Reasoning about Social Choice Functions

Transcription

1 Resoning bout Socil Choice Functions Nicols Troqurd Wiebe vn der Hoek Michel Wooldridge rxiv: v1 [cs.ma] 16 Feb 2011 Computer Science Deprtment, University of Liverpool, UK Abstrct We introduce logic specificlly designed to support resoning bout socil choice functions. The logic includes opertors to cpture strtegic bility, nd opertors to cpture gent preferences. We estblish correspondence between formule in the logic nd properties of socil choice functions, nd show tht the logic is expressively complete with respect to socil choice functions, i.e., tht every socil choice function cn be chrcterised s formul of the logic. We prove tht the logic is decidble, nd give complete xiomtiztion. To demonstrte the vlue of the logic, we show in prticulr how it cn be pplied to the problem of determining whether socil choice function is strtegy-proof. 1 Introduction Socil choice theory is concerned with collective decision mking in situtions where the preferences of the decision mkers my differ [3]. Socil choice theorists hve developed rnge of procedures, such s voting protocols, to support such collective decision mking, nd hve developed rnge of criteri with which to chrcterise the properties of such procedures. Such criteri re usully expressed xiomticlly, nd mjor concern of socil choice theory is to study the extent to which decision mking procedures do or do not stisfy these xioms [8, 2, 7, 12]. In short, the im of the present pper is to develop logic tht is explicitly intended for resoning bout socil choice procedures. We focus on socil choice functions, clss of socil choice procedures tht select single socil outcome s function of individul preferences. Voting procedures of the type used in politicl elections throughout the democrtic world re perhps the best-known exmples of socil choice functions. A voting procedure determines the winner of n election s function of the votes cst; votes cn be understood s n expression of voter preferences. One interesting issue tht rises in voting procedures is the extent to which voters re incentivised to truthfully report their preferences when voting. For exmple, suppose we hve two voters, 1 nd 2, who vote mong three cndidtes, x, y, nd z for role tht is currently filled by x. The voting procedure used in this exmple sys tht, if there is unnimously preferred cndidte, then tht will be chosen, otherwise the cndidte x remins. Suppose the true preferences of 1 re given by z < 1 x < 1 y nd those of 2 re x < 2 y < 2 z. If the socil choice function ws presented with these true preferences, cndidte x would be chosen (since there is no consensus). However, if 1

2 voter 2 would insted clim his preferences were x < 2 z < 2 y while 1 reveled its true preferences, then 2 would be better off, since y would be chosen, rther thn x, nd gent 2 prefers y over x. This issue suggests the following problem: Cn we design voting procedure tht is immune to such misrepresenttion, i.e., in which voter cn never do ny better thn by truthfully reporting its preferences? The term strtegy proof is used to refer to such voting procedures. In fct, fundmentl results in socil choice theory tell us tht there re severe limits to the development of strtegy-proof voting procedures [7, 12], nd for this reson, developing nd nlysing socil choice procedures is lively nd highly ctive reserch re. The long-term im of our work is to develop forml tools to ssist in the nlysis nd design of socil choice procedures. In prticulr, we hope to develop techniques tht will permit the utomted nlysis of socil choice procedures. To this end, we im to develop logics tht llow us to formlly express the properties of socil choice procedures, such tht these lnguges my be utomticlly processed. Our view is tht logic cn provide powerful tool for the nlysis of socil choice procedures [11, 16]. Such logics cn be used s query lnguges for socil choice procedures: given some property P of socil choice procedure, we im to be ble to encode the property P s n expression ρ P of our lnguge, which we then pose s query to n utomted nlysis system. Working towrds the long-term gol, the present pper presents logic for resoning bout socil choice procedures, nd in prticulr, for nlysing strtegy proofness. The reminder of the pper is structured s follows. In Section 2 we recll the min concepts from gme theory nd socil choice theory tht we use throughout the pper. We then introduce our logic in Section 3. The logic is bsiclly modl logic [5], which derives inspirtion from the Colition Logic of Propositionl Control (CL-PC) [15]. The ltter logic includes opertors to cpture strtegic bility. We extend this with opertors for cpturing gent preferences. The bsic ide is to model n gent s preferences vi tomic propositions: proposition p i x>y will be used to represent the fct tht gent i hs reported tht he prefers outcome x t lest s much s outcome y. The strtegic bilities of gents re cptured using CL-PC-like opertor: n gent cn choose ny ssignment of vlues for its preference vribles tht corresponds to preference ordering. After presenting the syntx nd semntics of the logic, we show how the logic cn be used to chrcterise socil choice functions, nd show tht the logic is expressively complete with respect to socil choice functions, i.e., tht every socil choice function cn be chrcterised s formul of the logic. We give complete xiomtiztion for the logic. To demonstrte the vlue of the logic, in Section 4 we formlise some properties of socil choice functions nd in prticulr, we show how it cn be pplied to the problem of determining whether socil choice function is strtegy-proof. We conclude in Section 5. 2 Bckground In this section, we present the bsic definitions of gme theory nd socil choice upon which we construct our frmework [6, 10]. We begin with some nottion. We ssume tht gme forms nd socil choice func- 2

3 tions (to be defined herefter) shre the sme domins of gents nd outcomes. We denote by N = {1,..., n} the finite set of gents (or plyers) nd by K the finite set of socil outcomes (outcomes herefter). We use the letters, b, c,... s constnts of K. We use vribles i, j,... to denote gents, nd outcomes will be denoted by the vribles x, y, z,.... Typiclly, one cn consider tht the gents re the voters nd the outcomes re the cndidtes in some election. We denote by L(K) the set of liner orders over K. (A liner order here is reltion tht is reflexive, trnsitive, ntisymmetric nd totl.) By using liner order, we re ssuming the plyers cnnot be indifferent between two distinct outcomes. A preference reltion is liner order of outcomes. Given K nd N, preference profile < is tuple (< i ) i N of preferences, where < i L(K) for every i. The set of preference profiles is denoted by L(K) N. Note tht we use the symbol < i for preference reltion for gents, which in this cse hppens to be reflexive (nd we do not write i for it). Also, we will use the symbol > i with the obvious mening, i.e., y > i x iff x < i y. Definition 1 (Socil choice function) Given K nd N, socil choice function (SCF) is single-vlued mpping from the set L(K) N of preference profiles into the set K of outcomes. For every preference profile, socil choice function describes the desirble outcome (from the point of view of the designer). Definition 2 (Strtegic gme form) Given the sets N nd K, strtegic gme form is tuple N, (A i ), K, o where: A i is finite nonempty set of ctions (or strtegies) for ech plyer i N; o : i N A i K ssigns n outcome for every combintion of ctions. A strtegic gme form is sometimes clled mechnism. It specifies the gents tking prt in the gme, their vilble ctions, nd wht outcome results from ech combintion of ctions. We refer to collection ( i ) i N, consisting of one ction for every gent in N, s n ction profile. Given n ction profile, we denote by i the ction of the plyer i. Remrk 1 There is direct link between strtegic gme forms nd socil choice functions. Any socil choice function cn be viewed s gme form in which the set of ctions of every gent is L(K) (think of this s the preference profiles the gent cn clim to be his), nd the function o represents the socil choice function (see [9]). For ny SCF F, we denote its ssocited gme form by g F. A strtegic gme is bsiclly the composition of strtegic gme form with collection of preference reltions (one for every gent) over the set of outcomes. Definition 3 (Strtegic gme) A strtegic gme is tuple N, (A i ), K, o, (< i ) where N, (A i ), K, o is strtegic gme form, nd for ech plyer i N, < i is preference reltion over K. 3

4 In our context, when the ctions A i in gme N, (A i ), K, o, (< i ) re preference reltions themselves, one should think of those s preferences tht i cn choose to report, wheres < i, encodes i s rel preferences. A solution concept defines for every gme set of ction profiles intuitively, those tht my be plyed through rtionl ction. Exctly which solution concept is used depends upon the ppliction t hnd: we will soon introduce well-celebrted solution concept of Nsh Equilibrium (see Exmple 1). Definition 4 (Solution concept) A solution concept SC is function tht mps strtegic gme form N, (A i ), K, o nd preference profile over K to subset of the ction profiles. We now introduce simple but fundmentlly importnt solution concept: Nsh equilibrium. Definition 5 (Nsh equilibrium) Given strtegic gme form g = N, (A i ), K, o nd preference profile < over K the set of Nsh equilibri NE(g, <) is given s the set of ction profiles in g such tht no plyer would benefit from deviting unilterlly from his current ction. More formlly, ( 1,... n ) NE(g, <) iff for every plyer k nd every k A k, we hve o( 1,... k... n) < k o( 1,... k... n ). We cn now introduce the notions of implementtion nd truthful implementtion. The problem of implementtion rises becuse plnner does not know the true preference profile of the plyers. Given socil choice function F involving set of plyers N nd set of outcomes K, the plnner only knows tht every plyer i N hs some preference < i, n element of L(K). We first define the cse of (stndrd) implementtion. Assuming pttern of behviour solution concept SC the role of the plnner is then to design mechnism (or gme form) g such tht for every possible preference profile < L(K) N, the strtegic gme g, < dmits t lest one SC-equilibrium, nd every SC-equilibrium leds to the outcome in K which is prescribed by the socil choice function for the preference profile t hnd, tht is, the vlue of F(<). Definition 6 (Implementtion) Given solution concept SC, we sy tht the gme form g = N, (A i ), K, o SC-implements the socil choice function F if for every preference profile < L(K) N we hve tht SC(g, <) nd SC(g, <) implies tht o( ) = F(<) In words: the gme form g SC-implements F if for ny gme form g, < bsed on g, ny outcome ssocited to strtegy profile in the solution concept SC is the sme s wht the socil choice function would yield for the preference <. Or, more loosely: the gme form g implements F if, for every preference profile < tht we cn ssocite with it, the outcomes in the gme g, < nd the result of F(<) gree t lest on those climed preferences tht re in the solution concept of the gme. The problem of implementtion is illustrted in Figure 1. We sy tht the socil choice function is SC-implementble if there is gme form tht SC-implements it. 4

5 < x y SC < 1 < 2 1 z x y G = g, (< 1, < 2 ) < 1 x x y 1 z x SC F G = g, (< 1, < 2 ) Figure 1: Implementtion. The preference profiles < nd < re two rbitrry members of L(K) N. The left prt represents the SCF F. F(< 1, < 2 ) = y nd F(< 1, < 2 ) = x. The right prt represents the strtegic gme form g instntited, in the upper prt with the preference profile (< 1, < 2 ) (gme G = g, (< 1, < 2 ) ) nd in the lower prt with the preference profile (< 1, < 2 ) (gme G = g, (< 1, < 2 ) ). All the SC-equilibri of G (nd possibly lso some others thn ( 1, 2 )) led to F(< 1, < 2 ). In like mnner, ll the SCequilibri of G led to F(< 1, < 2 ). This hs to be verified for every preference profile in L(K) N nd not only < nd < : if it holds, g is sid to SC-implement F. In some situtions however, n SCF cn be implemented by strtegic gme form of which the spce of ction profiles corresponds to the spce of preference profiles, nd telling the truth is n equilibrium. We cll strtegic gme form in which the set of strtegies of plyer i is the set of preferences over K direct mechnism. Hence, ech plyer is sked to report preference, but not necessrily the true one. An ppeling clss of direct mechnisms is tht in which reporting the true preference profile is n equilibrium of the gme consisting of the direct mechnism composed with the true preference profile. Tht is, for every < L(K) N, the ction profile where every plyer i reports its true preference < i is n equilibrium of the gme g, <. We cn define this notion for every solution concept SC. Definition 7 (Truthful implementtion) The direct mechnism g = N, (A i ), K, o truthfully SC-implements the SCF F if for every true preference profile < nd reported 5

6 < 2 < 2 < 2 < 2 < 1 < 1 SC y y < 1 x < 1 x SC G = g F, (< 1, < 2 ) G = g F, (< 1, < 2 ) Figure 2: Truthful implementtion. The preference profiles < nd < re two rbitrry members of L(K) N. The left prt represents the gme form g F ssocited to the SCF F when the preferences of the two plyers re < 1 nd < 2. The gme G = g F, (< 1, < 2 ) dmits n equilibrium t the ction profile (< 1, < 2 ). The right prt represents g F when the preferences of the two plyers re < 1 nd < 2. The gme G = g F, (< 1, < 2 ) dmits n equilibrium t the ction profile (< 1, < 2 ). This hs to be verified for every preference profile in L(K) N nd not only < nd < : if it holds, g F is sid to truthfully SC-implement F. profile with i =< i for every i: SC(g, <), nd o( ) = F(<) In words: g is truthful SC-implementtion of F if, for every profile <, whenever the gents declre tht to be their rel preferences, this solution concept SC, nd the outcome in the gme nd the function F re the sme. The problem of truthful implementtion is illustrted on Figure 2. We sy tht the socil choice function is truthfully SC-implementble if there is gme form tht truthfully SC-implements it. Note tht truthful implementtions only require tht the report of the true preference profile is n equilibrium, but it is not required tht this equilibrium is unique. In generl, other equilibri could be present tht would not led to the outcome prescribed by the SCF. However, this notion of implementtion cn be motivted. Indeed, it is ssumed tht plying direct mechnism, if csting the rel preference is n equilibrium strtegy, n gent would be sincere. We illustrte the differences between the problems of implementtion with some simple exmples ( miniml socil choice scenrio with only two voters nd two lterntives), which demonstrtes tht the two notions re contingent nd independent: gme form g cn be both truthful SC-implementtion nd n SC-implementtion of socil function F, it cn be both, nd it cn be either of them without being the other. 6

7 [, b] [b, ] [, b] [b, ] [, b] [, b] [b, ] b [b, ] b g H, ([, b], [, b]) g H, ([, b], [b, ]) [, b] [b, ] [, b] [b, ] [, b] [b, ] b g H, ([b, ], [, b]) [, b] [b, ] b g H, ([b, ], [b, ]) Figure 3: g H does not NE-implement H. But g H truthfully NE-implements H. Exmple 1 In this exmple we define some simple socil choice functions, for ll of them we set N = {1, 2} nd K = {, b}. Also, for the ske of comprison between stndrd nd truthful implementtions, we only consider direct mechnisms, since truthful implementtions re not defined otherwise. First, consider the function H for which we clim tht its ssocited gme form g H truthfully NE-implements H but g H does not NE-implement it. H is the socil choice function prescribing the outcome b if nd only both gents prefer b over. We write [, b] for the individul order of preferences of the outcome over the outcome b nd [b, ] for the individul preference of b over. Hence, we hve: H([, b], [, b]) = H([, b], [b, ]) = H([b, ], [, b]) = ; H([b, ], [b, ]) = b. Figure 3 represents the four possible gmes g H, < where < L({, b}) {1,2}. In ech of them, the circles indicte the ction profiles tht re Nsh equilibri. The outcomes in bold re the outcomes o( ) for which = <: in those outcomes, plyers hve reveled their true preferences. So for instnce, the outcome in the upper left corner of the gme g H, ([, b], [, b]) reds: the outcome in the gme here is nd the voters revel their true preferences. For every preference profile <, the ticks indicte tht the ction profile < leds to the outcome prescribed by the socil choice function H nd is Nsh equilibrium in the gme g H, < ; Hence the gme form g H truthfully NE-implements H: ll the bold outcomes re ticked. The cross designtes problem with the (stndrd) implementtion of H by g H : in the gme g H, ([b, ], [b, ]) the ction profile ([, b], [, b]) is Nsh equilibrium nd leds to the outcome, however H([, b], [, b]) = b. Hence, g H does not NE-implement H. 7

8 [, b] [b, ] [, b] [b, ] [, b] [, b] [b, ] b b [b, ] b b g J, ([, b], [, b]) g J, ([, b], [b, ]) [, b] [b, ] [, b] [b, ] [, b] [, b] [b, ] b b [b, ] b b g J, ([b, ], [, b]) g J, ([b, ], [b, ]) Figure 4: g J both NE-implements nd truthfully NE-implements J. Let us next consider the socil choice function J which is dicttoril for plyer 1, i.e., J is defined by J([, b], [, b]) = J([, b], [b, ]) = ; J([b, ], [, b]) = J([b, ], [b, ]) = b. The four possible gmes g J, < for J re depicted in Figure 4. It is esy to see tht the circled outcomes in those gmes re Nsh equilibri: they give the preferred outcome for 1 (so 1 cnnot improve by deviting) nd they re the sme in fixed row (so 2 cnnot chnge the outcome). Moreover, it is lso strightforwrd check tht for ll those Nsh equilibri, the outcome in the gme g J, < is the sme s J(<) (for instnce, in the top left gme, both equilibri yield which coincides with J([, b], [, b]), nd in the bottom left gme, both equilibri yield b = J([b, ], [, b]), etc): this justifies the ticks. So g NE-implements J. To show tht g lso truthfully NE-implements J, we need to check tht ll the bold outcomes in Figure 4 re circled nd ticked. Next, to give n exmple of n NE-implementtion tht is not truthful one, consider the gme form g J. It is mthemticlly equivlent to the gme form g J : the outcomes nd b re only inverted. Plying g J, the plyer 1 would simply ply the contrry to his true preference. This lwys yields Nsh equilibrium nd the outcomes re lwys s prescribed by J. Hence, like g J, the gme form g J is n NEimplementtion of the socil choice function J. However, since the plyer 1 needs to trick the gme in order chieve Nsh equilibrium, it is esy to see tht g J does not truthfully NE-implement J. The crosses on Figure 5 mrk the ction profiles tht correspond to the true preferences of the plyers, nd we cn see tht their respective outcome lwys fils to be s prescribed by J. 8

9 [, b] [b, ] [, b] [b, ] [, b] b b [, b] b b [b, ] [b, ] g J, ([, b], [, b]) g J, ([, b], [b, ]) [, b] [b, ] [, b] [b, ] [, b] b b [, b] b b [b, ] [b, ] g J, ([b, ], [, b]) g J, ([b, ], [b, ]) Figure 5: g J NE-implements J but does not truthfully NE-implement it. Finlly, we rgue tht it is possible for gme form to be neither NE-implementtion nor truthful implementtion of given function: tke P such tht P(<) = for ll profiles <. Moreover, for ll <, let ll outcomes in the mtrix for g P, < be b. For every <, every outcome in g M, < is Nsh equilibrium (no gent cn chnge the outcome, let lone improve it). At the sme time, for ll we hve b = o( ) P(<) =, which shows tht g does not NE-implement P. It does lso not truthfully NE-implement it: tke, for ny <, profile in the gme g M, < such tht = <. We hve lredy seen tht o( ) P(<), which proves our clim. 3 A Logic of socil choice functions Following the trdition in implementtion theory (cf. Remrk 1), we model socil choice functions s prticulr kind of strtegic gme form. In [13] we proposed logic for modelling strtegic gmes on the bsis of CL-PC. Every plyer controls set of propositionl vribles nd strtegy for plyer mounts to choosing truth vlue for the vribles he controls. We dpt the ides of [13] to gme forms where the strtegies of the plyers correspond to the reports of preferences. 3.1 Semntics Let X be n rbitrry set of propositions. We cn see vlution of X s subset V X where tt (i.e., true) is ssigned to the propositions in V nd ff (flse) is ssigned to the propositions in X \ V. We denote the set of possible vlutions over X by Θ X. 9

10 In the presence of set of plyers N nd set of outcomes K, the set of propositions controlled by plyer i N is defined s At[i, K] = {p i x>y x, y K}. Every p i x>y is proposition controlled by the gent i which mens tht i reports tht it vlues the outcome x t lest s good s y. We lso define At[N, K] = i N At[i, K], which is then the set of ll controlled propositions. We cn encode prticulr preference (or liner order) of plyer i s vlution of the propositions in At[i, K]. However, conversely, not ll vlutions correspond to liner order preference. A strtegy of plyer i consists of reporting vlution of At[i, K] encoding liner order over K. For every plyer i, we define strtegies[i, K] s set of vlutions V Θ At[i,K] such tht: (i) p i x>x V, (ii) if x y then p i x>y V iff p i y>x V, nd (iii) if p i x>y V nd p i y>z V then p i x>z V. Remrk 2 Every p i x>y could be seen s predictive expression p(i, x, y) tht would red tht gent i reported to prefer the outcome x over y. However, since N nd K re finite, we look t these expressions s finite collection of propositions. The constrints of control in Figure 6 will be their propositionl theory corresponding to the three preceding constrints on the vlutions. For every colition C N, let strtegies[c, K] be the set of tuples v C = (v i ) i C where v i strtegies[i, K]. It is the set of strtegies of the colition C. To put it nother wy, it corresponds to vlution of the propositions controlled by the plyers in C, encoding one preference over K for every plyer in C. A stte (or reported preference profile) is n element of strtegies[n, K], tht is, strtegy of the colition contining ll the plyers. We now define the models of socil choice functions. Definition 8 (Model of socil choice functions) A model of socil choice functions over N nd K is tuple M = N, K, out, (< i ), such tht: out : strtegies[n, K] K mps every stte to n outcome; For every i N, < i L(K) is the true order of preferences of i. Hence, every plyer i hs two levels of preferences: (i) true one, given by (< i ) nd (2) reported one, given by vlution in strtegies[i, K]. Tking out the true preference profile from model of SCF, we obtin mere instntition of pre-boolen gme [4]. It is required to ssign every vrible to one (ctul control) nd only one (exclusive control) plyer, but there re some constrints on the possible vlutions ( non-full control). In [4], ctul nd exclusive control re grsped by n ssignment function (mpping every propositionl vrible to exctly one plyer), nd the prtil control is modelled by set of constrints given s set of stisfible propositionl formule. The lnguge L scf [N, K] is inductively defined by the following grmmr: ϕ p x ϕ ϕ ϕ C ϕ i ϕ where p is tom of At[N, K], x is n tom of K, i N, nd C is colition. Given model M nd stte (i.e., reported profile v), formul C ϕ reds tht provided tht the plyers outside C hold on to their current strtegy v C, the colition C hs strtegy, 10

11 i.e., wy to nnounce their profiles, such tht ϕ holds. Formul i ϕ reds tht i loclly (t the current reported profile) considers reported profile where ϕ is true t lest s preferble. Definition 9 (Truth vlues of L scf [N, K]) Given model M = N, K, out, (< i ), we re going to interpret formule of L scf [N, K] in stte of the model. A stte v = (v 1,..., v n ) in M is tuple of vlutions v i strtegies[i, K], one for ech gent. The truth definition is inductively given by: M, v = p iff p v i for some i N M, v = x iff out(v) = x M, v = ϕ iff M, v = ϕ M, v = ϕ ψ iff M, v = ϕ or M, v = ψ M, v = C ϕ iff there is stte u such tht v i = u i for every i C nd M, u = ϕ M, v = i ϕ iff there is stte u such tht out(v) < i out(u) nd M, u = ϕ We ssume tht plyer i only mkes clims or nnouncements bout its own preferences, nd i controls nothing else, so the tomic cluse could equivlently hve red M, v = p i x>y iff p i x>y v i The truth of ϕ in ll models over set of plyers N nd set of outcomes K is denoted by = Λ scf [N,K] ϕ. The clssicl opertors,, cn be defined s usul. We lso define C ϕ C ϕ nd i ϕ i ϕ. Theorem 1 (Decidbility) The problem of deciding whether formul ϕ L scf [N, K] is stisfible is decidble. Proof. It suffices to remrk tht N nd K re finite. Hence, we cn enumerte every model of SCF over N nd K nd check whether ϕ is stisfible in one stte of one model. 3.2 Bllots We think of prticulr preference of L(K) encoded in the lnguge of the propositions s bllot. Definition 10 (Bllot) For every plyer i N, we cn see every < i L(K) s permuttion [x 1, x 2...] of the elements of K, where the more to the left the outcome is, the more it is preferred by the plyer i. We cn reify in the lnguge the reported preferences, tht is, the bllot csted by the plyer i: bllot i (<) p i x 1 >x 2 p i x 2 >x 3... p i x K 1 >x K. 11

12 Then, the formul bllot(<) bllot i (<) i N is reifiction of the reported preference profile < = (< 1,..., < n ), consisting of one bllot for every plyer i N. Remrk 3 Note tht for every < L(K), the formul bllot(<) is true t one nd only one stte. The reder fmilir with Hybrid Logic [1] my think of the formul bllot(<) s nominl, viz. stte lbel vilble in the object lnguge. Exmple 2 Suppose tht N = {1, 2} nd K = {, b, c}. Let preference profile (< ex 1, <ex 2 ) L(K)N given by the dt of the two permuttions [, c, b] nd [c,, b] representing respectively the preferences of plyer 1 nd 2. This reported preference profile cn be represented in the lnguge L scf [{1, 2}, {, b, c}] by the formul bllot(< ex ) p 1 >c p 1 c>b p2 c> p 2 >b. It is esy to verify tht the constrints on the elements of strtegies[1, K] nd strtegies[2, K] re sufficient for inferring complete chrcteristion of the preference profile. The following is vlid in the models of socil choice functions over {1, 2} nd {, b, c}: bllot(< ex ) p 1 > p 1 b>b p1 c>c p 1 >c p 1 c>b p1 >b p1 c> p 1 b>c p1 b> p 2 > p 2 b>b p2 c>c p 2 c> p 2 >b p2 c>b p2 >c p 2 b> p2 b>c 3.3 Chrcterising n SCF Recll tht model of socil choice functions is tuple M = N, K, out, (< i ), where < i re the rel preferences of the gents nd the outcome function o ssigns to every vlution n element of K. There is one-one correspondence between vlutions nd preference profiles: the preference profile P(v) ssocited with vlution v is the reltion < for which x > i y iff p i x>y v. Likewise, the vlution V(<) ssocited with < is the set {p i x>y x > i y}, which collect ll the toms form bllot(<). This mkes it possible to relte model M with socil choice function F s follows. We sy tht model M = N, K, out, (< i ) nd socil choice function F : L(K) N K correspond, if for every strtegy profile < nd its ssocited vlution v (i.e., for which V(<) = v nd P(v) =<), we hve o(v) = F(<). This correspondence cn be syntcticlly defined in formul ρ F : ρ F = N (bllot(<) F(<)) < L(K) N Note tht N plys the role of the universl/globl existentil modlity often noted E in the literture in modl logic: it llows us to quntify over ll the possible vlutions in Θ At[N,K], or bllots. Given the outcomes K, the gents N nd the socil choice function F, formul ρ F sys tht every profile < together with F(<) s n outcome ppers in the model. Since 12

13 the sttes of model re ll possible profiles in L(K) N, nd every profile occurs exctly once, we might s well hve defined ρ F s ρ F = (bllot(<) F(<)) < L(K) N It is esy to see tht the logic is expressively complete wrt. socil choice functions. Tht is, for every SCF F over set of plyers N nd set of outcomes K, there exists formul ρ F L scf [N, K] chrcterising it. Even though it my not be optiml in terms of succinctness, it suffices to consider the conjuncts of formule N (bllot(<) x), for < L(K) nd F(<) = x. The next exmple shows, using simple scenrio, tht we cn sometimes obtin less nïve nd more compct chrcteristions. Exmple 3 Consider the following model of SCF (or gme form) where plyer 1 chooses rows, plyer 2 chooses columns nd plyer 3 chooses mtrices. There re two outcomes nd b. Hence, every plyer i controls the set of toms {p i >, p i b>b, pi >b, pi b> }. Every plyer i hs two strtegies: p i > p i b>b pi >b pi b> nd pi > p i b>b pi >b pi b>, tht we denote respectively by [, b] nd [b, ]. (In the logic Λ scf [{1, 2, 3}, {, b}], they re in fct equivlent to the formule p i >b nd pi b>, respectively.) [, b] [b, ] [, b] [b, ] [, b] [, b] b [, b] [b, ] b [b, ] [b, ] b b We cn represent it in the logic Λ scf [{1, 2, 3}, {, b}] of socil choice functions by the formul: ρ F (p 1 >b p2 >b ) (p1 >b p3 >b ) (p2 >b p3 >b ). Note tht since out is functionl, in the models of socil choice functions with K = {, b} the outcome b will hold whenever does not. Going bck to the socil choice functions of Exmple 1, we invite the reder to check tht ρ H = b (p 1 b> p2 b> ) ρ J = p 1 >b ρ P = 3.4 True preferences In Section 3.2 we sw how to use the toms in At[i, K] to encode the reported preference or bllot of plyer i. These toms do not necessrily represent the true preferences of the gents. We hndle the true preferences of plyer i vi the i modlity. 13

14 From our bsic lnguge L scf [N, K], we cn lso define n opertor of interest concerning preferences. We cn define the globl binry opertor of preferences ψ i ϕ, corresponding to preference between propositions. It reds ll ϕ re better thn ll ψ. ψ i ϕ N < L(K) N (bllot(<) (ϕ N (ψ i bllot(<)). Agent i judges the proposition ϕ t lest s good s ψ iff when the reported preference profile is < nd ϕ holds t the stte lbeled by bllot(<), then, whenever ψ holds in stte, i would prefer the stte lbeled by bllot(<) (cf. Remrk 3). As in Definition 10 for reported preferences, we cn now reify the true preferences. Provided tht x nd y re two possible outcomes, the formul y i x cptures the fct the plyer i prefers (globlly) the outcome y over the outcome x. Hence, from preference profile < L(K) N, we reify the preference [x 1, x 2...] of the plyer i s follows: Then, the formul true i (<) (x K i x K 1 )... (x 3 i x 2 ) (x 2 i x 1 ). true(<) true i (<) is reifiction of the true preference profile <= (< 1,..., < n ). i N Remrk 4 Whenever in model of socil choice function M the true preference of plyer i is such tht x < i y, then the formul x i y is true t every stte of M. However, the other wy round does not hold. Indeed, when either x or y is not possible outcome of model, the formul x i y is lwys true for every i. From the definition, x i y N < L(K) (bllot(<) (y N(x N i bllot(<)). Hence, if y is not possible outcome, the min impliction y N (x i bllot(<)) is lwys true for y being lwys flse. Likewise, if x is not possible outcome, the impliction x i bllot(<) is lwys true for x being lwys flse. In turn, it mkes the min impliction lwys true. Also, < L(K) N bllot(<) will lwys be stisfied since stte of evlution represents bllot by definition. The object lnguge does not llow to tlk bout true preferences on impossible outcomes. This observtion will hve consequence in the wy we prove the completeness of the logic. 3.5 Axiomtics The xiomtiztion of the models of socil choice functions is presented in Figure 6. Constrints of control (refl), (ntisym-totl) nd (trns) sy tht every plyer csts n pproprite vlution of its controlled toms: vlution must encode liner order. (comp ) defines the locl bility of colitions in terms of locl bilities of subcolitions. The trnsitivity of the opertor C is the consequence of (comp ). Hence, together with (T(i)) nd (B(i)), it mkes of C n S5 modlity. (empty) mens tht the empty colition hs no power. (comp ) nd (confl) together mke sure tht the gents choices re independent. (exclu) mens tht if n tom is controlled by plyer i, the other plyers cnnot chnge its vlue. (bllot) mkes sure tht n gent is lwys 14

15 Constrints of control (refl) p i x>x (ntisym-totl) p i x>y p i y>x, where x y (trns) p i x>y p i y>z p i x>z Propositionl control (Prop) ϕ, where ϕ is propositionl tutology (K(i)) i (ϕ ψ) ( i ϕ i ψ) (T(i)) i ϕ ϕ (B(i)) ϕ i i ϕ (comp ) C1 C2 ϕ C1 C 2 ϕ (confl) i j ϕ j i ϕ (empty) ϕ ϕ (exclu) ( i p i p) ( j p j p), where j i (bllot) i bllot i (<) (comp-at) C1 δ 1 C2 δ 2 C1 C 2 (δ 1 δ 2 ) Outcomes nd preferences (func1) x K(x y K\{x} y) (func2) (bllot(<) ϕ) N (bllot(<) ϕ) (incl) N ϕ i ϕ (K(< i )) i (ϕ ψ) ( i ϕ i ψ) (4(< i )) i i ϕ i ϕ (ntisym ) (bllot(<) i bllot(< ) N (bllot(< ) i bllot(<) (totl ) (bllot(<) i bllot(< ) N (bllot(< ) i bllot(<) (unifpref ) (x i y) (x i y) Rules (MP) from ϕ ψ nd ϕ infer ψ (Nec( i )) from ϕ infer i ϕ Figure 6: Logic of socil choice functions Λ scf [N, K]. i rnges over N, C 1 nd C 2 over 2 N, x nd y re over K, nd < is over L(K) N. δ 1 nd δ 2 re two formule from L scf [N, K] tht do not contin common tom from At[N, K]. ϕ represents n rbitrry formul of L scf [N, K], nd p n rbitrry tom in At[N, K]. loclly ble to cst ny preference. From (comp-at), provided tht δ 1 nd δ 2 do not contin commonly controlled tom, if colition C 1 cn loclly enforce δ 1 nd C 2 cn loclly enforce δ 2 then they cn enforce δ 1 δ 2 together. Axiom (func1) forces the fct tht for every ction profile there is one nd only one outcome. (func2) ensures tht the outcomes re only determined by the vlutions. (incl) ensures tht if something is settled, plyer cnnot prefer its negtion. (4( i )) chrcterises trnsitivity. (ntisym ) nd (totl ) force tht the reltion of preference over sttes is ntisymmetric nd totl (nd hence, in prticulr, this reltion is reflexive). Finlly, (unifpref ) specifies fundmentl interction between preferences nd the outcomes. If the csted preference profile t hnd leds to x nd gent i prefers n ction profile leding to y, then t every ction profile leding to x, gent i will prefer every ction profile leding to y, tht is, ll y re better thn ll x. The logic hs cler flvour of norml modl logic [5]. The presence of (K(i)) with the necessittion rule (Nec( i )) gives to the opertor i the property of normlity. 15

16 The necessittion rule for the opertor i holds becuse of (Nec( i )) nd the xioms (comp ) nd (incl). The normlity of the modlity i then follows from (K(< i )). The xiomtics is lrgely inspired by the xiomtics of the logic of gmes nd propositionl control (henceforth LGPC) presented in [13]. The logic LGPC is designed to model strtegic gmes in generl. The gents hve rbitrry strtegies, nd preferences llowing for indifference between two different outcomes. On the other hnd, in this pper we focus on SCFs nd hence on prticulr strtegic gmes tht represent n SCF (cf. Remrk 1). While in LGPC we hd n xiom sying tht every tom ws ctully controlled by t lest one gent, here we re more specific s we know priori which toms re controlled by given gent. This is the role of the xiom (bllot). Constrints of controls re lso specific to the present study. The truth vlues of the controlled toms cnnot be independent of ech other s we use them to encode preferences. In LGPC, ll vlutions of the controlled toms were permitted. Theorem 2 (Soundness nd completeness) Λ scf [N, K] is sound nd complete with respect to the clss of models of socil choice functions. Proof. The proof of completeness first gives n equivlent but more stndrd semntics to the logic: the Kripke models of SCF. Then we build the cnonicl model. For every consistent formul ϕ, we show how to isolte sub-model M ϕ tht we prove is Kripke model of SCF tht stisfies ϕ. Further detils re given in the Appendix. 4 Applictions We hve lredy demonstrted tht the lnguge llows to completely chrcterise n SCF. In this section we show how we cn express properties of socil choice functions in the lnguge nd pply the logic to reson bout them. The lnguge cn be used to chrcterise requirements on socil choice functions. We first illustrte tht with some simple properties, nmely citizen sovereignty nd non-dicttorship. Next, we will chrcterise dominnt strtegy equilibrium. Finlly, we provide formlistion of monotonicity nd strtegy-proofness, nd use stndrd results of SCT to show how we cn use the logic to check whether n SCF is implementble in dominnt strtegy. 4.1 Citizen sovereignty nd non dicttorship We sy tht n SCF stisfies citizen sovereignty iff every outcome in K is fesible. Tht is, no outcome is rejected independently of the individul opinions. It is defined s follows. Definition 11 (Citizen sovereignty) An SCF F stisfies citizen sovereignty iff for every x K there is < L(K) N such tht F(<) = x. 16

17 The next formul is strightforwrd trnsltion of the definition of citizen sovereignty in the lnguge of socil choice functions. CITSOV N x. We sy tht n SCF stisfies non dicttorship iff no plyer cn lwys impose its fvourite outcome. Definition 12 (Non-dicttorship) An SCF F is non dicttoril iff for every plyer i N there is bllot < L(K) N such tht F(<) < i y for some y K \ {F(<)}. This sys tht for every plyer, there is bllot < whose outcome is F(<), nd i prefers n outcome tht is not F(<). We cn rewrite the definition of non dicttorship into the lnguge of socil choice functions s follows. NODICT N x p i y>x. i N The following proposition is immedite. x K x K y K\{x} Proposition 1 Consider socil choice function F nd ρ F formul chrcterising it. 1. F hs the property of citizen sovereignty iff = Λ scf [N,K] ρ F CITSOV. 2. F is non dicttoril iff = Λ scf [N,K] ρ F NODICT. 4.2 Dominnt strtegy equilibrium Citizen sovereignty nd non dicttorship re possible properties of socil choice function: their formultions in logic re globlly true (or flse) in model of SCF. However, the logic is lso ble to formlise solution concepts, which re properties of sttes. In [13], we chrcterised severl solution concepts (dominnt strtegy equilibrium, Nsh equilibrium, core membership... ) tht re directly pplicble in the logic of the present work. In order to formlise strtegy-proofness lter, we need to chrcterise dominnt strtegy equilibrium. A dominnt strtegy equilibrium cptures prticulrly importnt pttern of behviour. It rises when every plyer plys dominnt strtegy, tht is, strtegy tht would represent the best choice whtever the other gents ply. We define it directly in our models of SCF. Definition 13 (Dominnt strtegy equilibrium) Let v be stte in model of socil choice functions N, K, out, (< i ). v is dominnt strtegy equilibrium iff for every plyer i N nd every strtegy u N\{i} strtegies[n \ {i}, K], we hve out(u 0... u i... u n) < i out(u 0... v i... u n) for every u i strtegies[i, K]. 17

18 A dominnt strtegy equilibrium is strong solution concept: such n equilibrium does not depend on the knowledge of n gent i bout the strtegies or preferences of other plyers. It is convenient to introduce the notion of best response by n gent i. BR i (x i i x). x K A plyer i plys best response in stte if, x being the outcome, for every devition of i, i prefers x. We cn now define strtegy dominnce in terms of best response: DOM N\{i} BR i. i N We hve strtegy dominnt stte if the current choice of every plyer ensures them best response whtever other gents do. Proposition 2 Assume model of socil choice functions M nd stte v. We hve tht v is dominnt strtegy equilibrium iff M, v = DOM. 4.3 Monotonicity nd strtegy-proofness One importnt property of SCF is monotonicity, s this property cn ffect the implementbility of socil choice functions. Definition 14 (Monotonicity) An SCF F is monotonic iff for ll {<, < } L(K) N nd x K, if F(<) = x nd if for ll i N, for ll y K we hve tht tht y < i x implies tht y < i x, then, F(< ) = x. We propose to chrcterise monotonic socil choice functions. We define < L(K) < N L(K) x K[ N N (bllot(<) x) MON i N y K( N (bllot(<) p i x>y) N (bllot(< ) p i x>y) ) N (bllot(< ) x) ]. Although it my pper rther complex, the predicte MON is essentilly nothing more thn the expression of Definition 14 in our lnguge L scf [N, K]. The following proposition is immedite. Proposition 3 Consider socil choice function F nd ρ F formul chrcterising it. F is monotonic iff = Λ scf [N,K] ρ F MON. Monotonicity does not depend on the true preference profile of the plyers. Accordingly, our definition does not involve the modlities of preference i ϕ nd ϕ i ψ. Cpitlising on stndrd results from socil choice theory, we will show tht using the full expressivity of our lnguge (tht is, using true preference modlities) we cn obtin much simpler formultion. We sy tht n SCF is strtegy-proof if for every preference profile, telling the truth (reporting the true preference) is dominnt strtegy for every plyer. 18

19 Definition 15 (Strtegy-proofness) An SCF F is strtegy-proof iff F is truthfully DOMimplementble. Hence, choice function is strtegy-proof when it is truthfully implementble in dominnt strtegy: for every preference profile, reporting their true preference is dominnt strtegy for every plyer. The reveltion principle [7] is centrl result in implementtion theory. It sttes tht if n SCF is DOM-implementble, then it is truthfully DOM-implementble. It is true in generl even if L(K) is bsed on weker orders. The reveltion principle tells us tht if n SCF F is implementble in dominnt strtegies then there exists direct mechnism such tht for every preference profile <, truth telling (every plyer i reports < i ) is dominnt strtegy nd the outcome is F(<). Truthful implementtions re rther wek; it is esier in generl to implement choice function truthfully thn with stndrd implementtions. Indeed, in truthful implementtions there might be n equilibrium tht leds to n outcome different of the one prescribed by the SCF. But becuse in this pper we consider liner preferences, nd we ssume tht plyers cnnot be indifferent between two distinct outcomes, such sitution cnnot hppen. Thus, we cn be more specific thn the reveltion principle. Theorem 3 ([6, Corollry 4.1.4]) A direct mechnism g truthfully implements n SCF F in dominnt strtegies iff g DOM-implements F. Hence, when working in dominnt strtegies with liner preferences, the concepts of implementtion nd truthful implementtion coincide. We propose to chrcterise strtegy-proof socil choice functions s follows: STRPROOF [true(<) (bllot(<) DOM)] < L(K) N The formul STRPROOF is n immedite reformultion of the definition of strtegyproofness in our lnguge of socil choice functions. Proposition 4 Consider socil choice function F nd ρ F formul chrcterising it. F is strtegy-proof iff = Λ scf [N,K] ρ F STRPROOF. This Proposition provides us with generl procedure to check whether socil choice function is strtegy-proof. Moreover (becuse of Theorem 3), becuse we restrict our ttention to liner preferences, it llows us to check whether n SCF is DOMimplementble. Exmple 4 We cn verify for instnce tht the socil choice function chrcterised in Exmple 3 is strtegy-proof. = Λ scf [{1,2,3},{,b}] ( (p 1 >b p2 >b ) (p1 >b p3 >b ) (p2 >b p3 >b )) STRPROOF. Monotonicity sometimes implies implementbility nd this is ctully the cse in our setting. Since we re working with rich domins of preferences 1 nd liner orderings the following result holds. 1 The notion of rich domin is some tngentil to the purposes of this pper. Briefly, our domin of preferences is rich becuse we llow every liner order of K. See [6, Sec. 3.1] 19

20 Theorem 4 ([6, Cor , Th ]) An SCF is truthfully implementble in dominnt strtegies iff it is monotonic. This stndrd result of implementtion theory shows tht in our setting, the notions of monotonicity nd of strtegy-proofness mtch. Trivilly we re ctully ble to substntilly simplify the formul MON, our chrcteristion of monotonicity in the forml lnguge. Indeed, s consequence of Theorem 4, we hve the following. Proposition 5 = Λ scf [N,K] MON STRPROOF. 5 Discussion nd perspectives We hve presented the problem of direct implementtion in socil choice theory nd proposed logicl formlistion of it. We were ble to give sound nd complete xiomtiztion to the logic. We showed how we cn chrcterise socil choice functions nd properties of socil choice functions. And finlly, we hve demonstrted the vlue of the logic by proposing generl logicl procedure for checking whether socil choice function is strtegy-proof. Our logicl lnguge is forml counterprt of the lnguge of nturl mthemtics tht is typiclly used in socil choice theory. There re however two fetures tht mke it prticulrly useful: (i) it is supported by non mbiguous semntics; nd (ii) the resulting logic is decidble. Section 4 suggests logicl methodology for resoning bout problems of socil choice theory with the logic of socil choice functions. Let collection of properties of socil choice theory Pi, i {1,... n} be chrcterised in the logic Λ scf [N, K] by ρ Pi, respectively. 1. We cn use the logic in order to check whether n SCF stisfies certin property. An SCF F chrcterised by ρ F hs the property P1 iff ρ F ρ P1 is derivble in Λ scf [N, K]. 2. We cn use the logic in order to evlute the strength of constrints in SCT. P1 is property weker thn P2 iff the formul ρ P2 ρ P1 is derivble in Λ scf [N, K]. For instnce, insted of using result of SCT to prove Proposition 5, we could ctully use the logic to utomticlly verify tht monotonicity nd strtegyproofness coincide in the current setting. More interestingly, we could use it to prove new theorems. 3. We cn use the logic for mechnism design. Building mechnism tht implements socil choice procedure stisfying the properties P1, P2,... Pn consists of finding model for the formul ρ P1 ρ P2... ρ Pn. We believe these re exciting possibilities for socil choice theory nd logic, nd s the logic is decidble, they re in principle possible. 20

21 Acknowledgment An erlier bstrct of this pper ppered s [14]. We thnk the nonymous reviewers for their suggestions tht helped to improve the pper. We re lso grteful to the prticipnts of TARK 09. This reserch is funded by the EPSRC grnt EP/E061397/1 Logic for Automted Mechnism Design nd Anlysis (LAMDA). References [1] C. Areces nd B. ten Cte. Hybrid Logics, volume Hndbook of Modl Logic, chpter 14, pges Elsevier Science Inc., [2] K. Arrow. A difficulty in the concept of socil welfre. Journl of Politicl Economy, 58(4):328346, [3] K. J. Arrow, A. K. Sen, nd K. Suzumur, editors. Hndbook of Socil Choice nd Welfre, volume 1. Elsevier, [4] E. Bonzon, M.-C. Lgsquie-Schiex, nd J. Lng. Efficient colitions in Boolen gmes. In K. Apt nd R. vn Rooij, editors, New Perspectives on Gmes nd Interction, volume 4 of Texts in Logic nd Gmes, pges Amsterdm University Press, [5] B. F. Chells. Modl Logic: n introduction. Cmbridge University Press, [6] P. Dsgupt, P. Hmmond, nd E. Mskin. The Implementtion of Socil Choice Rules: Some Generl Results on Incentive Comptibiliy. Review of Economic Studies, 46: , [7] A. Gibbrd. Mnipultion of voting schemes: generl result. Econometric, 41(4): , [8] K. My. A set of independent, necessry nd sufficient conditions for simple mjority decision. Econometric, 20(4): , [9] H. Moulin. The Strtegy of Socil Choice. Advnced Textbooks in Economics. North Hollnd, [10] M. J. Osborne nd A. Rubinstein. A Course in Gme Theory. The MIT Press, [11] M. Puly. Logic for Socil Softwre. PhD thesis, University of Amsterdm, ILLC Disserttion Series [12] M. A. Stterthwite. Strtegy-proofness nd rrow s conditions: Existence nd correspondence theorems for voting procedures nd socil welfre functions. Journl of Economic Theory, 10: ,

22 [13] N. Troqurd, W. vn der Hoek, nd M. Wooldridge. A Logic of Gmes nd Propositionl Control. In Decker, Sichmn, Sierr, nd Cstelfrnchi, editors, 8th Interntionl Joint Conference on Autonomous Agents nd Multi Agent Systems (AAMAS-09), Budpest, Hungry, pges IFAAMAS, [14] N. Troqurd, W. vn der Hoek, nd M. Wooldridge. A logic of propositionl control for truthful implementtions. In TARK 09: Proceedings of the 12th conference on Theoreticl spects of rtionlity nd knowledge, pges ACM DL, [15] W. vn der Hoek nd M. Wooldridge. On the logic of coopertion nd propositionl control. Artificil Intelligence, 164(1-2):81 119, [16] M. Wooldridge, T. Ågotnes, P. E. Dunne,, nd W. vn der Hoek. Logic for utomted mechnism design progress report. In Proceedings of the Twenty-Second AAAI Conference on Artificil Intelligence (AAAI-2007), Vncouver, British Columbi, Cnd, Proof of Theorem 2 Λ scf [N, K] is sound nd complete with respect to the clss of models of socil choice functions. Proof. It is routine to verify tht ll principles of Figure 6 re vlid. We show tht if formul is consistent, it is provble in the system Λ scf [N, K]. We first introduce the Kripke models of SCF. A Kripke model of SCF is tuple M = N, K, S, (R i ), (P i ), V such tht: N nd K re prmeters; S = {V Θ At[N,K] i N, V i strtegies[i, K] s.t. V = i N V i }; V is vlution function of At[N, K] K where for every v S: p V(v) iff p v, p At[N, K]; there is unique x K s.t. x V(v); [ we sy tht the model is bsed on the outcome function out M when out M (v) = x iff x V(v)]. R i vu iff v j = u j for ll j i; there is < M L(K) N s.t. P i vu iff (if x V(v) nd y V(u) then x < M i sy tht the model is bsed on < M ]. y); [ we Truth vlues of i ϕ nd i ϕ in Kripke model of SCF re obtined in the stndrd wy from the reltions R i nd P i, respectively. Clerly, for every Kripke model M bsed on out M nd < M, we cn construct model of socil choice functions M scf = N, K, out M, (< M i ) nd reciproclly. By construction, there exists bijection f : S strtegies[n, K] tht ssocites stte s in M to stte v = (v 1... v n ) in M scf in such wy tht for every p At[i, K], we hve p V(s) iff p v i. The following is esy to see. 22

23 Clim 1 M, s = ϕ iff M scf, f (s) = ϕ. Hence, the proof of the theorem cn be reduced to proof of completeness of the logic wrt. to the clss of Kripke models of SCF. Let Ξ be the set of mximlly consistent sets (mcs.) of Λ scf [N, K]. We define the proper cnonicl model M cn = N, K, S, (R i ), (P i ), V s follows. N nd K re the prmeters of the logic. S = Ξ. R i Γ iff δ, i δ Γ. P i Γ iff δ, i δ Γ. p V(Γ) iff p. x V(Γ) iff x. Given n mcs. Γ 0 we define the set of mcs. describing the sme SCF nd where the plyers hve the sme true preferences (modulo the preferences concerning some outcome which is not fesible in the SCF): Cluster(Γ 0 ) {Γ 1 < L(K) N, x K, N (bllot(<) x) Γ 1 iff N (bllot(<) x) Γ 0 } {Γ 2 i N, {x, y} K, x i y Γ 2 iff x i y Γ 0 } Let ϕ be consistent formul of L scf [N, K]. There is n mcs. Γ ϕ s.t. ϕ Γ ϕ. The proof consists in constructing model from Γ ϕ such tht it is indeed Kripke model of SCF nd there is stte stisfying ϕ. We define M ϕ = N, K, S, R i, P i, V from M cn s follows: N = N nd K = K; S = Ξ Cluster(Γϕ ); R i = R i Cluster(Γ ϕ ); P i = P i Cluster(Γ ϕ ); p V ( ) iff p V( ), S. It is immedite tht the truth lemm holds. Clim 2 M ϕ, Γ = δ iff δ Γ. Hence, M ϕ, Γ ϕ = ϕ. The set of sttes in Kripke models of SCF is defined s the set of vlutions of At[N, K] encoding preference profile. We prove tht there exists bijection between S nd L(K) N. Clim 3 The following sttements re true: 1. S,! < L(K) N s.t. bllot(<) ; 2. < L(K) N,! S s.t. bllot(<). The first prt of the clim follows from the constrints of control (refl), (ntisym-totl) nd (trns). We now rgue tht for every < L(K) N, there is exctly one S such tht bllot(<). Let < L(K) N. We hve i bllot i (<) by (bllot). With (comp-at), we find tht N bllot(<). Hence, N bllot(<) Γ ϕ, nd there must be n mcs. s.t. bllot(<). Now suppose tht S lso contins bllot(<). By (func2), nd contin the sme formule. Then =, which proves the second prt of the clim. As consequence we will be llowed to use the formule of the form bllot(<) s world lbels in M ϕ. We now prove the min clim of this proof. 23