 6) that egalitaria optimality is ot i coflict with either proportioality or equitability. However, this is ot the case for evy: Propositio 9. There are cake-cuttig istaces where a egalitaria-optimal divisio ecessarily has evy arbitrarily close to 1, ad this is the maximum possible evy for such divisios. Proof. Let ɛ > 0 be a arbitrarily small costat, ad cosider players with the followig valuatio fuctios, which are fairly similar to those i the proof of Theorem 5. For i = 1,..., ( 1), player i assigs a value of 1 1 ɛ to the piece (i ɛ, i + ɛ ) (her favorite piece ), a value of 1 i+1 +ɛ to the piece (1 ɛ i+1, 1 + ɛ ) (her secod-favorite piece ), ad value of 0 to the rest of the cake. Fially, player assigs uiform value to the etire cake. It is clear that there is o way for the egalitaria value to exceed 1 + ɛ: I order for that to happe, player must get a coected piece of physical size larger tha 1 + ɛ, which must cotai the etire favorite piece of some player i <, ad so player i ca get utility at most 1 + ɛ. However, egalitaria welfare of 1 + ɛ ca be easily achieved, ad i such case player ideed devours the etire favorite piece of some player i < ; this player receives a piece worth (i her eyes) oly 1 + ɛ while she values the piece receives as worth 1 1 ɛ. The evy i every egalitaria-optimal divisio is therefore 1 ɛ 1+ɛ, which ca be arbitrarily close to 1 with a appropriate choice of ɛ. Sice the egalitaria-optimal divisio is always proportioal, every player must get at least 1 of the cake i it; therefore, i this player s view, aother player may get at most. It thus follows that i every such divisio the maximum possible evy is Coclusios ad Ope Problems I this work we aalyzed the possible degradatio i social welfare due to fairess requiremets, whe requirig that each player obtai a sigle coected piece. We obtai that the results vary cosiderably, depedig o the fairess criteria used, ad the social welfare fuctio i cosideratio. The bouds rage from provably o degradatio for proportioality ad equitability uder the egalitaria welfare, through a O( ) degradatio for evy-freeess ad proportioality uder the utilitaria welfare, to a O() degradatio for equitability uder the utilitaria welfare ad for evy-freeess uder the egalitaria welfare. We have also see that if we seek to trade fairess to achieve social optimality, the exchage

11 rate may (at the worst case) be ifiite for utilitaria welfare (for all three fairess criteria), or liear for egalitaria welfare ad evy-freeess. May ope questios await further research, icludig: Small umber of coected pieces. Oe motivatio for cosiderig cake cuttig with coected pieces is the desire to avoid situatios where a player receives a pile of crumbs for his fair share of the cake. O the other had, requirig that each player receives a sigle coected iterval may be too strict a requiremet. A atural middle groud is to require that each player receives oly a small umber of pieces, e.g. a costat umber. The questio thus arises to boud the degradatio to the social welfare uder such requiremets. I such a aalysis it would be iterestig to see how the bouds o degradatio behave as a fuctio of the umber of permissible pieces. The Egalitaria Price of Fairess with o-coected pieces. [CKKK09] provide bouds o the Price of Fairess usig the utilitaria welfare fuctio, for the settig that o-coected pieces are permissible. Boudig the egalitaria Price of Fairess i this settig remais ope. A trivial upper boud o the Price of Evy-freeess is, ad we have examples of istaces where this price is strictly larger tha 1, but obtaiig tight bouds seems to require additioal work ad techiques. The egalitaria Price of Proportioality ad Price of Equitability for idivisible goods. [CKKK09] provide aalysis for the utilitaria Price of Fairess for such goods. A simple example ca be costructed to show a tight boud of for the egalitaria Price of Evy-Freeess for this case. It thus remais ope to determie the egalitaria Price of Proportioality ad Equitability for such goods. The Price of Fairess for coected chores. As we already metioed, fair divisio of chores has a somewhat differet flavor from divisio of goods, ad may require somewhat differet techiques. Oe possible motivatio for requirig coected divisio of chores may be, for example, a case i which a group of gardeers eed to maitai a large garde, ad so would like to give each of them oe (coected) area to be resposible for. Ackowledgemet. We thak Ariel Procaccia for providig helpful commets o a earlier draft of this work. Refereces [ADK + 04] Elliot Ashelevich, Airba Dasgupta, Jo M. Kleiberg, Éva Tardos, Tom Wexler, ad Tim Roughgarde. The price of stability for etwork desig with fair cost allocatio. I FOCS, pages , 004. [Alo87] Noga Alo. Splittig ecklaces. Advaces i Mathematics, 63(3):47 53, [BT95] [BT96] Steve J. Brams ad Ala D. Taylor. A evy-free cake divisio protocol. The America Mathematical Mothly, 10(1):9 18, Steve J. Brams ad Ala D. Taylor. Fair Divisio: From cake cuttig to dispute resolutio. Cambridge Uiversity Press, New York, NY, USA, [CKKK09] Ioais Caragiais, Christos Kaklamais, Paagiotis Kaellopoulos, ad Maria Kyropoulou. The efficiecy of fair divisio. I WINE, pages , 009.

12 [DS61] L. E. Dubis ad E. H. Spaier. How to cut a cake fairly. The America Mathematical Mothly, 68(1):1 17, Ja [EP84] [EP06] S. Eve ad A. Paz. A ote o cake cuttig. Discrete Applied Mathematics, 7(3):85 96, Jeff Edmods ad Kirk Pruhs. Cake cuttig really is ot a piece of cake. I SODA 06: Proceedigs of the seveteeth aual ACM-SIAM symposium o Discrete algorithm, pages 71 78, New York, NY, USA, 006. ACM. Cake- [MIBK03] Malik Magdo-Ismail, Costas Busch, ad Mukkai S. Krishamoorthy. cuttig is ot a piece of cake. I STACS, pages , 003. [Mou04] Hervé J. Mouli. Fair Divisio ad Collective Welfare. Number i MIT Press Books. The MIT Press, 004. [Pro09] Ariel D. Procaccia. Thou shalt covet thy eighbor s cake. I IJCAI, pages 39 44, 009. [RW98] [Ste49] Jack Robertso ad William Webb. Cake-cuttig algorithms: Be fair if you ca. A K Peters, Ltd., Natick, MA, USA, H. Steihaus. Sur la divisio pragmatique. Ecoometrica, 17(Supplemet: Report of the Washigto Meetig): , Jul [Str80] Walter Stromquist. How to cut a cake fairly. The America Mathematical Mothly, 87(8): , [Str08] [SW03] Walter Stromquist. Evy-free cake divisios caot be foud by fiite protocols. Electroic Joural of Combiatorics, 15(1), Ja 008. Jiri Sgall ad Gerhard J. Woegiger. A lower boud for cake cuttig. I ESA, pages , Cotact Details Yoata Auma Departmet of Computer Sciece Bar-Ila Uiversity Ramat Ga, 5900, Israel auma@cs.biu.ac.il Yair Dombb Departmet of Computer Sciece Bar-Ila Uiversity Ramat Ga, 5900, Israel yair.biu@gmail.com