Choosing Combinatorial Social Choice by Heuristic Search

Transcription

1 Choosing Comintoril Soil Choie y Heuristi Serh Minyi Li 1 nd Quo Bo Vo 2 Astrt. This pper studies the prolem of omputing ggregtion rules in omintoril domins, where the set of possile lterntives is Crtesin produt of (finite) domin vlues for eh of given set of vriles, nd these vriles re usully not preferentilly independent. We propose very generl heuristi frmework SC* for omputing different ggregtion rules, inluding rules for rdinl preferene strutures nd Condoret-onsistent rules. SC* highly redues the serh effort nd void mny pirwise omprisons, nd thus it signifintly redues the running time. Moreover, SC* gurntees to hoose the set of winners in ggregtion rules for rdinl preferenes. With Condoret-onsistent rules, SC* hooses the outomes tht re suffiiently lose to the winners. 1 Introdution In mny multi-gent deision-mking senrios, the spe of lterntive hs omintoril struture: the set of possile lterntives is Crtesin produt of (finite) domin vlues for eh of given set of vriles (k. issues), nd usully these vriles re not preferentilly independent. In lssil soil hoie theory, ndidtes (k. lterntives, outomes) nd the gents preferenes re supposed to e listed expliitly s liner orders, nd then voting rule is pplied to selet one or set of winning lterntives. These trditionl methods rely on demnding ssumption tht the ndidtes should not e too numerous. However, when the domin hs omintoril struture, the numer of lterntives is exponentil in the numer of vriles, nd therefore, the gents preferenes re usully desried in some ompt representtion lnguges rther thn liner orders. This mkes the soil hoie prolem even more omplex nd hllenging, euse individul outome omprisons (nd thus pirwise omprisons etween outomes) in those lnguges might e omputtionlly diffiult. As most ommon ggregtion methods need numer of opertions t lest liner (sometimes even qudrti or exponentil) in the numer of possile lterntives, generting the whole reltion from those ompt lnguges nd diretly pplying rules to ompute soil hoie is imprtil [10, 9]. Severl wys of omputing rules in omintoril domins hve een onsidered. The most strightforwrd wy is to use issue-yissue (.k.. set-y-set) sequentil eletion. However, s soon s the vriles re not preferentilly independent, it is very likely tht deiding on the issues seprtely will led to suoptiml hoies []. Mny existing work onsider imposing domin restrition suh s seprility (whih mkes the issue-y-issue sequentil eletion work), see e.g., [11]; or weker restrition suh s O-leglity [17], whih llows for deiding on the issues one fter nother. Some lter works [12, 16] relx those restritions nd introdue notion of lo- 1 Swinurne University of Tehnology, Austrli, emil: myli@swin.edu.u 2 Swinurne University of Tehnology, Austrli, emil: BVO@swin.edu.u l Condoret winners ( lol Condoret winner ets every other lterntive tht differs in single vrile from tht lol Condoret winner). The uthors lso propose (nd implement) lgorithms for omputing them. However, this notion of winner differs from Condoret winner in the lterntive spe, s it only tkes into ount neighour lterntives. In this pper, we introdue very generl heuristi frmework SC* for omputing soil hoie in omintoril domins. SC* enles ggregting or voting on prtil ssignments (n ssignment of suset of vriles; nd possily ompring ssignments on different suset of vriles) until ll vriles hve een ssigned vlue. As result, it neither requires ounting of ndidtes s tht in most deision-mking instnes, nor imposes ny restrition on the gents preferene strutures. The proposed heuristi pproh llows serhing in muh smller su-spe of the lterntives, nd thus requires signifintly less pirwise outome omprisons. It is generl enough to e pplile to oth ggregtion rules for rdinl preferenes nd severl Condoret-onsistent rules. Most importntly, SC* gurntees optiml soil hoie in ggregtion rules for rdinl preferene strutures. With Condoret-onsistent rules, SC* hooses the ndidtes tht re suffiiently lose to the winners of the rule. Lst ut not lest, the proposed lgorithm is independent to the preferene representtions, nd thus it is pplile to most representtion lnguges in omintoril domins. Notie tht we omit ll the theorem proofs in this pper due to spe limittion, while longer version ontining ll detil proofs n e found t http: // 2 Preliminries Let V = {X 1,..., X m } e set of vriles, where eh vrile X k tkes vlues in finite domin D Xk. An lterntive is uniquely identified y its vlues of ll vriles. The set of lterntives is denoted y X, suh tht X = D X1,..., D Xm. If X = {X σ1,..., X σl } V, with σ 1 < < σ l then D X denotes D Xσ1 D Xσl nd x denotes n ssignment of vrile vlues of X, i.e., x D X. If X = V, x is omplete ssignment (orresponds to n outome); otherwise x is lled prtil ssignment. If x nd y re ssignments to disjoint sets X nd Y, respetively (X Y = ), we denote the omintion of x nd y y x y. If X Y = V, we ll x y ompletion of ssignment x. We denote y Comp( x) the set of ompletions of x. A vote p is liner order on X, i.e., trnsitive, ntisymmetri, nd totl reltion on X. We denote L(X) s the set of ll possile liner orders on X. An n-gent profile P is olletion of n votes, tht is, P = (p 1,..., p n ), where p i L(X). Let P(X) e the set of ll possile profiles over X, (voting or ggregtion) rule r : P(X) 2 X mps ny profile p P(X) to suset of lterntives (winners). Sine diret ssessment of the preferene reltions in ominto-

2 ril domins is usully infesile, severl ompt lnguges hve een proposed to void the exponentil low up. or instne, some lnguges re rdinl, e.g., utility networks [1] nd soft onstrints [5, 7]; some lnguges re purely qulittive, e.g., CP-nets [2] nd CI-nets []. In this pper, we ssume tht eh gent hs liner preferene over the lterntive spe, whih my e ompletely or prtilly represented y ompt lnguge. Therefore, vote p might e (prtilly) represented y ompt lnguge L. Tht is, L (prtilly) indues n order p over the lterntive spe. The proposed heuristi frmework.1 The serh tree We identify optiml soil hoie using serh tree T. A serh tree T n e onsidered s n ssignment tree. or omintoril soil hoie prolem over m vriles V = {X 1,..., X m }, let k e the mximum size of the vrile domin: X V, D X k, then T is k-ry tree. The depth of T is m with the root eing t depth 0. Suppose tht serh tree T is generted following n order over vriles σ = X σ1 > > X σm, then eh level l onsiders the vlue ssigns to vrile X σl. If node Φ from the upper level is eing expnded with the vlue of vrile X t the urrent level, then Φ hs D X rnhes nd eh rnh ssigns different vlue x to X (x D X ). The root node represents n empty ssignment. A node Φ t depth l represents unique vlue ssignment (speified y the pth from the root to tht node) ssg D Xσ1 D Xσl to the set of vriles {X σ1,..., X σl }. Eh node t depth m orresponds to n lterntive, s ll the vriles hs een ssigned vlue..2 The SC* serh lgorithm Similr to other heuristi serh lgorithms [1], SC* retes serh tree y itertively seleting node tht ppers to e most likely to led towrds n optiml soil hoie ( winner of the rule). We first give generl definition of the evlution funtion of the serh strtegy, nd then we would speify different evlution funtions for different ggregtion rules in the next setion. Definition 1 (Evlution funtion) In n itertion, given set of lef nodes L in the serh tree T, n evlution funtion mps eh node Φ in L into numer ( : L R), whih indites how promising (good) the node is, i.e., how likely the node will eventully led to winner of the given rule. In this work, (Φ) is modelled s kinds of disvlue of Φ (e.g., ost, distne to gol, penlty, disutility). Therefore, the smller the vlue of node Φ, the more promising Φ is. SC* (see Algorithm 1) is dpted from the A* heuristi serh lgorithm [6] with eing the heuristi funtion. We first rndomly generte n order over vriles σ = X σ1 >... X σm, following whih the serh tree would e reted (line 1). Strting with the root node with n empty ssignment (logilly expressed s T rue) (line 2), SC* mintins priority queue of nodes to e expnded, known s the f ringe. The lower vlue of node Φ, the higher its priority is, i.e., the more upfront it is in the f ringe. We emphsize here tht for two nodes with the sme vlue, the one t level l < m must e ordered efore (given higher priority thn) the one t level l = m. In eh itertion of SC*, the first node Φ, i.e. the node with the lowest vlue, is removed from the f ringe (line ). If it does not represent omplete ssignment (ssg denotes the ssignment of Φ), it will e expnded (line 6-9). Let X e the next vrile to e ssigned ording to the order σ, for eh x D X, hild node expnded from Φ with the ssignment ssg x will e reted nd Algorithm 1: SC*(P) Input: P, preferene profile of n gents over m issues Output: outs, set of outomes 1 Rndomly generte n order over vriles σ = X σ1 > > X σm ; 2 f ringe INSERT(MAKE-NODE(T rue), f ringe); while f ringe do Φ REMOVE-IRST( f ringe); ssg ASSIGNMENT(Φ); 5 if ssg is not omplete ssignment then 6 X NEXT-VARIABLE(ssg, σ); 7 foreh x D X do 8 INSERT(MAKE-NODE(ssg x), f ringe); 9 end 10 Compute for eh lef node; 11 SORT-RINGE-ASC( f ringe); 12 else 1 AppendTo(outs, ssg); 1 Φ REMOVE-IRST(fringe); 15 while (Φ ) = (Φ) do 16 ssg ASSIGNMENT(Φ ); 17 AppendTo(outs, ssg); 18 Φ REMOVE-IRST(fringe); 19 end 20 end 21 end 22 return outs; 2 Notie tht funtion SORT-RINGE-ASC sorts the f ringe ording to n sending order of the vlues; for two nodes with the sme vlue, the one with prtil ssignment (t level l < m) must e ordered efore the one with omplete ssignment (t level l = m). dded into the f ringe. After reting ll possile hild nodes, the vlue of eh existing lef node is omputed ordingly, nd the f ringe is sorted in the sending order of the vlues (line 10-11). SC* ontinues until the urrent hosen node Φ for expnsion is lef node t level m, i.e. the ssignment of Φ is omplete ssignment to the set of domin vriles V (line 12). Notie tht there might e more thn one node with the sme vlue in n itertion. As we mentioned efore, with the nodes tht hve the sme vlue, we give higher priority to node t n upper level (l < m) thn the one t the deepest level (l = m). Consequently, when the first node Φ in the f ringe is t level m, the other nodes tht hve the sme vlue s Φ must lso e t level m. Then, SC* lgorithm ollets the nodes with the smllest vlue, nd put their ssignments (eh orresponds to unique outome) into set outs (line 1-19). inlly, SC* returns outs s the set of soilly preferred lterntives (line 22).. The sis of heuristi All the heuristi serh strtegies introdued in this pper re sed on the following definition of est possile lterntive of node: Definition 2 (Best possile lterntive) At eh node Φ of the serh tree T, eh gent i hs est possile lterntive (BPA) on Φ, denoted y BPA i (Φ), whih is the optimisti outome tht gent i n otin with the vrile vlues ssigned from the root to Φ eing fixed, i.e., the est outome tht gent i n otin from the sutree of Φ. ormlly, BPA i (Φ) = OPTIMIZE(ssg, L i )

3 where ssg = ASSIGNMENT(Φ) represents the ssignment speified y the pth from the root to Φ; funtion OPTIMIZE optimizes the vlues of the remining vriles tht re not in ssg, ording to gent i s preferene L i. Let Comp(ssg) represents the ompletions of ssg, then BPA i (Φ) is the est lterntive mong Comp(ssg) for gent i. Consequently, the BPA of the root node for n gent i orresponds to the optiml (est) outome for gent i in the entire outome spe, i.e. ll vriles re ssigned the preferred vlues ording to p i. or node Φ t level m of T, i, j {1,..., n} nd i j, BPA i (Φ) = BPA j (Φ) = ASSIGNMENT(Φ), s the ssignment of Φ is omplete. or instne, onsider n ordering over three inry vriles A, B nd C: ā ā ā ā, nd node Φ with the ssignment ssg = (resp. ssg = ā ), the ompletions of ssg is {,,, } (resp. {ā, ā }). Aording to the preferene order, the BPA of Φ is (resp. ā ), euse (resp. ā ā ). Evlution funtion of soil hoie rules.1 Aggregtion rules for rdinl preferenes When preferenes re rdinl, the preferene profile P = f 1,..., f n onsists of every gent s soring funtion. A soring funtion f i of n gent i mps every lterntive x ( x X) into rel numer R ( f i : X R), sed on the penlty (or distne) tht x used (or from the gol) ording to gent i s preferene. Typil exmples of rdinl preferene strutures inlude penlty soring funtions, dis-utility funtions nd some logil preferene lnguges like weighted gols nd distne gols. To model the preferene of the group, for eh lterntive x, the sore f i ( x) of eh gent i re synthesized y n ggregtion opertor : R n R into so-lled soil welfre funtion s( x) refleting the preferene of the group of gents [1]. ormlly, given rdinl preferene profile P = f 1,..., f n, the soil welfre soring funtion s mpping from X to R is defined y: x X, s( x) = { f i ( x) i = 1,..., n} Clssilly, is n opertor tht stisfies non-deresingness for eh of its rgument nd ommuttivity. As disussed in [8], the most nturl hoies for re sum nd mx. sum is utilitrin ggregtion opertor, stting tht the olletive sore of n outome is the sum of the sores of the gents in the group. On the other hnd, mx sttes tht the mximum sore mong ll the gents should e onsidered. Thus, the mx ggregtion opertor orresponds to the eglitrin soil welfre. inlly, n outome x is -optiml iff s( x) is minimized. or eh node Φ in the serh tree T, eh gent i hs est possile lterntive BPA i (Φ), nd ordingly, n optimisti sore f i (BPA i (Φ)), whih is the est (the smllest) possile sore tht gent i my otin from the sutree of Φ. Applying SC* to ompute n optiml soil hoie with rdinl preferenes, the heuristi evlution funtion Crdinl is defined s follows. Definition (Evlution funtion of rdinl rules) The evlution funtion Crdinl, mpping from node Φ to R, is defined y: rdinl (Φ) = { f i (BPA i (Φ)) i = 1,..., n} In eh itertion, SC* hooses node tht hs the minimum Crdinl vlue to expnd, until the node hosen for expnsion orresponds to omplete ssignment. Notie tht with rdinl preferene strutures, the vlue of node is fixed. Therefore, the vlue of node will only need to e lulted one. igure 1. () gent 1 () gent 2 () gent Agents s preferenes Theorem 1 Given tht the soil welfre funtion stisfies nonderesingness nd ommuttivity, the evlution funtion Crdinl is dmissile. Proof: On eh node Φ of the serh tree, the evlution funtion Crdinl is ggregted from set of optimisti (i.e. the possile minimum) sores of the gents, eh of whih results from the est possile lterntive (BPA) of n gent. Trvel long the pth of the tree, oth the individul nd the olletive sores re non-deresing. Consequently, for ny node Φ nd ssg = ASSIGNMENT(Φ), it stisfies tht y Comp(ssg) nd i 1,..., n, f i ( x i ) f i( y) where x i = BPA i (Φ). As stisfies non-deresingness, { f i ( x i ) i = 1,..., n} { f i ( y) i = 1,..., n}, tht is, Crdinl (Φ) s( y). Thus, Crdinl never overestimtes the olletive sores of the lef nodes nd is dmissile. Theorem 2 If Crdinl is dmissile, SC* hooses the set of winners ording to the rule. Proof: SC* trverses the tree serhing ll neighours; it follows the lowest evluted vlue pth nd keeps sorted priority queue of lternte pth segments long the wy. If t ny point the pth eing followed hs higher evluted vlue thn other enountered pth segments, the higher evluted vlue pth is kept in the f ringe nd the proess is ontinued with the lower vlue su-pth. This ontinues until the urrent node hosen for expnsion is lef node. Consequently, if the evlution funtion Crdinl never overestimtes the olletive sores of the lef nodes, the first lef node hosen for expnsion tht orresponds to n lterntive is gurnteed to e optiml. Exmple. Consider three gents preferenes over three inry domin vriles V = {A, B, C} depited in igure 1. The soring funtion f i of n gent i is defined y the position of the lterntive in the preferene ordering, i.e., the optiml outome eing t 1 nd the worst outome eing t X. or instne, onsider gent 1 s preferene ordering in igure 1(), f 1 () = 1, f 1 (ā) = 2,..., f 1 (ā ) = 8. The tle elow shows the BPA i nd f i of eh reted node Φ of eh gent i, nd then the Crdinl vlues. In this exmple, we onsider mx rule ( =mx). Therefore, the Crdinl vlue of node Φ (in olumn) is the mximum f i mong the three rows. Φ Agent 1 Φ 2 Φ Φ Φ 5 Φ 6 BPA i f i BPA i f i BPA i f i BPA i f i BPA i f i BPA i f i gent 1 1 ā 2 ā 2 ā 6 ā 2 ā gent 2 5 ā 1 ā 1 ā ā 1 ā 2 gent ā 1 ā 1 ā 2 ā 7 ā 1 Crdinl igure 2 shows the serh tree of this exmple. The 1 st itertion retes two hild nodes Φ 1 nd Φ 2 of the root. BPA 1 (Φ 1 ) =, f 1 () = 1; BPA 2 (Φ 1 ) =, f 2 ( ) = 5; BPA (Φ 1 ) =, f ( ) =. Therefore Crdinl (Φ 1 ) = mx{1, 5, } = 5. Similrly,

4 =7 root =5 1 2 igure = =6 Serh tree for rdinl preferenes Crdinl (Φ 2 ) = 2. As Crdinl (Φ 2 ) < Crdinl (Φ 1 ), Φ 2 is given higher priority in the f ringe nd is expnded in the 2 nd itertion. This proess ontinues until Φ 6 (speifies omplete ssignment) is hosen for expnsion. As Φ 6 is the only node with minimum Crdinl vlue mong the existing lef nodes (Φ 1, Φ, Φ 5, Φ 6 ), SC* returns ā (ASSIGNMENT(Φ 6 )) s the olletive deision..2 Condoret-onsistent rules Give preferene profile P, Condoret winner (CW) is ndidte x suh tht when ompred with ny other ndidte y ( x y), more thn hlf of the gents prefer x over y. Let #{i n : x i y} denote the numer of gents who prefer n lterntive x over nother lterntive y, formlly, n lterntive x is Condoret winner iff y X, #{i n : x i y} > #{ j n : y j x}. When the Condoret winner exists, it is unique. However, it is possile for prdox to form, in whih olletive preferenes n e yli (i.e. not trnsitive), even if the preferenes of individul gents re not. In omintoril domins, however, the numer of lterntives is often muh lrger thn the numer of gents, nd therefore, there lmost never exists Condoret winner in prtil ses. Hene, in the next susetions, we onsider two Condoretonsistent rules, nmely Copelnd rule nd Minimx rule. A rule is Condoret-onsistent, if it hooses the Condoret winner when one exists. Before going to define the evlution funtion of these two rules, we first introdue some relted onepts s follows. In eh itertion, we denote L s the set of existing lef nodes in the serh tree. In order to pply SC* to ompute Condoretonsistent rules, we first define n upper pproximtion of preferene reltions over the set of lef nodes L, sed on the gent s optimisti evlution (BPA) of eh lef node. Definition (Preferene reltions etween lef nodes) Given pir of lef nodes Φ nd Φ, we sy n gent i prefers Φ over Φ (written s Φ i Φ ) iff gent i prefers the BPA of Φ over the BPA of Φ. ormlly, i {1,..., n} nd Φ, Φ L, Φ i Φ iff BPA i (Φ) i BPA i (Φ ). Proposition 1 Let Φ e node t level m (represents omplete ssignment) nd Φ e node t level l (l m), x e the lterntive orresponds to Φ ( x = ASSIGNMENT(Φ)) nd ssg = ASSIGNMENT(Φ ), if n gent i prefers Φ to Φ, then y Comp(ssg ), x i y. Proof: y Comp(ssg ), BPA i (Φ ) i y or BPA i (Φ ) = y. As individul preferene is trnsitive nd x i BPA i (Φ ), x i y. Definition 5 (Mjority domintion etween lef nodes) In n itertion, lef node Φ mjority domintes nother lef node Φ, written s Φ m j Φ if there is mjority numer of gents who prefer Φ over Φ. Corollry 1 Let Φ e node t level m nd Φ e node t level l (l m); x = ASSIGNMENT(Φ) nd ssg = ASSIGNMENT(Φ ), if Φ m j Φ, then y Comp(ssg ), x m j y..2.1 Copelnd rule Given profile P, the Copelnd sore of n lterntive is the numer of lterntives it ets in pirwise omprisons. ormlly, for n lterntive x, let s( x) denotes the Copelnd sore of x, then s( x) = #{ y X : x m j y} A Copelnd winner x is n lterntive tht mximizing s( x). In eh itertion, we ondut pirwise omprison over the set of lef nodes L (sed on Definition nd Definition 5). or onsisteny, we define the evlution funtion s sort of disvlue of node nd the smller Copelnd is, the more promising the node is. Therefore, in n itertion, the Copelnd vlue of node Φ is defined y the numer of lef nodes tht mjority domintes Φ. Definition 6 (Copelnd evlution funtion) The evlution funtion Copelnd, mpping from n existing lef node Φ (Φ L) to [0, + ], is defined y: Copelnd (Φ) = #{Φ L : Φ m j Φ} In the se with Copelnd rule, the evlution vlue Copelnd of node vries during the serh (s some lef nodes re expnded nd new nodes re reted). Therefore, in eh itertion, we not only need to lulte the Copelnd vlue of the new reted nodes, ut lso need to updte the Copelnd vlue of other remining lef nodes. Theorem The evlution funtion Copelnd is not dmissile. Proof: or node Φ, Copelnd is equl to the numer of existing lef nodes tht domintes Φ. The domintion reltions etween nodes is sed on the gents BPAs on nodes: i {1,..., m} nd Φ, Φ L, Φ i Φ iff BPA i (Φ ) i BPA i (Φ). Moreover, Φ m j Φ if there is mjority numer of gents who prefer Φ over Φ. However, if Φ m j Φ, exept the se tht Φ is t level m, it does not neessrily men tht there must exist n lterntive in the sutree of Φ tht n mjority domintes Φ. or simple exmple, onsider three gents {1, 2, } nd two inry vriles X 1, X 2, D X 1 = {x 1, x 1 }, D X 2 = {x 2, x 2 }, the gents preferenes re s follows: gent 1: x 1 x 2 1 x 1 x 2 1 x 1 x 2 1 x 1 x 2 ; gent 2: x 1 x 2 2 x 1 x 2 2 x 1 x 2 2 x 1 x 2 ; gent : x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2. Initilly, there re only two nodes: Φ with pth ssignment x 1, nd Φ with pth ssignment x 1. Bsed on the gents preferenes, then Copelnd (Φ ) = 0 nd Copelnd (Φ) = 1 s oth gent 1 nd 2 prefer Φ over Φ. However, BPA 1 (Φ ) = x 1 x 2 nd BPA 2 (Φ ) = x 1 x 2. Neither x 1 x 2 or x 1 x 2 n mjority domintes the ny ompletion of ASSIGNMENT(Φ) (i.e., x 1 x 2 nd x 1 x 2 ). In ft, ording to the gents preferenes, the Copelnd sore for eh outome is s follows: s(x 1 x 2 ) = 2 (it is mjority dominted y oth x 1 x 2 nd x 1 x 2 ); s(x 1 x 2 ) = (it is mjority dominted y x 1 x 2, x 1 x 2 nd x 1 x 2 ); s( x 1 x 2 ) = 1 (it is mjority dominted y x 1 x 2 ) nd s( x 1 x 2 ) = 0. Consequently, x 1 x 2 Comp(Φ) ut Copelnd (Φ) > s( x 1 x 2 ). The evlution funtion Copelnd my overestimte the numer of lterntives tht dominted node, nd is not dmissile. Exmple (Cont.) We ontinue with the three gents preferenes in igure 1. In the tle elow we give pirwise omprison mtrix

5 1 =1 2 =2 =2 root =0 2 =0 =1 5 6 =1 igure. 2 =1 =2 Serh tree for Copelnd rule (Φ in olumn nd Φ in row). The vlue in the ell of olumn Φ row Φ is 1 iff Φ m j Φ, otherwise is -1. We denote h Copelnd s the Copelnd vlue of lef node in the h itertion. In eh itertion h, we lulte the h Copelnd vlue y only looking t the rows of the existing lef nodes: the h Copelnd (Φ) is the sum of those rows whih is equl to -1 in the olumn of Φ. There is ross in ell of the tle iff: i) Φ = Φ (ter-orner); or ii) Φ or Φ hs een expnded (is no longer lef node) in tht itertion. root =1 2 = 2 = 7 8 5= =1 = = = 9 10 =2 5 =2 5 igure. Serh tree for Minimx rule Definition 7 (Minimx evlution funtion) The evlution funtion Minimx, mpping from n existing lef node Φ (Φ L) to [0, + ], is defined y: Minimx (Φ) = mx Φ L, Φ ΦN(Φ, Φ ) Φ Φ Φ 1 Φ 2 Φ Φ Φ 5 Φ 6 Φ Φ 2-1 Copelnd Φ -1-1 Φ Copelnd Φ Φ Copelnd igure illustrtes the serh tree with this exmple (In the serh tree h Copelnd is written shortly s h ). In the 1 st itertions, Φ 1 nd Φ 2 re reted s the hildren of the root node. As Φ 2 m j Φ 1, 1 Copelnd (Φ 1) = 1 nd 1 Copelnd (Φ 2) = 0. The ordering in the fringe would e Φ 2 Φ 1. In the 2 nd itertion, the first node Φ 2 in the f ringe is popped out nd expnded. Two hild nodes Φ nd Φ re reted. We run pirwise omprison etween three existing lef nodes Φ 1, Φ nd Φ. As Φ m j Φ 1, Φ m j Φ 1, Φ m j Φ, 2 Copelnd (Φ 1) = 2, 2 Copelnd (Φ ) = 0, 2 Copelnd (Φ ) = 1. Therefore, the order of the nodes in the f ringe is Φ Φ Φ 1 nd Φ will e expnded in the next itertion. In the rd itertions, two hild node Φ 5 nd Φ 6 of Φ re reted. Copelnd (Φ 1) = 2, Copelnd (Φ ) = 2, Copelnd (Φ 5) = 1 nd Copelnd (Φ 6) = 1. The urrent f ringe is Φ 5 Φ 6 Φ 1 Φ. In the th itertion, Φ 5 is hosen for expnsion. As Φ 5 is omplete ssignment nd Φ 6 hs the sme Copelnd vlue s Φ 5, the orresponding lterntives of Φ 5 nd Φ 6 (ā nd ā ) will e put into outs. SC* returns outs s the finl hosen set of lterntives..2.2 Minimx Given profile P nd pir of lterntives x nd y, let N( x, y) denote the numer of gents tht rnk y hed of x in the profile P: N( x, y) = #{i n : y i x}. The Minimx sore s( x) of n lterntive x is defined y: s( x) = mx y X, y x N( x, y) A Minimx winner x is n lterntive tht minimizing s( x). Similr s the Copelnd rule, in eh itertion, we ondut pirwise omprison over the set of lef nodes L. or pir of lef nodes Φ nd Φ, we define N(Φ, Φ ) s the numer of gents tht prefer Φ over Φ: N(Φ, Φ ) = #{i n : Φ i Φ}. Then we n define the following evlution funtion of Minimx rule: Theorem The evlution funtion Minimx is not dmissile. Proof: or node Φ, Minimx is the mximum numer of defets when ompred with every other existing lef node, nd the numer of defets is ounted sed on the BPAs of the nodes. The resoning is similr to the se with Copelnd rule. or ny nodes Φ nd Φ, even if mjority numer of gents prefer node Φ to Φ, it does not neessrily men tht this mjority hs the sme BPA on Φ, i.e., it does not gurntee tht there exists t lest one lterntive tht is soilly (mjority) prefer to ny lterntive from the sutree of Φ. Applied the sme exmple in the proof for Copelnd, we will lso hve Minimx (Φ) > s( x 1 x 2 ) (Φ is node with pth ssignment x 1 nd x 1 x 2 Comp(Φ)). Consequently, Minimx my overestimte the numer of worst se defets of node, nd thus is not dmissile. Exmple (Cont.) We pply SC* to ompute the Minimx rule with our running exmple (igure 1). In the tle elow we give the numer of defets in pirwise omprison (Φ in olumn nd Φ in row) The vlue in the ell of olumn Φ row Φ is the numer of defets of Φ vs. Φ, i.e., the numer of gents tht prefer Φ over Φ (N(Φ, Φ )). The Minimx h vlue of node Φ in itertion h is then the mximum mong the rows of the urrent existing nodes in the olumn Φ. Similrly, there is ross in ell of the tle iff: Φ = Φ ; or Φ or Φ hs een expnded. Φ Φ Φ 1 Φ 2 Φ Φ Φ 5 Φ 6 Φ 7 Φ 8 Φ 9 Φ 10 Φ Φ 2 2 Minimx Φ 2 Φ Minimx Φ Φ Minimx Φ Φ Minimx Φ Φ Minimx

6 igure illustrtes the serh tree with this exmple (In the serh tree h Minimx is written shortly s h ). The serh proess is similr to the se with Copelnd rule. However, in the rd itertion, Minimx (Φ 1) = Minimx (Φ 5) = Minimx (Φ 6) = 2 (the minimum). Φ 1 is prtil ssignment, it will e given higher priority thn Φ 5 nd Φ 6 in the f ringe. Therefore, Φ 1 will e expnded in the th itertion. This proess ontinues, until Φ 9 is hosen for expnsion (speifies omplete ssignment). As, Φ 9, Φ 10, Φ 5 nd Φ 6 hve the sme Minimx vlue, SC* then returns four lterntives,, ā, nd ā (speified y Φ 9, Φ 10, Φ 5, Φ 6, respetively). 5 Experiment In the experiments, we ompre the proposed SC* pproh with two other lgorithms: (i) stndrd diret eletion method DirE. DirE runs diret eletion mong ll possile lterntives. It gurntees to find the winners of rule. (ii) sequentil voting lgorithm SeqV. SeqV runs sequentil issue-y-issue voting following rndom order over the set of vriles. When the gents vote for the vlues of remining vrile, they vote sed on the est possile lterntive with the vriles tht lredy ssigned vlue eing fixed. In these experiments, we fous on inry vriles nd onsider the representtion lnguge SLO SCPnet [5] to represent the gents preferenes. The SLOSCPnets re generted with topology orders over vriles tht follow norml distriution. The struture of eh gent s preferene network is ritrry yli, nd the gents re not required to hve ommon preferene strutures. or eh numer of vriles, we run 5000 rounds of experiments. Tle 1. Experimentl results with mx rule for rdinl preferenes Vr. Visited Time Su. R. Distne Nodes DirE SeqV SC* SeqV SeqV % % % With ggregtion rules for rdinl preferenes, we onsider = mx. Tle 1 shows the experimentl results with 5 to 15 vriles. It n e lerly seen tht SC* limits the numer of visited nodes nd the running time of SC* is redued y severl orders of mgnitude ompred to the DirE lgorithm when the numer of vriles is lrge. On the other hnd, the running time for SeqV lgorithm is the lest mong the three lgorithms. However, the perentge of ses it hooses winner is very low. The lst two olumns show the suess rte (Su. R.) nd the distne etween the winners nd the outome hosen y SeqV lgorithm. The distne is defined y the differene etween the soil welfre sores of the winner nd the outome hosen y SeqV lgorithm. When the numer of vriles is lrge, for instne 15 vriles, in less thn % ses of these experiments, the SeqV n find out winner. Also, the verge distne to the winners is quite lrge, in the se of 15 vriles. or the experiments with Condoret-onsistent rules, Tle 2 nd Tle shows the results with Copelnd nd Minimx rule, respetively. On the one hnd, SC* is muh fster thn the DirE lgorithm, whih eomes infesile when the numer of vriles is lrger thn 8. On the other hnd, SC* provides muh higher suess rte (Su. R.) nd smller distne to the winners thn the SeqV lgorithm. Here, the distne to Copelnd winner (resp. Minimx winner) is the differene of the Copelnd sores (resp. Minimx sores) etween the winners nd the outomes hosen y the lgorithms. Tle 2. Experimentl results with Copelnd rule Vr. Avg. PC Time Su. R. Distne Redution DirE SeqV SC* SeqV SC* SeqV SC* 20.% % 95.9% % % 80.2% % % 72.1% PC Redution: redution in the numer of pirwise omprisons. Tle. Experimentl results with Minimx rule Vr. Avg. PC Time Su. R. Distne Redution DirE SeqV SC* SeqV SC* SeqV SC* 1.9% % 97.5% % % 9.9% % % 85.% PC Redution: redution in the numer of pirwise omprisons. 6 Conlusion nd future work We hve studied the prolem of omintoril soil hoie in this pper. As the size of lterntive spe is huge in omintoril domins, we proposed very generl heuristi frmework SC* for omputing different ggregtion rules. SC* gurntees to hoose the winners of the rules for ggregting rdinl preferenes. In the ses with Condoret-onsistent rules, SC* hooses the lterntives tht re suffiiently lose to the winners. When ggregting rdinl preferenes, y onsidering eh gent s sore vlue s n ojetive, SC* proesses in wy similr to the heuristi lgorithm U* [15] for multi-ojetive optiml pth serhing in yli OR-grphs. However, there re signifint differenes etween the two: i) In the prolem of optiml pth serhing in OR-grph, U* is est-first serh lgorithm in whih the heuristi informtion of node is ssumed given. U* lgorithm ggregtes the given ojetive vlues (lled rewrd vetors) of the pth from the strt node to the urrent node nd the estimted rewrd vetors from the urrent node to the end (the ssumed given heuristi informtion) into utility vlue u to guide the serh. On the other hnd, our proposed SC* lgorithm is designed to ggregte multi-gent preferenes in omintoril domins. SC* lgorithm defines the dmissile heuristi sed on the est possile lterntive (BPA) of eh gent. Consequently, we hve proposed method to otin n dmissile heuristi vlue sed on the gents preferenes insted of ssuming tht heuristi funtion is given. ii) The deisions for node expnsion in the lgorithms U* nd SC* re different. U* lgorithm selets mong the nodes sed on funtion IE, whih lultes n upper ound utility mong ll possile options of node (e.g., possile next rs to tke or possile pths from the node to the end). On the other hnd, insted of hving to onsider ll possile options, the evlution funtion used y SC* (see Definition ) ggregtes the gents sores for their optimisti pths from the urrent node (i.e., the sores of their BPAs). Note tht for given node, lulting the BPA (i.e., optiml pth) for n individul gent is omputtionlly esy for yli preferene strutures. Hene, SC* essentilly does not onsider ll possile pths from the urrent node to the end node. In sted, it omputes the evlution vlue sed on the individul gents sores. iii) With preferenes in omintoril domins, vriles re interdependent, nd n individul gent s preferene (nd thus the olletive preferene) over the vlue of vrile my depend on the vlues ssigned to some other vriles. As result, different from the prolem disussed in [15], the ost (or rewrd vetor) for node is not fixed vetor. In omintoril soil hoie prolem, the evlution

7 vlue of node depends on its nestors on the serh tree s well s the vriles to e ssigned efore gol node is rehed. iv) inlly, the opertor used in [15] for ggregting rewrd vetors of multiple rs or su-pths is ssumed to e orderpreserving. On the other hnd, due to the interdependeny etween vriles in the ontext of preferene, it is not the se when ggregting preferenes in omintoril domins. SC* is sed on the gents optimisti evlutions of lterntives, i.e., est possile lterntives of the nodes. As n issue of future reserh, it would e interesting to further investigte the pessimisti evlutions, i.e., worst possile lterntives of nodes. A further extension of this work my tke into ount glol onstrints (whih mkes some lterntives not ville). Lst ut not lest, other wys to model dmissile evlution funtions for Condoret-onsistent rules re lso n importnt topi of our future reserh. [15] Chelse C. White, Brdley S. Stewrt, nd Roert L. Crrwy, Multiojetive, preferene-sed serh in yli OR-grphs, Europen Journl of Opertionl Reserh, 56(), 57 6, (1992). [16] Lirong Xi, Vinent Conitzer, nd Jérôme Lng, Voting on multittriute domins with yli preferentil dependenies, in AAAI 08: Proeedings of the 2rd ntionl onferene on Artifiil intelligene, pp AAAI Press, (2008). [17] Lirong Xi, Jérôme Lng, nd Mingsheng Ying, Strongly deomposle voting rules on multittriute domins, in AAAI 07: Proeedings of the 22nd ntionl onferene on Artifiil intelligene, pp AAAI Press, (2007). 7 Aknowledgement We would like to thnk the nonymous reviewers for fruitful disussions nd omments. This work ws prtilly supported y the ARC Disovery Grnts DP nd DP REERENCES [1] hiem Bhus nd Adm Grove, Grphil models for preferene nd utility, in Proeedings of the Proeedings of the Eleventh Conferene Annul Conferene on Unertinty in Artifiil Intelligene (UAI- 95), pp. 10, Sn rniso, CA, (1995). Morgn Kufmnn. [2] Crig Boutilier, Ronen I. Brfmn, Crmel Domshlk, Holger H. Hoos, nd Dvid Poole, CP-nets: tool for representing nd resoning with onditionl eteris prius preferene sttements, J. Artif. Int. Res., 21, , (erury 200). [] Sylvin Bouveret, Ulle Endriss, nd Jérôme Lng, Conditionl importne networks: A grphil lnguge for representing ordinl, monotoni preferenes over sets of goods, in IJCAI, pp , (2009). [] Steven J. Brms, D. Mr Kilgour, nd Willim S. Zwiker, The prdox of multiple eletions, Soil Choie nd Welfre, 15, , (1998). [5] Crmel Domshlk, Steven Dvid Prestwih, rnes Rossi, Kristen Brent Venle, nd Toy Wlsh, Hrd nd soft onstrints for resoning out qulittive onditionl preferenes, J. Heuristis, 12(- 5), , (2006). [6] Peter Hrt, Nils Nilsson, nd Bertrm Rphel, A forml sis for the heuristi determintion of minimum ost pths, IEEE Trnstions on Systems Siene nd Cyernetis, (2), , (erury 1968). [7] Piero L Mur nd Yov Shohm, Expeted utility networks, in Proeedings of the ifteenth onferene on Unertinty in rtifiil intelligene, UAI 99, pp. 66 7, Sn rniso, CA, USA, (1999). Morgn Kufmnn Pulishers In. [8] Celine Lfge nd Jérôme Lng, Logil representtion of preferenes for group deision mking, in KR, pp , (2000). [9] Jérôme Lng, rom preferene representtion to omintoril vote, in KR, pp , (2002). [10] Jérôme Lng, Logil preferene representtion nd omintoril vote, Annls of Mthemtis nd Artifiil Intelligene, 2, 7 71, (Septemer 200). [11] Jérôme Lng nd Lirong Xi, Sequentil omposition of voting rules in multi-issue domins, Mthemtil Soil Sienes, 57(), 0 2, (My 2009). [12] Minyi Li, Quo Bo Vo, nd Ryszrd Kowlzyk, Mjority-rulesed preferene ggregtion on multi-ttriute domins with CP-nets, in Proeedings of the 10th Interntionl Conferene on Autonomous Agents nd Multigent Systems, volume 1, pp , Tipei, Tiwn, (2011). Rihlnd, SC. Interntionl oundtion for Autonomous Agents nd Multigent Systems. [1] Hervi Moulin, Axioms of Coopertive Deision Mking (Eonometri Soiety Monogrphs), Cmridge University Press, July [1] Jude Perl, Heuristis: intelligent serh strtegies for omputer prolem solving, Addison-Wesley Longmn Pulishing Co., In., Boston, MA, USA, 198.