The Computational Rise and Fall of Fairness

Transcription

1 Proceedigs of the Twety-Eighth AAAI Coferece o Artificial Itelligece The Coputatioal Rise ad Fall of Fairess Joh P Dickerso Caregie Mello Uiversity dickerso@cscuedu Joatha Golda Caregie Mello Uiversity jagolda@adrewcuedu Jerey Karp Caregie Mello Uiversity jkarp@adrewcuedu Ariel D Procaccia Caregie Mello Uiversity arielpro@cscuedu Tuoas Sadhol Caregie Mello Uiversity sadhol@cscuedu Abstract The fair divisio of idivisible goods has log bee a iportat topic i ecooics ad, ore recetly, coputer sciece We ivestigate the existece of evyfree allocatios of idivisible goods, that is, allocatios where each player values her ow allocated set of goods at least as highly as ay other player s allocated set of goods Uder additive valuatios, we show that eve whe the uber of goods is larger tha the uber of agets by a liear fractio, evy-free allocatios are ulikely to exist We the show that whe the uber of goods is larger by a logarithic factor, such allocatios exist with high probability We support these results experietally ad show that the asyptotic behavior of the theory holds eve whe the uber of goods ad agets is quite sall We deostrate that there is a sharp phase trasitio fro oexistece to existece of evy-free allocatios, ad that o average the coputatioal proble is hardest at that trasitio Itroductio The allocatio of goods to iterested agets is a cetral teet of society Soe goods, like lad, are divisible: a echais ca split a sigle good aogst ultiple agets Others, like the houses or cars i a estate sale or divorce proceedigs, are idivisible: a echais ust allocate each good to exactly oe aget A chief cocer i the assiget of divisible ad idivisible goods to agets ad i the eployet of divorce lawyers cocers defiig ad guarateeig the fairess of the fial allocatio Oe foral otio of fairess is evy-freeess A allocatio of goods is evy free EF if each player values her ow allocated set of goods at least as highly as ay other player s allocated set of goods While EF divisios exist for ay uber of players i the divisible goods case see, eg, Procaccia 3, ad the refereces therei, it is ot guarateed that such fair allocatios exist whe idivisible goods are cosidered Ideed, cosider the siple case of a sigle Copyright c 4, Associatio for the Advaceet of Artificial Itelligece wwwaaaiorg All rights reserved good ad two agets, both of which have positive value for the good Allocatig the good to either aget will result i evy fro its epty-haded parter I this paper, we ivestigate the coditios uder which EF allocatios of idivisible goods exist, whe agets values of goods are draw at rado Uder additive valuatios ad rather geeral coditios o the distributios over values of idividual goods, we characterize coditios for oexistece, showig that eve whe the uber of goods is larger tha the uber of agets by a liear fractio, a EF allocatio is ulikely to exist Theore We the show that whe the uber of goods is larger by a logarithic factor tha the uber of agets, a EF allocatio exists with high probability Theore Thus, these asyptotic existece results are alost tight We support our theoretical results, which apply asyptotically, with a epirical exploratio of the EF allocatio proble o differet distributios over valuatios ad differet objectives over EF allocatios usig two iteger prograig odels The theory applies to each of these experiets eve whe the uber of agets ad goods is quite sall We also ucover a pheoeo coo to ay probles i artificial itelligece: that the hardest coputatioal EF allocatio probles o average occur durig the sharp trasitio fro oexistece to existece Related Work Fair divisio occupies a iportat place i AI research; see, eg, Chevaleyre, Edriss, ad Maudet 7, Bouveret ad Lag 8, Che et al, Cohler et al ad the survey by Chevaleyre et al 6 Aog the ay AI papers that study the EF allocatio of idivisible goods, the work of Bouveret ad Lag 8 is of particular iterest They showed that deteriig the existece of a EF allocatio is coputatioally hard I cotrast, we focus o typical istaces, ad show that EF allocatios exist, or do ot exist, with high probability Siilarly, the phase trasitio pheoeo is a staple of AI research Cheesea, Kaefsky, ad Taylor 99; Hogg, Hubera, ad Willias 996 I a utshell, co- 45

2 strait satisfactio probles CSPs typically have the curious property that as the proble becoes ore costraied, the probability of the existece of a feasible solutio sharply drops fro to Aroud the sae poit where this phase trasitio occurs kow as the critical value of the order paraeter, search algoriths experiece a sharp spike i ruig tie a steep coputatioal rise, the a equally steep fall as the proble becoes ore costraied so rulig out the existece of a solutio becoes easier We show that a siilar pheoeo occurs i the cotext of the existece ad coputatio of EF allocatios The proble of coalitioal aipulatio i electios is aother popular topic i coputatioal social choice that has ru the gautlet fro i worst-case coplexity Coitzer, Sadhol, ad Lag 7, through ii probabilistic existece ad oexistece results Coitzer ad Sadhol 6; Procaccia ad Roseschei 7; Xia ad Coitzer 8, to iii ivestigatios of the phase trasitio at the threshold betwee oexistece ad existece Walsh ; Mossel, Procaccia, ad Rácz 3 Fro the techical ad coceptual viewpoits, though, our proble is copletely differet Note that we tackle ii ad iii siultaeously, ad for the first tie i the cotext of fair divisio Bras, Kilgour, ad Klaler 4 desig a echais for the EF allocatio of idivisible goods, which also satisfies other desirable properties While their schee guaratees evy-freeess, it ay ot allocate all the goods To aeliorate this shortcoig, they show that whe the ordial prefereces of agets over goods are draw uiforly at rado, their schee will allocate all goods with high probability, as the uber of goods goes to ifiity Our existece result, Theore, is sigificatly stroger i several ways: i it gives a exact relatio betwee the uber of agets ad uber of goods, istead of assuig that oe is costat ad the other goes to ifiity, ii it holds uder far ilder assuptios o the probability distributio over istaces, ad iii it relies o a ituitively desirable allocatio echais that gives each good to the aget that wats it the ost, thereby axiizig social welfare as we discuss below Bras ad Fishbur also work i a probabilistic odel, but with oly two agets; our results hold for ay uber of agets Our Model Deote the set of agets by N = {,, }, ad the set of goods by G, where G = Aget i has utility u i g [, for good g; ote that costraiig the utilities to a iterval is without loss of geerality We ake the very coo assuptio that utilities are additive, that is, for a subset of goods G G ad aget i N, it holds that u i G = u ig A allocatio is a partitio A = A,, A of the goods, where A i is the budle of goods allocated to aget i N The allocatio A is said to be evy free EF if ad oly if for ay two agets i, j N, u i A i u i A j, that is, each aget weakly prefers its ow budle to the budle allocated to ay other aget Distributios Over Utilities For every aget i ad good g G, the utilities u g,, u g are draw fro a joit, o-atoic distributio D over [,, that is, for every x [,, Pr[u i g = x = Let us state two assuptios o D, which hold for every g G; Theore will require the first, ad Theore will require the secod: [A For all i, j N such that i j, u i g ad u j g are idepedet ad idetically distributed [A For all i, j N, Pr[arg ax u k g = {i} = /, ad there exist costats µ, µ such that < E[u i g arg ax u k g = {j} µ < µ E[u i g arg ax u k g = {i} Let us illustrate these assuptios usig two atural distributios that will be featured i our epirical results: UNIFORMx, y: For each aget i N ad good g G, draw u i g U[x, y, where U is the uifor distributio CORRELATEDx, y: Idepedetly assig each good g a itrisic base value µ g U[x, y The, for each aget i, draw u i g N µ g, σ g, where N is the trucated oral distributio ad σ g µ g First, cosider UNIFORM Clearly it satisfies [A Assuptio [A sees techical, but is actually quite ild To be cocrete, take UNIFORM,, so the utilities are draw uiforly at rado i [, The first part of [A holds due to syetry Moreover, i this case, E[u i g =, ad E[ax u k g = + 3 see, eg, Boutilier et al, Corollary 45 Clearly E[u i g arg ax u k g = {j} E[u i g, ad due to syetry E[u i g arg ax u k g = {i} = E[ax u kg, so we ca set µ = /3 ad µ = / Assuptio [A still holds if the utilities are draw fro a iterval [x, y [, by scalig ad shiftig µ ad µ Siilarly, CORRELATEDx, x satisfies both assuptios for ay x, utilities are siply draw iid fro the sae oral distributio But whe x < y, CORRE- LATEDx, y oly satisfies assuptio [A This distributio does ot satisfy [A, because for a fixed g G, u i g ad u j g are ot idepedet A Sall Nuber of Goods If there are fewer goods tha agets, ie, <, the clearly o EF allocatio is possible there will be a aget with o goods Coceivably, though, it could be that if is slightly larger tha say, = + the a EF allocatio 46

3 is likely to exist I this sectio we show that this is ot the case: the uber of extra goods ust be liear i Recall that our distributios over utilities are o-atoic Therefore, for i N ad two goods g g, Pr[u i g = u i g = So, we ca safely assue that each aget has a uique favorite good, ad defie a fuctio f : N G that aps each aget to its favorite good, that is, fi = argax u i g We are ow ready to state our first result Theore Assue that [A holds Let δ, e be a costat If the probability that there exists a EF allocatio is at least δ the +cδ, where cδ > is a costat that depeds oly o δ We require the followig lea, which gives a ecessary coditio for evy-freeess that depeds o the uber of collisios of the fuctio f Lea Let u,, u be utility fuctios for the agets such that u i g u i g for all g g For each good g, let X g = f g be the set of agets whose favorite good is g If there is a EF allocatio the + ax{ X g, } Proof of Lea Fix a allocatio A, ad let i N such that g A i I order to avoid evyig i, every aget i X g \ {i} ust receive at least two goods Hece, X g = : X g > : X g > X g + : X g > = X g + ax{ X g, } = + ax{ X g, } X g Proof of Theore Let C be a rado variable that couts the uber of collisios betwee agets top prefereces Specifically, the value of C is deteried as follows Startig fro C =, for each i =,,, if there exists j < i such that fi = fj the icreet C by Usig the otatios of Lea, it is easy to see that C = ax{ X g, } Our first goal is to copute E[C Let Y ij be a Beroulli rado variable that takes the value if fi = fj, ad otherwise The for all i j, Pr[Y ij = = /, due to assuptio [A Let Z i be aother Beroulli rado variable that takes the value if fi = fj for soe j < i, ad otherwise Z i = if ad oly if there exists j < i such that Y ij = Furtherore, for a fixed i the variables Y ij are idepedet due to assuptio [A Therefore E[Z i = i Now we ca siply write C = i N Z i Usig the liearity of expectatio: [ E[C = E Z i = i i= i N e i i= i= + e i e, where the third trasitio follows fro the well-kow fact that x e x for all x R, ad the fourth trasitio assues that is eve purely for ease of expositio Now, suppose that C k with probability δ Also usig the fact that C, we get δk + δ E[C e, ad therefore k δ e δ We wat k to be lower-bouded by a costat fractio of, which is true if ad oly if e < δ Deotig = α, we ca write e α < δ; equivaletly, α < l δ, ad by rearragig we get α < l δ Usig our assuptio that δ <, we see that the e right had side of Equatio is a costat greater tha Let us therefore set + βδ = l δ the β is a costat greater tha that depeds oly o δ We ow have all the igrediets i place to coplete the theore s proof O the oe had, if > βδ the we are doe O the other had, if βδ the Equatio gives us a costat lower boud o k that depeds oly o δ, say γδ, which was derived uder the assuptio that Pr[C k δ Next, set γ δ = γδ/ So, if it holds that γ δ k < γδ, ie, k is a costat fractio of that is strictly saller tha the lower boud, the the assuptio does ot hold, ie, Pr[C k > δ By Lea, i those cases where C k, there is a EF allocatio oly if the uber of goods is at least + k + γ δ I other words, if < + γ δ, a EF allocatio would ot exist with probability δ for the precedig choices of paraeters We coclude that it ust be the case that i{βδ, +γ δ} Settig cδ = i{βδ, γ δ} copletes the proof A Large Nuber of Goods Next, we exaie the case where the uber of goods is sigificatly larger tha the uber of agets by a logarithic factor, to be precise I this case, a EF allocatio exists with high probability, 47

4 Theore Assue that [A holds Let = O l The a EF allocatio exists with probability as Before provig the theore, two coets are i order First, why are we writig = O l istead of the ore ituitive = Ω l? The reaso is that we wat to ephasize that oly the uber of goods has to go to ifiity; the uber of agets ca stay sall, eve costat The theore holds eve if the uber of agets goes to ifiity, as log as this happes at ost at the specified rate copared to the uber of goods Secod, Theore states that there exists a EF allocatio, but the proof shows soethig stroger: that this allocatio ca be obtaied by givig each good to the aget that values it the ost, ie, to arg ax i N u i g This is, i fact, the allocatio that axiizes the utilitaria social welfare, which is the su of utilities So, a alterative forulatio is that, uder the theore s coditio, the socialwelfare-axiizig allocatio is EF with high probability Turig to the theore s proof, we require the followig well-kow result Lea Cheroff Let X,, X be idepedet rado variables i [, Deote X = i= Xi The for all ɛ [,, Pr[X + ɛe[x exp ɛ3 E[X Pr[X ɛe[x exp ɛ E[X Proof of Theore We explicitly costruct a allocatio by givig each good g G to the aget that likes it ost, that is, to arg ax i N u i g This algorith iduces a allocatio A = A,, A, where each A i ca be forally thought of as a rado variable that takes values i G We prove the allocatio A is EF with high probability Let X g i be a rado variable that takes the value u i g if {i} = arg ax u k g, ad otherwise It holds that u i A i = Xg i ; we will use this observatio to calculate E[u i A i Usig [A twice, for all i N ad g G it holds that E[X g i [ = Pr {i} = arg ax = [u E ig [ u k g E {i} = arg ax u k g u ig {i} = arg ax µ Therefore, usig the liearity of expectatio, E[u ia i = E[X g i µ u k g Next, for all i j ad g G let Y g ij be rado variables that take the value u i g if {j} = arg ax u k g, ad otherwise It holds that u i A j = Y g ij Furtherore E[Y g ij = E [u ig {j} = arg ax u k g µ, by assuptio [A However, a techicality is that our assuptios do ot provide a lower boud for [ E u i g {j} = arg ax u k g which is required to use Lea We therefore defie variables Z g ij such that E[Zg ij = µ/, Zg ij [,, ad Zg ij stochastically doiates Y g ij I particular, E[ Zg ij = µ, ad due to stochastic doiace, for all x R+, [ [ Pr Z g ij x Pr Y g ij x We ca therefore use the Z g ij variables to reaso about u i A j Let E ij be the evet that aget i evies aget j For E ij to happe, it ust be the case that Y g ij > Xg i, which happes oly if X g i µ µ µ = µ µ µ µ [ = µ µ E X g µ i, or Let us set Y g ij µ µ µ = = + µ µ µ + µ µ µ = µ + µ µ µ [ E Z g ij { } ɛ = i, µ µ µ Because µ < µ, it also holds that ɛ µ µ µ The variables X g i ad X g i are idepedet for g g, ad siilarly Z g ij ad Zg ij are idepedet Usig Lea, we have that [ [ Pr X g i ɛe X g i exp ɛ µ, ad [ Pr Pr Settig [ [ Y g ij + ɛe Z ij Z g ij + ɛe [ ɛ µ 3 l 3, Z ij exp ɛ 3 µ ad usig the uio boud, we coclude that Pr[E ij exp ɛ µ + exp ɛ 3 µ =

5 The allocatio A is EF if ad oly if E ij does ot occur for all i j Usig Equatio 3 ad the uio boud over pairs of agets, the probability that A is ot EF is at ost Pr E ij Pr[E ij i j i j 3 Thus, the probability that A is ot EF goes to zero as grows I Betwee: A Phase Trasitio I this sectio, we support our theoretical results with a epirical exploratio of the trasitio fro oexistece to existece of evy-free allocatios as a fuctio of the uber of goods ad agets We fid that the ost difficult allocatio probles occur durig the sharp phase trasitio fro oexistece to existece We show that this behavior, which is coo to ay discrete feasibility probles, holds uder both of two atural optiizatio odels oe with ad oe without a objective fuctio ad uder differet distributios over agets utility values Experietal Setup We geerate istaces with agets ad goods as follows by saplig valuatios for each aget ad each good fro a give distributio over utility fuctios I our experietal setup, we draw fro two distributios CORRELATED4, 6 ad UNIFORM, defied earlier Ituitively, the UNIFORM distributio radoly assigs a value to each good for each aget, while the CORRELATED distributio first draws a itrisic value for each good, the assigs a rado value to each aget draw fro a trucated oegative oral distributio aroud that itrisic value UNIFORM satisfies both distributioal assuptios ad thus aligs with both Theores ad, while our istatiatio of CORRELATED oly satisfies assuptio [A, or the assuptio eeded for Theore Still, we will show that both theoretical results hold experietally for both distributios, eve whe the uber of agets ad goods is quite sall Give a istace as geerated above, we search for a evy-free allocatio usig oe of two ixed iteger progras MIPs Both forulatios use biary variables x ig that are activated if ad oly if aget i is allocated good g Model #, a feasibility proble, is defied as follows: fid x ig i N, g G st xig = g G i N vigx i g vigxig i i N x ig {, } i N, g G Ituitively, the first set of costraits esures that each good is allocated to exactly oe aget, while the secod set of costraits esures that each aget values its allocatio at least as highly as ay other aget s allocatio For this feasibility proble, o explicit objective fuctio is ecessary; ideed, the feasible regio defied by the costraits is exactly the space of all evy-free allocatios We ow defie Model #, a optiizatio versio of the evy-free allocatio proble, as follows: i e st xig = g G i N vigx i g vigxig e i i N x ig {, } i N, g G e R oeg This secod MIP odel iiizes a real-valued oegative variable e represetig the axiu evy betwee ay two agets; thus, a EF allocatio exists if ad oly if the objective value is zero at the optiu This is a iteger prograig-based ipleetatio of the evy iiizatio proble described by Lipto et al 4 Model # ay see like the ore geeral odel sice it is aeable to the additio of various objective fuctios For exaple, addig a objective fuctio that axiizes i N v igx ig would produce a evy-free allocatio that also axiizes social welfare It is ot obvious how to adapt Model # to iclude arbitrary objective fuctios Still, there is soe evidece that relaxig the feasible regio ad the re-castig the feasibility proble as a optiizatio proble ay result i better rutie perforace For exaple, Sadhol, Gilpi, ad Coitzer 5 saw speedups usig a optiizatio odel istead of a feasibility odel i specific proble classes whe explorig various MIP odels for fidig Nash equilibria i two-player gaes although they did ot see a overall speedup We copare the perforace of both odels i the coig sectio All experiets were perfored i Pytho usig IBM ILOG CPLEX 6 i sigle-threaded ode uder its default cofiguratio Rus were coducted o Blacklight, 3 a ccnuma supercoputer with 8GB of RAM per core; each experiet was ru at least 6 ties with a tie liit of hours per ru For solve tie copariso, rus that tied out were coservatively cosidered to have copleted i hours Whe tieouts were igored or pealized heavily eg, couted as a = hour ru, our experiets exhibited the sae qualitative behavior Phase Trasitios We ow explore the existece of phase trasitios i various istatiatios of the evy-free allocatio proble Figure shows a exaple phase trasitio for the existece of, ad hardess of fidig, a evy-free allocatio i a proble with = agets valuig {,, 3} goods Results are preseted for both the UNI- FORM ad CORRELATED distributios over utility fuctios usig Model # without ad with a social welfare axiizig objective fuctio The thick red lie correspodig to the left y-axis plots the fractio of istaces with goods ad agets such that a evy-free allocatio existed Aligig with Theore, Figure shows that the probability of a EF allocatio existig is sall whe the uber of goods is ot uch larger tha the uber of agets Siilarly, aligig with Theore, whe the uber of goods is ibco/software/coerce/optiizatio/cplex-optiizer/ Source code & data: 3 blacklightpscedu 49

6 Fractio Feasible =, Existece, U[, 6 Frac Feasible Solve Tie s Solve Tie feas 5 Solve Tie ifeas 4 3 Media Rutie s Fractio Feasible =, Existece, Correlated 4 Frac Feasible Solve Tie s 35 Solve Tie feas Solve Tie ifeas Media Rutie s Fractio Feasible =, Social Welfare Max, U[, Frac Feasible Solve Tie s Solve Tie feas Solve Tie ifeas Media Rutie s Fractio Feasible =, Social Welfare Max, Correlated 8 Frac Feasible Solve Tie s 6 Solve Tie feas Solve Tie ifeas Media Rutie s Figure : Phase trasitio for = uder either UNIFORM or CORRELATED, with or without axiizig social welfare ore but ot ecessarily substatially ore, the probability of a EF allocatio existig is essetially oe Figure explores this trasitio quatitatively for icreasig ubers of agets by plottig the iiu value where at least 99% of the geerated istaces were feasible Fittig a / l fuctio for either UNIFORM or CORRELATED shows that the asyptotically-stated Theore holds eve whe the uber of goods ad agets is quite sall Uifor Correlated 78 / l / l Figure : The iiu value of where at least 99% of the istaces were feasible as icreases Figure also plots rutie as a fuctio of the uber of goods The thick dashed lie correspodig to the right y-axis plots the edia rutie to either prove the oexistece of a solutio or fid ad prove the optiality of a feasible solutio The two dotted lies also correspodig to the right y-axis plot the edia ruties for oly the feasible ad ifeasible istaces, respectively We see a classical hardess bup aroud the phase trasitio, with edia solutio tie beig uch higher whe the probability of a feasible istace is sall but ot trivial Here, provig ifeasibility takes sigificat coputatioal effort Figure 3 shows that this hardess behavior is ot just a artifact of the feasibility Model #; ideed, the optiizatio proble defied by Model # exhibits a eve ore stark hardess bup aroud the phase trasitio This roughly aligs with the experieces of Sadhol, Gilpi, ad Coitzer 5, who foud that relaxig the feasible regio while ovig soe costraits ito the objective did ot result i a overall speedup Discussio & Future Research I this paper, we theoretically ad epirically ivestigated the existece of evy-free allocatios of idivisible goods Uder additive valuatios ad geeral assuptios o the distributios over values of idividual goods, we theoretically characterized the coditios for oexistece ad existece of evy-free allocatios We supported these asyptotic results with experiets o two value distributios usig two MIP odels ad foud, epirically, that the theoretical coditios for oexistece of evy-free allocatios apply eve whe the uber of agets ad goods is quite sall Furtherore, we discovered that the hardest coputatioal probles i this space o average exist durig the phase trasitio betwee oexistece ad existece I typical phase trasitio work, what is icreased o the x-axis is the uber of costraits while keepig the uber of variables costat Our phase trasitio is, i that sese, differet because as we icrease the uber of goods while keepig the uber of agets fixed, both the uber of variables ad costraits icreases Our phase trasitio is evertheless siilar to prior oes i that i there is a sharp trasitio fro ifeasibility to feasibility, ii the coplexity peak occurs at that trasitio, iii the coplexity peak is drive aily by ifeasible istaces, ad iv the ifeasible istaces get harder ad rarer as we ove to the side of the phase trasitio where istaces are typically feasible While the theoretical results we preseted are essetially tight, it would be useful to copletely characterize the phase trasitio betwee oexistece ad existece of a evyfree allocatio We showed experietally that this phase trasitio is quite sharp, but either provig that the logarithic factor i Theore is ecessary or further whittlig dow this boud toward Theore would be helpful Results of this ature are actively beig pursued with rado 3-SAT probles Kaporis, Kirousis, ad Lalas 6; Maeva ad Siclair 8 Furtherore, relaxig the distributioal assuptios especially o Theore would, if possible, be useful toward this ed Alog the lies of ehaced MIP techiques, it would be iterestig to try to flatte the hardess bup we saw i the experiets through the use of custo brachig ad fathoig rules, variable prioritizatio schees, ad other heuristics that aitai search copleteess Ackowledgets This aterial was fuded by NSF grats IIS-36, CCF- 668, CCF-5883, ad IIS , by a NDSEG fellowship, ad used the Pittsburgh Supercoputig Ceter i partership with the XSEDE, which is supported by NSF grat OCI We thak Itel Corporatio for achie gifts 4

7 Media Rutie s = 6, Existece, U[, Model Model Media Rutie s = 6, Existece, Correlated Model Model Media Rutie s =, Existece, U[, Model Model Media Rutie s =, Existece, Correlated Model Model Figure 3: Rutie copariso of Model # feasibility ad Model # optiizatio for = 6 ad = agets Refereces Boutilier, C; Caragiais, I; Haber, S; Lu, T; Procaccia, A D; ad Sheffet, O Optial social choice fuctios: A utilitaria view I Proceedigs of the 3th ACM Coferece o Electroic Coerce EC, 97 4 Bouveret, S, ad Lag, J 8 Efficiecy ad evyfreeess i fair divisio of idivisible goods: logical represetatio ad coplexity Joural of Artificial Itelligece Research 3: Bras, S J, ad Fishbur, P C Fair divisio of idivisible ites betwee two people with idetical prefereces: Evy-freeess, Pareto-optiality, ad equity Social Choice ad Welfare 7:47 67 Bras, S J; Kilgour, M; ad Klaler, C 4 Twoperso fair divisio of idivisible ites: A efficiet, evyfree algorith Notices of the AMS 6:3 4 Cheesea, P; Kaefsky, B; ad Taylor, W 99 Where the really hard probles are I Proceedigs of the th Iteratioal Joit Coferece o Artificial Itelligece IJ- CAI, Che, Y; Lai, J K; Parkes, D C; ad Procaccia, A D Truth, justice, ad cake cuttig I Proceedigs of the 4th AAAI Coferece o Artificial Itelligece AAAI, Chevaleyre, Y; Due, P E; Edriss, U; Lag, J; Leaître, M; Maudet, N; Padget, J; Phelps, S; Rodríguez-Aguilar, J A; ad Sousa, P 6 Issues i ultiaget resource allocatio Iforatica 3:3 3 Chevaleyre, Y; Edriss, U; ad Maudet, N 7 Allocatig goods o a graph to eliiate evy I Proceedigs of the d AAAI Coferece o Artificial Itelligece AAAI, 7 75 Cohler, Y J; Lai, J K; Parkes, D C; ad Procaccia, A D Optial evy-free cake cuttig I Proceedigs of the 5th AAAI Coferece o Artificial Itelligece AAAI, Coitzer, V, ad Sadhol, T 6 Noexistece of votig rules that are usually hard to aipulate I Proceedigs of the Natioal Coferece o Artificial Itelligece AAAI Coitzer, V; Sadhol, T; ad Lag, J 7 Whe are electios with few cadidates hard to aipulate? Joural of the ACM 543: 33 Hogg, T; Hubera, B A; ad Willias, C P 996 Phase trasitios ad the search proble Artificial Itelligece 8 : 5 Kaporis, A C; Kirousis, L M; ad Lalas, E G 6 The probabilistic aalysis of a greedy satisfiability algorith Rado Structures & Algoriths 84: Lipto, R; Markakis, E; Mossel, E; ad Saberi, A 4 O approxiately fair allocatios of idivisible goods I ACM Coferece o Electroic Coerce EC, 5 3 Maeva, E, ad Siclair, A 8 O the satisfiability threshold ad clusterig of solutios of rado 3-sat forulas Theoretical Coputer Sciece 47: Mossel, E; Procaccia, A D; ad Rácz, M Z 3 A sooth trasitio fro powerlessess to absolute power Joural of Artificial Itelligece Research 48:93 95 Procaccia, A D, ad Roseschei, J S 7 Averagecase tractability of aipulatio i electios via the fractio of aipulators I Proceedigs of the 6th Iteratioal Coferece o Autooous Agets ad Multi-Aget Systes AAMAS, 78 7 Procaccia, A D 3 Cake cuttig: Not just child s play Couicatios of the ACM 567:78 87 Sadhol, T; Gilpi, A; ad Coitzer, V 5 Mixediteger prograig ethods for fidig Nash equilibria I Proceedigs of the th AAAI Coferece o Artificial Itelligece AAAI, Walsh, T Where are the hard aipulatio probles? Joural of Artificial Itelligece Research 4: 9 Xia, L, ad Coitzer, V 8 Geeralized scorig rules ad the frequecy of coalitioal aipulability I Proceedigs of the 9th ACM Coferece o Electroic Coerce EC, 9 8 4