Proeedings of the Thirty-First AAAI onferene on Artifiil Intelligene (AAAI-17) Envy-Free Mehnisms with Minimum Numer of uts Rez Alini, Mid Frhdi, Mohmmd Ghodsi, Msoud Seddighin, Ahmd S. Tik Shrif University of Tehnology, Duke University, University of Mihign Ann Aror Institute for Reserh in Fundmentl Sienes (IPM) Shool of omputer Siene lini@s.duke.edu, {m frhdi, mseddighin}@e.shrif.edu, ghodsi@shrif.edu, tik@umih.edu Astrt We study the prolem of fir division of heterogeneous resoure mong strtegi plyers. Given divisile heterogeneous ke, we wish to divide the ke mong n plyers in wy tht meets the following riteri: (I) every plyer (wekly) prefers his lloted ke to ny other plyer s shre (suh notion is known s envy-freeness), (II) the mehnism is strtegy-proof (truthful), nd (III) the numer of uts mde on the ke is miniml. We provide methods, nmely expnsion proess nd expnsion proess with unloking, for dividing the ke under different ssumptions on the vlution funtions of the plyers. 1 Introdution The prolem of dividing ke mong set of individuls hs een widely studied in the pst 60 yers. The suet ws first defined y Steinhus (1948). The desription of the prolem is s follows: given heterogeneous ke nd set of plyers, with potentilly different tendenies to different prts of the ke, how to ut the ke nd distriute it mong the plyers in fir mnner? Severl notions re defined for mesuring the firness of n llotion (see (Proi 2014) for detils). One of the most importnt notions is envy-freeness. An llotion of the ke is envy-free if eh plyer (wekly) prefers its lloted shre to ny other plyer s shre. Envy-free resoure llotion hs een vstly studied in the literture. For two plyers, the fmous method of ut nd hoose gurntees envy-freeness of the llotion. For three plyers, Selfridge nd onwy designed protool for finding n envy-free division of the ke. In their method, plyer my reeive more thn one piee (see (Proi 2013) for detils). Brms nd Tylor generlized this method to n ritrry numer of plyers (1995). However, their method doesn t gurntee ny upper ound on the numer of uts. Reently, in (2016), Aziz nd Mkenzie suggested ounded envy-free protool for ny numer of plyers. In some settings, the numer of uts is lso importnt. In severl ppers (e.g. (Stromquist 1980), (Brnel nd Brms 2004), (Stromquist 2007), (Bei et l. 2012)) the ke utting with minimum numer of uts hs een studied. Eh opyright 2017, Assoition for the Advnement of Artifiil Intelligene (www.i.org). All rights reserved. ut might hve n dditionl ost. As n exmple, suppose the ke models proessing time tht must e firly lloted mong set of tsks. Every tsk-swith imposes n overhed; minimizing totl mount of overhed would e equivlent to minimizing the numer of uts on the ke. In ddition, plyers my not hve ny vlue for very smll piees mde y lrge numer of uts. In (rginnis, Li, nd Proi 2011), this issue ws illustrted y the dvertisement exmple: think of the ke s time nd onsider the llotion of dvertising time. In suh setting, lrge numer of uts n yield so smll periods of time tht re not useful for dvertising. In n llotion with smll numer of uts this prolem is unlikely. Stromquist, in (1980), proved the existene of n envyfree division of the ke mong n plyers with n 1 uts whih is the minimum numer of uts required to divide ke mong n plyers. However, the proof is not onstrutive nd does not yield polynomil time lgorithm. In (2007), he showed tht no finite protool n find n envyfree llotion with minimum numer of uts for n 3. (Deng, Qi, nd Seri 2012) proved tht the prolem of finding n envy-free llotion of the ke, with minimum numer of uts is PPAD-omplete. They lso proposed n FPTAS for the se of three plyers. In numer of the reent ppers (e.g. (rginnis, Li, nd Proi 2011), (Brms et l. 2012), (Bei et l. 2012), (My nd Nisn 2012), (hen et l. 2013), (Aziz nd Ye 2014)) some restrited lsses of vlution funtions hve een studied. Pieewise onstnt nd pieewise uniform vlution funtions re two importnt speil lsses of vlution funtions whih re very importnt in prtie. One of the importnt properties of these vlution funtions is tht they n e desried onisely. (Kurokw, Li, nd Proi 2013) proved tht finding n envy-free llotion (in Roertson-We model) when the vlution funtions re pieewise-uniform is s hrd s solving the prolem without ny restrition on the vlution funtions. Reently, some studies onsidered the prolem from gme theoreti viewpoint. Mny ke utting lgorithms re not truthful. For exmple, even the ut nd hoose method whih is reltively simple does not gurntee truthfulness. In (Brânzei et l. 2016), the strtegi outome of the ke utting lgorithms hs een studied. They proved the existene of n pproximte sugme perfet Nsh equilirium for 312
lss of protools. Another line of reserh whih is more relted to our work, ttempts to find truthful mehnisms. Similr to firness, there re different notions for the onept of truthfulness. In (Brms et l. 2006), wek notion of truthfulness is defined: plyers don t risk telling lie, if there exists senrio (for other plyers vlutions) in whih lying results in lower pyoff. As n exmple, they showed tht ut nd hoose protool is wekly truthful. My nd Nisn (2012) designed truthful nd Preto-effiient mehnisms to divide the ke etween two plyers where eh plyer is interested in suset of the ke, uniformly. In (2013), hen et l. onsidered strong notion of truthfulness (denoted y strtegy-proofness), in whih the plyers dominnt strtegies re to revel their true vlutions over the ke. They presented strtegy-proof mehnism for the se when the vlution funtions re pieewise uniform. They lso designed rndomized lgorithm tht is envy-free nd truthful in expettion, for pieewise liner vlution funtions. However, their method for dividing the ke uses Ω(n 2 m) uts, where m is the numer of piees in eh vlution funtion. Aziz nd Ye (2014) onsidered the prolem when vlution funtions re pieewise onstnt/uniform. Bsed on prmetri network flows, they designed n envy-free lgorithm tht is group strtegy-proof 1 for pieewise uniform vlutions. It is notle tht their method eomes equivlent to mehnism 1 from (hen et l. 2013), in the se of pieewise uniform vlutions. 1.1 Our work We investigte the prolem of finding envy-free nd truthful mehnisms with smll numer of uts. By smll, we men tht the numer of uts does not exeed O(nm), where m is the numer of steps of eh plyer s (pieewise onstnt) vlution funtion. To the est of our knowledge, this is the first study tht ims to pproximte the numer of uts. The sis of our method is simple nd elegnt proess lled expnsion proess. After desriing the proess, we strt with the se, where eh plyer s vlution funtion is pieewise onstnt with only one step nd preserves speifi property tht we nme ordering property. For this se, we propose EFISM whih is polynomil time, strtegyproof nd envy-free llotion with n 1 uts (Theorem 2). Next, we remove the ordering ssumption nd show tht generlized form of the expnsion proess n find n envyfree llotion tht uts the ke into t most 2n 1 piees in polynomil time (Theorem 3). Furthermore, using more omplex form of this proess, we propose EFGISM, whih is polynomil time lgorithm tht is truthful, envy-free nd uts the ke into t most 2n 1 piees (Theorem 4). In ddition, we onsider the se where the vlution funtions re pieewise onstnt with m piees. When the numer of plyers is onstnt, we provide poly(m) time lgorithm for envy-free division of the ke with n 1 uts. Finlly, we onsider the se tht the plyers possess prtiulr property, nmely intersetion property nd show tht 1 Group strtegy-proof mens no group of plyers n misreport their vlutions, suh tht in the resulting llotion ll of them ern more pyoff under this ssumption, modifition of the expnsion proess yields poly(m, n) time, envy-free lgorithm tht uts the ke in O(nm) lotions. 2 Model Desription nd Preliminries In this pper, we use the term intervl for two purposes: vlution funtions nd the shres lloted to the plyers. For revity, denote the former type of intervls y I nd the ltter y I. Also, we Suppose tht for every vlution intervl I i, I i =[α i,β i ] nd for every shre intervl I i, I i =[ i, i ]. Given set N of n plyers nd ke. We represent the ke y the intervl [0, 1]. For every plyer p i N, vlution funtion ν i :[0, 1] R is given. For eh p i Nnd intervl I =[, ], we define V i (I) s ν i(x)dx. Our ssumption is tht the vlues of the plyers re normlized, suh tht V i () =1, for eh plyer p i. A piee of the ke, is set of mutully disoint su-intervls of [0, 1]. For piee P, we define V i (P ) s I P V i(i). A vlution funtion ν is pieewise onstnt, if there exists set S ν = {I ν1, I ν2,...,i νk } of mutully disoint intervls, suh tht for ny two points x, x in I νi, ν(x) = ν(x )=r i nd for ny point x tht does not elong to ny intervl in S v, ν(x) =0. To put it simply, pieewise onstnt vlution onsists of finite numer of intervls, suh tht ll the points in the sme intervl hve the sme vlue, nd for the points tht do not elong to ny intervl, the vlution is 0. Wesyν hs k steps, if S ν = k. A division of the ke mong set N of n plyers is set D = {P 1,P 2,...,P n } of piees, with eh piee P i = {I i,1,i i,2,...,i i, Pi } eing set of intervls with the following two properties: (I) every pir of intervls re mu- tully disoint nd (II) no piee of the ke is left ehind: i, I i, =. The numer of uts in division D is ( i P i ) 1. A division D = {P 1,P 2,...,P n } is envy-free, if for every plyer p i nd piee P D the inequlity V i (P i ) V i (P ) holds. The mority of this pper is foused on the se, where eh vlution funtion is single intervl. For this se, we suppose tht for every plyer p i N, S vi = {I i }, where I i =[α i,β i ]. Furthermore, denote y T the set of vlution intervls, i.e., T = {I 1, I 2,...,I n }. In this setting, the envy-free notion for division D n e interpreted s follows: for eh plyer p i nd k i we hve I i, I i I k, I i. For set of intervls X, we define DOM(X) s the miniml intervl tht inludes ll memers of X s su-intervls; e.g., in the se tht eh vlution funtion is single intervl, for set T T we hve: DOM(T )=[min I T α, mx I i T β i]. Furthermore, we define the density of X, denoted y Φ(X) s: λ(x)/ X where λ(x) is the totl length of DOM(X) tht is overed y t lest one intervl in X. We ll set X of intervls solid, if for every point x DOM(X), there 313
d T = {,,, d} DOM(T ) Φ(T ) = DOM(T ) /4 Figure 1: Domin nd density exists n intervl I in X suh tht x I. For exmple, in Fig 1, the set T is solid. When T is solid, we hve: λ(t )= DOM(T ) =mx I i T β i min I T α Our ssumption is tht every piee of the ke is vlule for t lest one plyer. In the Appendix 2, we show tht slightly modified versions of our lgorithms n hndle the ses where this ssumption does not hold. 3 The Expnsion proess The min tool in our method for dividing the ke is proedure lled expnsion proess. The expnsion proess expnds some ssoited intervls to the plyers, inside their desired re. We use exp(t ) to refer to the expnsion proess on the set T of vlution intervls. We initite the expnsion proess for T y ssoiting zero length intervl I i t the eginning of its orresponding I i T, i.e., I i =[ i = α i, i = α i ]. Denote y S, the set of these Intevls. We expnd the intervls in S onurrently, ll from the endpoint. The expnsion is performed in wy tht preserves two invrints:(i) The expnsion hs the sme speed for ll the intervls so s the lengths of the intervls remins the sme nd (II) I i lwys remins within I i. During the expnsion, the endpoint of n intervl I i my ollide with the strting point of nother intervl I. In this se, I i pushes the strting point of I forwrd during the expnsion. The push ontinues to the end of the proess. If I i pushes I,wesyI i is stuk in I. Note tht y the wy we initite the proess, the intervls remin sorted ording to their orresponding α i s. Also in the speil se of equl α i for two plyers, the one with smller β i omes first. Definition 1. During the expnsion, n intervl I i eomes loked, if the endpoint of I i rehes β i. Definition 2. A hin is sequene of intervls I σ1,i σ2,...,i σk, with the property tht for 1 i<k, I σi is stuk in I σi+1. A hin is loked, if I σk is loked. The size of hin is the numer of intervls in tht hin. By definition, single intervl is hin of size 1. The expnsion ends when n intervl eomes loked. The termintion ondition ensures tht the seond invrint is lwys preserved. In the Appendix, you n see detiled exmple of the expnsion proess. 2 The long version with ppendix is ville t www.s.duke.edu/ lini/emn-aaai2017.pdf Definition 3. The expnsion proess for T is perfet, if the ssoited intervls over the entire DOM(T ). If the proess termintes due to loked intervl efore entirely overing DOM(T ), the proess is imperfet. Note tht if n expnsion proess on T ends perfetly, then for every ssoited intervl I i,wehve I i =Φ(T ). Despite the ft tht we desried the expnsion proess ontinuously, it n e effiiently implemented vi swiping of the events (see the Appendix for more detils). Oservtion 1. During the expnsion proess, every intervl I i is either eing pushed y nother intervl, or its strting point is still on α i. 4 EFISM: Speil Intervl Sheduling In this setion, we ssume tht the vlution funtion of eh plyer is single intervl. In ddition, we suppose tht the intervls hve the following property: i, α i α β i β (1) In other words, no intervl is su-intervl of nother (unless they strt or end in the sme ple). For this se, we present polynomil time, envy-free, nd truthful llotion with n 1 uts. We nme this lgorithm s EFISM. The ide in EFISM is repetedly expnding the intervls nd removing the loked hins. Let T e the vlution intervls orresponding to the plyers in N. We egin y lling exp(t ). As desried in Setion 3, the proedure termintes either perfetly or imperfetly. In the first se we re done. Otherwise, t lest one hin is loked. Let = I σ1,i σ2,...,i σk e loked hin in S with mximl size. Sine is mximl, no intervl gets stuk in I σ1.by Oservtion 1, σ1 is extly on α σ1. Let T e the set of vlution intervls orresponding to the intervls in. Lemm 1. DOM(T )=[α σ1,β σk ]. Now, we llote eh I σi to p σi. Lemm 2 sttes tht suh n llotion is envy-free for p σ1,p σ2,...,p σk. Lemm 2. For every intervl I σi nd I σ in, we hve V σi (I σi ) V σi (I σ ). Next, we remove plyers p σ1,p σ2,...,p σk from N.We lso remove DOM(T ) from. By removing DOM(T ), the ke is divided into two su-kes: the piee to the right of DOM(T ) nd the piee to the left of DOM(T ), respetively l nd r. Let N l (N r ) e the set of plyers with their shre inside l ( r ). Also, let T l nd T r e the sets of vlution intervls orresponding to N l nd N r. Now, we updte the vlution funtions of the plyers in l nd r. Speifilly, for every plyer p i N l with β i > α σ1 we hnge the vlue of β i to α σ1. Similrly, for every plyer p N r with α <β σk we hnge α to β σk. After removing the lloted piee long with its orresponding plyers nd updting the vlutions, we perform this expnsion nd removl independently for oth T l nd T r. The proess ontinues until ll the plyers re removed. In Algorithm 1, you n find psudoode for EFISM. In ddition, you n find detiled exmple in the Appendix. 314
Algorithm 1 EFISM lgorithm funtion EFISM(, N, T ) orresponds to the intervl [, ] if then exp(t ) Expnsion proess on T = I σ1,i σ2,...,i σk : mximl loked hin for 1 i k do Allote I σi to p σi l =[, α σ1 ] r =[β σk,] for every p k Ndo if k < σ1 then β k =min(β k,α σ1 ) Add p k, I k to N l, T l respetively else if k > σk then α k = mx(α k,β σk ) Add p k, I k to N r, T r respetively EFISM( l, N l, T l ) EFISM( r, N r, T r ) Theorem 2. EFISM is envy-free, truthful nd uts the ke in extly n 1 lotions. Remrk tht removing the ordering property desried in the eginning of this setion my result in n inpproprite llotion. For exmple, onsider the input desried in Figure 2. lerly, running EFISM on this input does not yield n envy-free llotion; here p envies p. In ddition, the llotion does not llote the entire ke, euse piee etween I nd I is left over. I I I Figure 2: EFISM for intervls without ordering property 5 Expnsion Proess with Unloking In this setion, we introdue more generl form of the expnsion proess. The ide is the ft tht during the expnsion proess, there might e some ses tht loked hin n eome unloked y re-permuting some of its intervls. Definition 4. Let = I σ1,i σ2,...,i σk e mximl loked hin. A permuttion I δ1,i δ2,...,i δr of the intervls in is sid to e -unloking, if the following onditions re held: (I) i,i δi nd δ r = σ k,(ii) For ll 1 r 1, δ α δ+1 nd δ <β δ+1,(iii) α δ1 δr nd β δ1 > δr. The intuition ehind the definition of unloking permuttion is s follows: let I δ1,i δ2,...,i δr e -unloking permuttion, where = I σ1,i σ2,...,i σk. Then, we n hnge the order of intervls in y pling I δ in the lotion of I δ 1 for 1 < r nd pling I δ1 in the lotion of I δr. By the definition of unloking permuttion, fter suh opertions I δr (I σk ) is no longer loked. Thus, I σk is not rrier for the expnsion nd the proess n e ontinued. I I I Id Ie d e Figure 3: Exmple of Permuttion Grph. Here the loked hin I,I,I,I d,i e n e unloked y permuttion ( I I I I d I e I I e I I I d ) Definition 5. A loked hin = I σ1,i σ2,...,i σk is strongly loked, if dmits no unloking permuttion tht ontins I σk. Definition 6. The expnsion proess with unloking U- exp(.) is strongly loked, if t lest one of its hins is strongly loked. For set T of vlution intervls, we use U-exp(T ) to refer to the expnsion proess with unloking. The expnsion proess with unloking is in ft, the sme s expnsion proess with the exeption tht when the proess is fed with loked hin, it tries to unlok the hin y n unloking permuttion. If the hin eomes unloked, the expnsion goes on. The proess runs until either the entire DOM(T ) is lloted (perfet) or strongly loked hin ours (imperfet). In the Appendix you n find detiled exmple. It is worth mentioning tht there my e multiple loked intervls in moment. In suh situtions, we seprtely try to unlok eh intervl. Definition 7. A permuttion grph for loked hin is direted grph G V,E. For every intervl I σi, there is vertex v σi in V. The edges in E re in two types E l nd E r, i.e., E = E l E r. The edges in E l nd E r re determined s follows: (I) For eh I σi nd I σ, the edge (v σi,v σ ) is in E l,ifi>nd α σi σ.(ii) For eh I σi nd I σ, the edge (v σi,v σ ) is in E r,ifi<nd β σi > σ. See Figure 3 for n exmple of permuttion grph. A trivilly neessry nd suffiient ondition for hnin to e strongly loked is tht G ontins no yle ontining v σk. However, regrding the speil struture of G, we n define stronger neessry nd suffiient ondition for strongly loked sitution. Definition 8. A direted yle in G is one-wy, ifit ontins extly one edge from E r. Note tht no yle in G n ontin only the edges from one of E l or E r. In Lemm 3, we use one-wy yles to give neessry nd suffiient ondition for hin to e strongly loked. Lemm 3. A hin = I σ1,i σ2,...,i σk is strongly loked, iff G dmits no one-wy yle tht ontins v σk. 315
6 EFGISM: Generl Intervl Sheduling In this setion, we ssume tht the vlution funtion for eh plyer is n intervl, without ny restrition on the strting nd ending points of the intervls. For this se, we propose n envy-free nd truthful llotion tht uses less thn 2n uts. Our lgorithm for finding proper llotion is sed on the expnsion proess with unloking. Generlly speking, we itertively run U-exp(.) proess on the remining plyers shres. This proess llotes the entire ke, or stops in n strongly loked sitution. We prove some desirle properties for this sitution nd leverge those properties to llote piee of the ke to the plyers in the loked hin. Next, we remove the stisfied plyers nd shrink the lloted piee (s defined in Definition 9) nd solve the prolem reursively for remining plyers nd the remining prt of the ke. Definition 9 (shrink). Let e ke nd I =[I s,i e ] e n intervl. By the term shrinking I, we men removing I from nd glueing the piees to the left nd right of I together. More formlly, every vlution intervl [α i,β i ] turns into [f(α i ),f(β i )], where x x < I s f(x) = I s I s x I e x I e + I s I e <x (see Figure 4). As wrm-up, we ignore the truthfulness property nd show tht the expnsion proess with unloking yields n envy-free llotion with 2(n 1) uts. e e d d x Figure 4: The ke nd the intervls,,, d nd e efore nd fter shrinking intervl x 6.1 Envy-free llotion with 2(n 1) uts For this se, our lgorithm is s follows: In the eginning, we run U-exp(T ). The proess either ends perfetly nd the llotion is found, or strongly loked hin ppers. By the definition of strongly loked, we know tht there exists loked hin with no unloking permuttion. Let = I σ1,i σ2,...,i σk e mximl strongly loked hin. Now, onsider G. By Lemm 3, G ontins no onewy yle. Let l e the minimum index, suh tht there is direted pth from v σk to v σl using the edges in E l. Lemm 4. There is direted pth from v σk to every vertex v σl with l >l, using edges in E l. I I I I(ont) rel shre I Figure 5: n inrese his shre y misreporting Bsed on Lemm 4 nd the ft tht G ontins no onewy yle, there is no edge from v σl to v σk in E r for ny l l, whih mens: l l β σl σk (2) On the other hnd, there is no pth from v σk to v σl for l <l, tht is: l l α σl > σl 1 (3) We now llote every intervl I σl to p σl for l l k, remove {p σl,p σl+1,...,p σk } from N, nd shrink the intervl [ σl, σk ]. Next, we ontinue the expnsion proess with the remining plyers nd ke. The itertion etween expnsion proess with unloking nd lloting the ke in the strongly loked sitution goes on, until the entire ke is lloted. Theorem 3. The lgorithm desried ove is envy-free, nd uts the ke in t most 2(n 1) lotions. 6.2 EFGISM Method It is worth mentioning tht the llotion desried in setion 6.1 is not truthful. onsider the exmple in Figure 5. It n e oserved tht plyer n inrese his shre y misreporting α. In this setion, we try to resolve this issue. Our strtegy to del with this diffiulty is to run U-exp(.) only for speil suset of plyers in every step. Lemm 5 plys the key role in our method. Lemm 5. Let T e set of intervls, with the property tht for every T T, Φ(T ) > Φ(T ) (we ll suh set s irreduile). Then we n divide DOM(T ) into t most 2 T 1 piees nd ssoite them to the intervls, suh tht:(i) the totl length of the piees ssoited with ny intervl is extly Φ(T ), (II) the piees lloted to ny intervl is totlly within the intervl. Proof. We use indution on T. For T =1the lim trivilly holds: we n ssoite DOM(T ) to the intervl in T tht needs no ut. Suppose tht the proposition is true for T <k. We prove it for T = k. onsider U-exp(T ). If U-exp(T ) ends perfetly, then we re done. Otherwise, let = I σ1,i σ2,...,i σk e mximl strongly loked hin fter the proess. onsidering G, let l e the minimum index, suh tht there is direted pth from v σk to v σl. Lemm 6. l>1. 316
By Lemm 4, we know tht equtions (2) nd (3) re held for the hin.now,let x = β σk (k l +1) Φ(T ). (4) Lemm 7. σl 1 <x< σl. We show tht the piee [x, β σk ] n e lloted to plyers p σl,p σl+1,...,p σk using 2(k l +1) 2 uts. For this, onsider the vlution intervls T = I σ l, I σ l+1,...,i σ k suh tht: l i k I σ i =(mx(x, α σi ),β σi ) Note tht DOM(T )=[x, β σk ] nd hene, Φ(T )= β σ k x k l +1 = σ k x k l +1 Regrding Eqution (4), Φ(T )=Φ(T ). Lemm 8. For ll T T, we hve Φ(T ) > Φ(T ). Lemm 8 shows tht the set of intervls in T dmit the properties desried in Lemm 5. Furthermore, regrding Lemm 6, T is proper suset of T. By indution hypothesis, we know tht we n ut DOM(T ) into t most 2(k l +1) 2 piees nd ssign them to plyers p σl,p σl+1,...,p σk suh tht oth of the properties in Lemm 5 re stisfied. Denote y N T the plyers with vlutions in T. We shrink DOM(T ) nd remove the plyers p σl,p σl+1,...,p σk from N T. Lemm 9 ssures tht the onditions in Lemm 5 re held for the remining ke nd remining plyers. Lemm 9. Let T e the intervls relted to the plyers in N T = N T \{p σl,p σl+1,...,p σk } fter shrinking DOM(T ). Then, T is irreduile with Φ(T )=Φ(T ). Aording to Lemm 9, we n use indution hypothesis to show tht the set T n e lloted to the plyers in N T with 2(l 1) 2 uts. The totl numer of uts would e 2(l 1) 2+2(k l +1) 2=2k 4 uts plus two uts on x nd β σk tht results in 2k 2 uts. Bsed on lemm 5, we introdue the lgorithm EFGISM s follows: mong ll susets of N, we find suset suh tht their orresponding intervls hs the minimum density (nd the set with minimum size, if there were multiple options). Let N e this suset nd let T e the intervls orresponding to the plyers in N. In Lemm 10, we show tht T (nd onsequently N) n e found in polynomil time. Lemm 10. T n e found in polynomil time. Sine T hs the minimum possile density, T is irreduile. Hene, we n llote to every plyer in N, piee from DOM(T ) with the properties defined in Lemm 5. Next, we remove the plyers in N from N nd shrink DOM(T ) from. Now, we reursively ssign the remining piee of the ke to remining plyers using EFGISM. In Algorithm 2 you n find psudoode for EFGISM. (5) Algorithm 2 EFGISM lgorithm funtion EFGISM(N, T, ) if then T =rgmin T T Φ(T ) N = plyers with intervl in T Allote(N,DOM(T )) By Lemm 5 Shrink(, DOM(T )) T is lso updted EFGISM(N \N,T, ) Theorem 4. EFGISM is envy-free nd truthful nd uses t most 2(n 1) uts. We redit the proof for truthfulness of EFGISM to (hen et l. 2013). 7 Pieewise onstnt funtions In this setion, we onsider more generl se in whih the vlution funtions of the plyers re pieewise onstnt. Denote y m the mximum numer of intervls tht every vlution funtion n hve, tht is, for every plyer p i, S i m. Here, we ssume tht for every p i, S i = m. This is without loss of generlity, sine we n rek n intervl into severl su-intervls. Thus, for every plyer p i, we suppose S i = {I i,1, I i,2,...,i i,m }. This setion onsists of two prts. In the first prt, we show tht for onstnt numer of plyers, one n find the envy-free llotion with n 1 uts in time poly(m). Next, in the seond prt, we utilize the expnsion proess with unloking to devise poly(n, m) time, envy-free lgorithm with O(nm) uts on the ke. Rell tht finding n envy-free llotion with n 1 uts for n plyers is PPAD omplete even for the se of n =3 (Deng, Qi, nd Seri 2012). In Theorem 5, we show tht for onstnt numer of plyers with pieewise onstnt vlution, the prolem n e solved in time poly(m). Theorem 5. An envy-free llotion with n 1 uts n e found for onstnt numer of plyers whose vlution funtions re pieewise onstnt with m steps in time poly(m). Proof. Firstly, note tht from ((Stromquist 1980)) we know there exists n envy-free llotion with n 1 uts. In suh n llotion there re n 1 utting points. Let 0 1 2 n 1 1 e those utting points nd 0 =0, n =1e the strt nd end of the ke. In ddition, for eh plyer, their vlution funtion n e desried y 2m onstnt points (2 onstnt points for eh step) nd m onstnt vlues whih re the density vlue of eh step. Therefore, there re t most 2mn onstnt points on the ke in wy tht eh plyer likes the ke etween two onseutive onstnt points uniformly. In other words, the density vlue of the ke etween two onseutive onstnt points is onstnt vlue, for eh of the plyers. Now, if we know the rnge of eh utting point (it n e etween whih of the two onseutive onstnt points) then we n write the vlue of the ith piee reted y utting points ([ i 1, i ]) for eh plyer s liner funtion of the utting points. However, in order to stisfy the envy-freeness 317
we lso need to know how the piees will e lloted to the plyers. If we know ll of these informtions then we n formulte the prolem s liner progrm (n(n 1) onstrints for envy-freeness, n 1 onstrints gurntees 0 1 2 n 1 1, nd other onstrints fix the rnge of the utting points). Any fesile solution of the liner progrm is n envy-free llotion with n 1 uts. If we n t find fesile solution for one liner progrm then we need to hek the next possiility of the rnge of the utting points nd llotion of the piees. In the worst se, we need to hek every possiility whih mens tht we = O(m n ) different liner progrms. Finlly, we know tht suh n llotion exists nd one of the liner progrms finds fesile solution. Hene, for onstnt n, y solving polynomil numer of different liner progrms, we n find n envy-free llotion. need to solve n (2mn+n 1)! (2mn)! In the seond prt, we exploit expnsion method with unloking to find proper llotion. Here, we ssume tht the vlution funtions hve speil property, nmely, intersetion property. Denote y R i,,k the set of intervls in S k tht hve non-empty intersetion with I i,. We suppose tht for every vlution intervl I i, nd every plyer p k (k i), R i,,k =1. For this se, we prove Theorem 6. Theorem 6. Let N e set of plyers whose vlution funtions re pieewise onstnt with m steps. Assuming tht the intersetion property holds, there exists poly(m, n) time llotion lgorithm tht is envy-free nd uts the ke in O(nm) lotions. Proof. onsider n instne of the prolem with nm plyers, where the vlution funtion of plyer p i, is I i,.now, we exeute EFGISM for this instne. By the properties of EFGISM, we know tht the resulting llotion is envy-free nd uts the ke in t-most 2(nm 1) ples. Let P i, e the set of intervls lloted to p i, in EFGISM. We show tht the llotion tht llotes P i = 1 m P i, to plyer p i is lso envy-free. To prove envy-freeness, we use n struturl property of the expnsion proess: y the first invrint of the expnsion proess, the finl llotion would llote to every plyer p i, set of piees tht re totlly within I i,.in ddition, note tht for intervl I i,, R i,,k = 1 for every plyer p k.wehvev i (P i ) = 1 m V i(p i, ) nd V i (P k )= 1 m V i(p k, ). Furthermore, y intersetion property, t most one vlution intervl of p k, sy I k,l hs non-empty intersetion with I i,. By the envy-freeness of EFGISM, we know tht p i, prefers his shre to the shre lloted to p k,l, Tht is V i, (P i, ) V i, (P k,l ). Regrding the ft tht I i, I k,l = for ll l l, wehve V i, (P i, ) l V i,(p k,l ). Thus, V i, (P i, ) V i, (P k,l ) l V i (P i ) V i, (P k,l ). The right hnd side of ove eqution is t lest V i (P k ). l Referenes Aziz, H., nd Mkenzie, S. 2016. A disrete nd ounded envy-free ke utting protool for four gents. In Proeedings of the 48th Annul AM SIGAT Symposium on Theory of omputing, 454 464. AM. Aziz, H., nd Ye,. 2014. ke utting lgorithms for pieewise onstnt nd pieewise uniform vlutions. In Interntionl onferene on We nd Internet Eonomis, 1 14. Springer. Brnel, J. B., nd Brms, S. J. 2004. ke division with miniml uts: envy-free proedures for three persons, four persons, nd eyond. Mthemtil Soil Sienes 48(3):251 269. Bei, X.; hen, N.; Hu, X.; To, B.; nd Yng, E. 2012. Optiml proportionl ke utting with onneted piees. In AAAI. Brms, S. J., nd Tylor, A. D. 1995. An envy-free ke division protool. Amerin Mthemtil Monthly 9 18. Brms, S. J.; Jones, M. A.; Klmler,.; et l. 2006. Better wys to ut ke. Noties of the AMS 53(11):1314 1321. Brms, S. J.; Feldmn, M.; Li, J. K.; Morgenstern, J.; nd Proi, A. D. 2012. On mxsum fir ke divisions. In AAAI. Brânzei, S.; rginnis, I.; Kurokw, D.; nd Proi, A. D. 2016. An lgorithmi frmework for strtegi fir division. In Thirtieth AAAI onferene on Artifiil Intelligene. rginnis, I.; Li, J. K.; nd Proi, A. D. 2011. Towrds more expressive ke utting. In IJAI. hen, Y.; Li, J. K.; Prkes, D..; nd Proi, A. D. 2013. Truth, ustie, nd ke utting. Gmes nd Eonomi Behvior 77(1):284 297. Deng, X.; Qi, Q.; nd Seri, A. 2012. Algorithmi solutions for envy-free ke utting. Opertions Reserh 60(6):1461 1476. Kurokw, D.; Li, J. K.; nd Proi, A. D. 2013. How to ut ke efore the prty ends. In AAAI. My, A., nd Nisn, N. 2012. Inentive omptile two plyer ke utting. In Interntionl Workshop on Internet nd Network Eonomis, 170 183. Springer. Proi, A. D. 2013. ke utting: not ust hild s ply. ommunitions of the AM 56(7):78 87. Proi, A. D. 2014. ke utting lgorithms. Steinhus, H. 1948. The prolem of fir division. Eonometri 16(1). Stromquist, W. 1980. How to ut ke firly. Amerin Mthemtil Monthly 640 644. Stromquist, W. 2007. Envy-free ke divisions nnot e found y finite protools. In Fir Division. 318