On the Axiomatic Foundations of Ranking Systems

Transcription

1 On the Axiomti Fountions of Rnking Systems Alon Altmn n Moshe Tennenholtz Fulty of Inustril Engineering n Mngement Tehnion Isrel Institute of Tehnology Hif Isrel Astrt Resoning out gent preferenes on set of lterntives, n the ggregtion of suh preferenes into some soil rnking is funmentl issue in resoning out multi-gent systems. When the set of gents n the set of lterntives oinie, we get the rnking systems setting. A fmous type of rnking systems re pge rnking systems in the ontext of serh engines. In this pper we present n extensive xiomti stuy of rnking systems. In prtiulr, we onsier two funmentl xioms: Trnsitivity, n Rnke Inepenene of Irrelevnt Alterntives. Surprisingly, we fin tht there is no generl soil rnking rule tht stisfies oth requirements. Furthermore, we show tht our impossiility result hols uner vrious restritions on the lss of rnking prolems onsiere. Eh of these xioms n e iniviully stisfie. Moreover, we show omplete xiomtiztion of pprovl voting using one of these xioms. 1 Introution The rnking of gents se on other gents input is funmentl to multi-gent systems (see e.g. [Resnik et l., 2000]). Moreover, it hs eome entrl ingreient of vriety of Internet sites, where perhps the most fmous exmples re Google s PgeRnk lgorithm[pge et l., 1998] n eby s reputtion system[resnik n Zekhuser, 2001]. This si prolem introues new soil hoie moel. In the lssil theory of soil hoie, s mnifeste y Arrow[1963], set of gents/voters is lle to rnk set of lterntives. Given the gents input, i.e. the gents iniviul rnkings, soil rnking of the lterntives is generte. The theory stuies esire properties of the ggregtion of gents rnkings into soil rnking. In prtiulr, Arrow s elerte impossiility theorem[arrow, 1963] shows tht there is no ggregtion rule tht stisfies some miniml requirements, while y relxing ny of these requirements pproprite soil ggregtion rules n e efine. The novel feture of the rnking systems setting is tht the set of gents n the set of lterntives oinie. Therefore, in suh setting one my nee to onsier the trnsitive effets of voting. For exmple, if gent reports on the importne of (i.e. votes for) gent then this my influene the reiility of report y on the importne of gent ; these iniret effets shoul e onsiere when we wish to ggregte the informtion provie y the gents into soil rnking. Notie tht nturl interprettion/pplition of this setting is the rnking of Internet pges. In this se, the set of gents represents the set of Internet pges, n the links from pge p to set of pges Q n e viewe s two-level rnking where gents in Q re preferre y gent(pge) p to the gents(pges) whih re not in Q. The prolem of fining n pproprite soil rnking in this se is in ft the prolem of (glol) pge rnking. Prtiulr pprohes for otining useful pge rnking hve een implemente y serh engines suh s Google[Pge et l., 1998]. The theory of soil hoie onsists of two omplementry xiomti perspetives: The esriptive perspetive: given prtiulr rule r for the ggregtion of iniviul rnkings into soil rnking, fin set of xioms tht re soun n omplete for r. Tht is, fin set of requirements tht r stisfies; moreover, every soil ggregtion rule tht stisfies these requirements shoul oinie with r. A result showing suh n xiomtiztion is terme representtion theorem n it ptures the ext essene of (n ssumptions ehin) the use of the prtiulr rule. The normtive perspetive: evise set of requirements tht soil ggregtion rule shoul stisfy, n try to fin whether there is soil ggregtion rule tht stisfies these requirements. Mny efforts hve een investe in the esriptive pproh in the frmework of the lssil theory of soil hoie. In tht setting, representtion theorems hve een presente to mjor voting rules suh s the mjority rule[my, 1952]. Reently, we hve suessfully pplie the esriptive perspetive in the ontext of rnking systems y proviing representtion theorem[altmn n Tennenholtz, 2005] for the wellknown PgeRnk lgorithm [Pge et l., 1998], whih is the sis of Google s serh tehnology. An exellent exmple for the normtive perspetive is Arrow s impossiility theorem mentione ove. In [Tennenholtz, 2004], we presente some preliminry results for rnking systems where the set of voters n the set of lterntives oinie. However, the xioms presente in tht work onsist

2 of severl very strong requirements whih nturlly le to n impossiility result. In this pper we provie n extensive stuy of rnking systems. We introue two funmentl xioms. One of these xioms ptures the trnsitive effets of voting in rnking systems, n the other pts Arrow s well-known inepenene of irrelevnt lterntives(iia) xiom to the ontext of rnking systems. Surprisingly, we fin tht no generl rnking system n simultneously stisfy these two xioms! We further show tht our impossiility result hols uner vrious restritions on the lss of rnking prolems onsiere. On the other hn, we show tht eh of these xioms n e iniviully stisfie. Moreover, we use our IIA xiom to present positive result in the form of representtion theorem for the well-known pprovl voting rnking system, whih rnks the gents se on the numer of votes reeive. This xiomtiztion shows tht when ignoring trnsitive effets, there is only one rnking system tht stisfies our IIA xiom. This pper is struture s follows: Setion 2 formlly efines our setting n the notion of rnking systems. Setions 3 n 4 introue our xioms of Trnsitivity n Rnke Inepenene of Irrelevnt Alterntives respetively. Our min impossiility result is presente in Setion 5, n further strengthene in Setion 6. Our positive result, in the form of n xiomtiztion for the Approvl Voting rnking system in presente in Setion 7. Finlly, some onluing remrks re given in Setion 8. 2 Rnking Systems Before esriing our results regring rnking systems, we must first formlly efine wht we men y the wors rnking system in terms of grphs n liner orerings: Definition 2.1. Let A e some set. A reltion R A A is lle n orering on A if it is reflexive, trnsitive, n omplete. Let L(A) enote the set of orerings on A. Nottion 2.2. Let e n orering, then is the equlity preite of, n is the strit orer inue y. Formlly, if n only if n ; n if n only if ut not. Given the ove we n efine wht rnking system is: Definition 2.3. Let G V e the set of ll grphs with vertex set V. A rnking system F is funtionl tht for every finite vertex set V mps grphsg G V to n orering F G L(V ). If F is prtil funtion then it is lle prtil rnking system, otherwise it is lle generl rnking system. One n view this setting s vrition/extension of the lssil theory of soil hoie s moele y [Arrow, 1963]. The rnking systems setting iffers in two min properties. First, in this setting we ssume tht the set of voters n the set of lterntives oinie, n seon, we llow gents only two levels of preferene over the lterntives, s oppose to Arrow s setting where gents oul rnk lterntives ritrrily. 3 Trnsitivity A si property one woul ssume of rnking systems is tht if n gent s voters re rnke higher thn those of gent, Figure 1: Exmple of Trnsitivity then gent shoul e rnke higher thn gent. This notion is formlly pture elow: Definition 3.1. Let F e rnking system. We sy tht F stisfies strong trnsitivity if for ll grphs G = (V, E) n for ll verties v 1, v 2 V : Assume there is 1-1 mpping f : P(v 1 ) P(v 2 ) s.t. for ll v P(v 1 ): v f(v). Further ssume tht either f is not onto or for some v P(v 1 ): v f(v). Then, v 1 v 2. Consier for exmple the grph G in Figure 1 n ny rnking system F tht stisfies strong trnsitivity. F must rnk vertex elow ll other verties, s it hs no preeessors, unlike ll other verties. If we ssume tht F G, then y strong trnsitivity we must onlue tht F G s well. But then we must onlue tht F G (s s preeessor is rnke lower thn s preeessor, n hs n itionl preeessor ), whih les to ontrition. Given F G, gin y trnsitivity, we must onlue tht F G, so the only rnking for the grph G tht stisfies strong trnsitivity is F G F G F G. In [Tennenholtz, 2004], we hve suggeste n lgorithm tht efines rnking system tht stisfies strong trnsitivity y itertively refining n orering of the verties. Note tht the PgeRnk rnking system efine in [Pge et l., 1998] oes not stisfy strong trnsitivity. This is ue to the ft tht PgeRnk reues the weight of links (or votes) from noes whih hve higher out-egree. Thus, ssuming Yhoo! n Mirosoft re eqully rnke, link from Yhoo! mens less thn link from Mirosoft, euse Yhoo! links to more externl pges thn oes Mirosoft. Noting this ft, we n weken the efinition of trnsitivity to require tht the preeessors of the ompre gents hve n equl outegree: Definition 3.2. Let F e rnking system. We sy tht F stisfies wek trnsitivity if for ll grphs G = (V, E) n for ll verties v 1, v 2 V : Assume there is 1-1 mpping f : P(v 1 ) P(v 2 ) s.t. for ll v P(v 1 ): v f(v) n S(v) = S(f(v)). Further ssume tht either f is not onto or for some v P(v 1 ): v f(v). Then, v 1 v 2. Inee, n ielize version of the PgeRnk rnking system efine on strongly onnete grphs stisfies this wekene version of trnsitivity. Furthermore, the result in the exmple ove oes not hnge when we onsier wek trnsitivity in ple of strong trnsitivity. 4 Rnke Inepenene of Irrelevnt Alterntives A stnr ssumption in soil hoie settings is tht n gent s reltive rnk shoul only epen on (some property of) their immeite preeessors. Suh xioms re usully lle inepenene of irrelevnt lterntives(iia) xioms.

3 e f 1 2 Figure 2: An exmple of RIIA. 1 In our setting, we require the reltive rnking of two gents must only epen on the pirwise omprisons of the rnks of their preeessors, n not on their ientity or rinl vlue. Our IIA xiom, lle rnke IIA, iffers from the one suggeste y [Arrow, 1963] in the ft tht we o not onsier the ientity of the voters, ut rther their reltive rnk. For exmple, onsier the grph in Figure 2. Furthermore, ssume rnking system F hs rnke the verties of this grph s following: e f. Now look t the omprison etween n. s preeessors, n, re oth rnke eqully, n oth rnke lower thn s preeessor f. This is lso true when onsiering e n f e s preeessors n re oth rnke eqully, n oth rnke lower thn f s preeessor e. Therefore, if we gree with rnke IIA, the reltion etween n, n the reltion etween e n f must e the sme, whih inee it is oth n e f. However, this sme sitution lso ours when ompring n f ( s preeessors n re eqully rnke n rnke lower thn f s preeessor e), ut in this se f. So, we n onlue tht the rnking system F whih proue these rnkings oes not stisfy rnke IIA. To formlly efine this onition, one must onsier ll possiilities of ompring two noes in grph se only on orinl omprisons of their preeessors. We ll these possiilities omprison profiles: Definition 4.1. A omprison profile is pir, where = ( 1,..., n ), = ( 1,..., m ), 1,..., n, 1,..., m N, 1 2 n, n 1 2 m. Let P e the set of ll suh profiles. A rnking system F, grph G = (V, E), n pir of verties v 1, v 2 V re si to stisfy suh omprison profile, if there exist 1-1 mppings f 1 : P(v 1 ) {1...n} n f 2 : P(v 2 ) {1...m} suh tht given f : ({1} P(v 1 )) ({2} P(v 2 )) N efine s: f(1, v) = f1(v) f(2, u) = f2(u), f(i, x) f(j, y) x F G y for ll (i, x), (j, y) ({1} P(v 1 )) ({2} P(v 2 )). For exmple, in the exmple onsiere ove, ll of the pirs (, ), (, f), n (e, f) stisfy the omprison profile (1, 1), (2). We now require tht for every suh profile the rnking system rnks the noes onsistently: Definition 4.2. Let F e rnking system. We sy tht F stisfies rnke inepenene of irrelevnt lterntives (RIIA) if there exists mpping f : P {0, 1} suh tht for every grph G = (V, E) n for every pir of verties v 1, v 2 V () Grph G 1 2 () Grph G 2 Figure 3: Grphs for the proof of Theorem 5.1 n for every omprison profile p P tht v 1 n v 2 stisfy, v 1 F G v 2 f(p) = 1. As RIIA is n inepenene property, the rnking system F =, tht rnks ll gents eqully, stisfies RIIA. A more interesting rnking system tht stisfies RIIA is the pprovl voting rnking system, efine elow. Definition 4.3. The pprovl voting rnking system AV is the rnking system efine y: v 1 AV G v 2 P(v 1 ) P(v 2 ) A full xiomtiztion of the pprovl voting rnking system is given in setion 7. 5 Impossiility Our min result illustrtes the impossiility of stisfying (wek) trnsitivity n RIIA simultneously. Theorem 5.1. There is no generl rnking system tht stisfies wek trnsitivity n RIIA. Proof. Assume for ontrition tht there exists rnking system F tht stisfies wek trnsitivity n RIIA. Consier first the grph G 1 in Figure 3(). First, note tht 1 n 2 stisfy some omprison profile p = ((x, y), (x, y)) euse they hve ientil preeessors. Thus, y RIIA, 1 F G F G 1 1, n therefore 1 F G 1 2. By wek trnsitivity, it is esy to see tht F G 1 1 n F G 1. If we ssume F G 1 1, then y wek trnsitivity, 1 F G 1 whih ontrits our ssumption. So we onlue tht F G 1 1 F G 1. Now onsier the grph G 2 in Figure 3(). Agin, y RIIA, 1 F G 2 2. By wek trnsitivity, it is esy to see tht 1 F G 2 n F G 2. If we ssume 1 F G 2, then y wek trnsitivity, F G 2 1 whih ontrits our ssumption. So we onlue tht F G 2 1 F G 2. Consier the omprison profile p = ((1, 3), (2, 2)). Given F, 1 n stisfy p in G 1 (euse F G 1 1 F G 1 2 F G 1 ) n in G 2 (euse F G 2 1 F G 2 2 F G 2 ). Thus, y

4 RIIA, 1 F G 1 1 F G 2, whih is ontrition to the ft tht 1 F G 1 ut F G 2 1. x2 y This result is quite surprise, s it mens tht every resonle efinition of rnking system must either onsier rinl vlues for noes n/or eges (like [Pge et l., 1998]), or operte orinlly on glol sle (like [Altmn n Tennenholtz, 2005]). x1 x2 () Grph G 1 x1 () Grph G 2 6 Relxing Generlity A hien ssumption in our impossiility result is the ft tht we onsiere only generl rnking systems. In this setion we nlyze severl speil lsses of grphs tht relte to ommon rnking senrios. 6.1 Smll Grphs A nturl limittion on preferene grph is p on the numer of verties (gents) tht prtiipte in the rnking. Inee, when there re three or less gents involve in the rnking, strong trnsitivity n RIIA n e simultneously stisfie. An pproprite rnking lgorithm for this se is the one we suggeste in [Tennenholtz, 2004]. However, when there re four or more gents, strong trnsitivity n RIIA nnot e simultneously stisfie (the proof is similr to tht of Theorem 5.1, ut with vertex remove in oth grphs). When five or more gents re involve, even wek trnsitivity n RIIA nnot e simultneously stisfie, s implie y the proof of Theorem Single Vote Setting Another nturl limittion on the omin of grphs tht we might e intereste in is the restrition of eh gent(vertex) to extly one vote(suessor). For exmple, in the voting prigm this oul e viewe s setting where every gent votes for extly one gent. The following proposition shows tht even in this simple setting wek trnsitivity n RIIA nnot e simultneously stisfie. Proposition 6.1. Let G 1 e the set of ll grphs G = (V, E) suh tht S(v) = 1 for ll v V. There is no prtil rnking system over G 1 tht stisfies wek trnsitivity n RIIA. Proof. Assume for ontrition tht there is prtil rnking system F over G 1 tht stisfies wek trnsitivity n RIIA. Let f : P {0, 1} e the mpping from the efinition of RIIA for F. Let G 1 G 1 e the grph in Figure 4. By wek trnsitivity, x 1 F G 1 x 2 F G 1 F G 1. (, ) stisfies the omprison profile (1, 1, 2), (3), so we must hve f (1, 1, 2), (3) = 0. Now let G 2 G 1 e the grph in Figure 4. By wek trnsitivity x 1 F G 2 x 2 F G 2 y F G 2 F G 2. (, ) stisfies the omprison profile (2, 3), (1, 4), so we must hve f (2, 3), (1, 4) = 0. Let G 3 G 1 e the grph in Figure 4. By wek trnsitivity it is esy to see tht x 1 F G 3 F G 3 x 7 F G 3 y 1 F G 3 y 2 F G 3 F G 3. Furthermore, y wek trnsitivity we onlue tht F G 3 n F G 3 from F G 3 ; n x4 y2 x5 x6 x7 x3 x1 x2 y1 () Grph G 3 Figure 4: Grphs from the proof of proposition 6.1 y 1 F G 3 from x 3 F G 3. Now onsier the vertex pir (, ). We hve shown tht x 1 F G 3 x 2 F G 3 y 1 F G 3. So, (, ) stisfies the omprison profile (1, 1, 2), (3), thus y RIIA F G 3. Now onsier the vertex pir (, ). We hve lrey shown tht F G 3 F G 3 F G 3. So, (, ) stisfies the omprison profile (2, 3), (1, 4), thus y RIIA F G 3. However, we hve lrey shown tht F G 3 ontrition. Thus, the rnking system F nnot exist. 6.3 Biprtite Setting In the worl of reputtion systems[resnik et l., 2000], we frequently oserve istintion etween two types of gents suh tht eh type of gent only rnks gents of the other type. For exmple uyers only intert with sellers n vie vers. This type of limittion is pture y requiring the preferene grphs to e iprtite, s efine elow. Definition 6.2. A grph G = (V, E) is lle iprtite if there exist V 1, V 2 suh tht V = V 1 V 2, V 1 V 2 =, n E (V 1 V 2 ) (V 2 V 1 ). Let G B e the set of ll iprtite grphs. Our impossiility result extens to the limite omin of iprtite grphs. Proposition 6.3. There is no prtil rnking system over G B G 1 tht stisfies wek trnsitivity n RIIA. 6.4 Strongly Connete Grphs The well-known PgeRnk rnking system is (ielly) efine on the set of strongly onnete grphs. Tht is, the

5 set of grphs where there exists irete pth etween ny two verties. Let us enote the set of ll strongly onnete grphs G SC. The following proposition extens our impossiility result to strongly onnete grphs. Proposition 6.4. There is no prtil rnking system over G SC. 7 Axiomtiztion of Approvl Voting In the previous setions we hve seen mostly negtive results whih rise when trying to ommote (wek) trnsitivity n RIIA. We hve shown tht lthough eh of the xioms n e stisfie sepertely, there exists no generl rnking system tht stisfies oth xioms. We hve previously shown[tennenholtz, 2004] nontrivil rnking system tht stisfies (wek) trnsitivity, ut hve not yet shown suh rnking system for RIIA. In this setion we provie representtion theorem for rnking system tht stisfies RIIA ut not wek trnsitivity the pprovl voting rnking system. This system rnks the gents se on the numer of votes eh gent reeive, with no regr to the rnk of the voters. The xiomtiztion we provie in this setion shows the power of RIIA, s it shows tht there exists only one (interesting) rnking system tht stisfies it without introuing trnsitive effets. In orer to speify our xiomtiztion, rell the following lssil efinitions from the theory of soil hoie: The positive response xiom essentilly mens tht if n gent reeives itionl votes, its rnk must improve: Definition 7.1. Let F e rnking system. F stisfies positive response if for ll grphs G = (V, E) n for ll (v 1, v 2 ) (V V ) \ E, n for ll v 3 V : Let G = (V, E (v 1, v 2 )). If v 3 F G v 2, then v 3 F G v 2. The nonymity n neutrlity xioms men tht the nmes of the voters n lterntives respetively o not mtter for the rnking: Definition 7.2. A rnking system F stisfies nonymity if for ll G = (V, E), for ll permuttions π : V V, n for ll v 1, v 2 V : Let E = {(π(v 1 ), v 2 ) (v 1, v 2 ) E}. Then, v 1 F (V,E) v 2 v 1 F (V,E ) v 2. Definition 7.3. A rnking system F stisfies neutrlity if for ll G = (V, E), for ll permuttions π : V V, n for ll v 1, v 2 V : Let E = {(v 1, π(v 2 )) (v 1, v 2 ) E}. Then, v 1 F (V,E) v 2 v 1 F (V,E ) v 2. Arrow s lssil Inepenene of Irrelevnt Alterntives xiom requires tht the reltive rnk of two gents e epennt only on the set of gents tht preferre one over the other. Definition 7.4. A rnking system F stisfies Arrow s Inepenene of Irrelevnt Alterntives (AIIA) if for ll G = (V, E), for ll G = (V, E ), n for ll v 1, v 2 V : Let P G (v 1 ) \ P G (v 2 ) = P G (v 1 ) \ P G (v 2 ) n P G (v 2 ) \ P G (v 1 ) = P G (v 2 )\P G (v 1 ). Then, v 1 F G v 2 v 1 F G v 2. Our representtion theorem sttes tht together with positive response n RIIA, ny one of the three inepenene onitions ove (nonymity, neutrlity, n AIIA) re essentil n suffiient for rnking system eing AV 1. In ition, we show tht s in the lssil soil hoie setting when only onsiering two-level preferenes, positive response, nonymity, neutrlity, n AIIA re n essentil n suffiient representtion of pprovl voting. This result extens the well known xiomtiztion of the mjority rule ue to [My, 1952]: Proposition 7.5. (My s Theorem) A soil welfre funtionl over two lterntives is mjority soil welfre funtionl if n only if it stisfies nonymity, neutrlity, n positive response. We n now formlly stte our theorem: Theorem 7.6. Let F e generl rnking system. Then, the following sttements re equivlent: 1. F is the pprovl voting rnking system(f = AV ) 2. F stisfies positive response, nonymity, neutrlity, n AIIA 3. F stisfies positive response, RIIA, n either one of nonymity, neutrlity, n AIIA Proof. (Sketh) It is esy to see tht AV stisfies positive response, RIIA, nonymity, neutrlity, n AIIA. It remins to show tht (2) n (3) entil (1) ove. To prove (2) entils (1), ssume tht F stisfies positive response, nonymity, neutrlity, n AIIA. Let G = (V, E) e some grph n let v 1, v 2 V e some gents. By AIIA, the reltive rnking of v 1 n v 2 epens only on the sets P G (v 1 ) \ P G (v 2 ) n P G (v 2 ) \ P G (v 1 ). We hve now nrrowe our onsiertion to set of gents with preferenes over two lterntives, so we n pply Proposition 7.5 to omplete our proof. To prove (3) entils (1), ssume tht F stisfies positive response, RIIA n either nonymity or neutrlity or AIIA. As F stisfies RIIA we n limit our isussion to omprison profiles. Let f : P {0, 1} e the funtion from the efinition of RIIA. We will use the nottion to men f, = 1, to men f, = 0, n to men n. By the efinition of RIIA, it is esy to see tht for ll. By positive response it is lso esy to see tht (1, 1,..., 1) (1, 1,..., 1) iff n m. Let P = }{{}}{{} n m ( 1,..., n ), ( 1,..., m ) e omprison profile. Let G = (V, E) e the following grph : V = {x 1,..., x mx{n, m}} {v 1,..., v n, v 1,...,v n, v} {u 1,..., u m, u 1,..., u m, u} E = {(x i, v j ) i j } {(x i, u j ) i j } {(v i, v) i = 1,...,n} {(u i, u) i = 1,...,m}. 1 In ft, n even weker onition of eoupling, tht in essene llows us to permute the grph struture while keeping the eges nmes is suffiient in this se.

6 It is esy to see tht in the grph G, v n u stisfy the profile P. Let π e the following permuttion: v i x = v i v i x = v i π(x) = u i x = u i u i x = u i x Otherwise. The reminer of the proof epens on whih itionl xiom F stisfies: If F stisfies nonymity, let E = {(π(x), y) (x, y) E}. Note tht in the grph (V, E ) v n u stisfy the profile (1, 1,...,1), (1, 1,...,1), n thus v }{{}}{{} F (V,E ) n m u n m. By nonymity, u F (V,E) v u F (V,E ) v, thus proving tht f(p) = 1 n m for n ritrry omprison profile P, n thus F = AV. If F stisfies neutrlity, let E = {(x, π(y)) (x, y) E}. Note tht in the grph (V, E ) v n u stisfy the profile (1, 1,...,1), (1, 1,...,1), n thus v }{{}}{{} F (V,E ) n m u n m. By neutrlity, u F (V,E) v u F (V,E ) v, gin showing tht f(p) = 1 n m for n ritrry omprison profile P, n thus F = AV. If F stisfies AIIA, let E = {(x, π(y)) (x, y) E} s efore. So, lso v F (V,E ) u n m. Note tht P G (v) = P (V,E )(v) n P G (u) = P (V,E )(u), so y AIIA, u F (V,E) v u F (V,E ) v, n thus s efore, F = AV. 8 Conluing Remrks Resoning out preferenes n preferene ggregtion is funmentl tsk in resoning out multi-gent systems (see e.g. [Boutilier et l., 2004; Conitzer n Snholm, 2002; LMur n Shohm, 1998]). A typil instne of preferene ggregtion is the setting of rnking systems. Rnking systems re funmentl ingreients of some of the most fmous tools/tehniques in the Internet (e.g. Google s pge rnk n eby s reputtion systems, mong mny others). Our im in this pper ws to tret rnking systems from n xiomti perspetive. The lssil theory of soil hoie ly the fountions to lrge prt of the rigorous work on multi-gent systems. Inee, the most lssil results in the theory of mehnism esign, suh s the Gir- Stterthwite Theorem (see [Ms-Colell et l., 1995]) re pplitions of the theory of soil hoie. Moreover, previous work in AI hs employe the theory of soil hoie for otining fountions for resoning tsks [Doyle n Wellmn, 1989] n multi-gent oorintion [Kfir-Dhv n Tennenholtz, 1996]. It is however interesting to note tht rnking systems suggest novel n new type of theory of soil hoie. We see this point s espeilly ttrtive, n s min reson for onentrting on the stuy of the xiomti fountions of rnking systems. In this pper we ientifie two funmentl xioms for rnking systems, n onute si xiomti stuy of suh systems. In prtiulr, we presente surprising impossiility results, n representtion theorem for the wellknown pprovl voting sheme. Referenes [Altmn n Tennenholtz, 2005] Alon Altmn n Moshe Tennenholtz. Rnking systems: The pgernk xioms. In Proeeings of the 6th ACM onferene on Eletroni ommere (EC-05), [Arrow, 1963] K.J. Arrow. Soil Choie n Iniviul Vlues (2n E.). Yle University Press, [Boutilier et l., 2004] Crig Boutilier, Ronen I. Brfmn, Crmel Domshlk, Holger H. Hoos, n Dvi Poole. Cpnets: A tool for representing n resoning with onitionl eteris prius preferene sttements. J. Artif. Intell. Res. (JAIR), 21: , [Conitzer n Snholm, 2002] V. Conitzer n T. Snholm. Complexity of mehnism esign. In Proeeings of the 18th onferene on unertinity in Artifiil Intelligene (UAI-02), pges , [Doyle n Wellmn, 1989] J. Doyle n M.P. Wellmn. Impeiments to Universl Preferene-Bse Defult Theories. In Proeeings of the 1st onferene on priniples of knowlege representtion n resoning, [Kfir-Dhv n Tennenholtz, 1996] No E. Kfir-Dhv n Moshe Tennenholtz. Multi-Agent Belief Revision. In Proeeings of the 6th onferene on theoretil spets of rtionlity n knowlege (TARK), [LMur n Shohm, 1998] P. LMur n Y. Shohm. Conitionl, Hierrhil Multi-Agent Preferenes. In Proeeings of Theoretil Aspets of Rtionlity n Knowlege, pges , [Ms-Colell et l., 1995] A. Ms-Colell, M.D. Whinston, n J.R. Green. Miroeonomi Theory. Oxfor University Press, [My, 1952] Kenneth O. My. A set of inepenent, neessry n suffiient onitions for simple mjority eision. Eonometri, 20(4):680 84, [Pge et l., 1998] L. Pge, S. Brin, R. Motwni, n T. Winogr. The pgernk ittion rnking: Bringing orer to the we. Tehnil Report, Stnfor University, [Resnik n Zekhuser, 2001] P. Resnik n R. Zekhuser. Trust mong strngers in internet trnstions: Empiril nlysis of ey s reputtion system. Working Pper for the NBER workshop on empiril stuies of eletroni ommere, Jnury [Resnik et l., 2000] P. Resnik, R. Zekhuser, R. Friemn, n E. Kuwr. Reputtion systems. Communitions of the ACM, 43(12):45 48, [Tennenholtz, 2004] M. Tennenholtz. Reputtion systems: An xiomti pproh. In Proeeings of the 20th onferene on unertinity in Artifiil Intelligene (UAI-04), 2004.