Txtion Srch in Booln Gms Vdim Lvit Dpt. of Computr Scinc Bn-Gurion Univrsity Br-Shv, Isrl lvitv@cs.bgu.c.il Tl Grinshpoun Dpt. of Computr Scinc Bn-Gurion Univrsity Br-Shv, Isrl grinshpo@cs.bgu.c.il An L. C. Bzzn PPGC / UFRGS Cix Postl 15064 91501-970, P. Algr, Brzil bzzn@inf.ufrgs.br Amnon Misls Dpt. of Computr Scinc Bn-Gurion Univrsity Br-Shv, Isrl m@cs.bgu.c.il ABSTRACT Agnts in Booln gm hv prsonl gol rprsntd s propositionl logic formul ovr st of Booln vribls, whr som of ths vribls r not ncssrily hld by th gnt. Th ctions vilbl to ch gnt r ssumd to hv som cost, nd th gnt s scondry gol is to minimiz its costs. An intrsting problm is to find txtion schm tht imposs dditionl costs on th gnts ctions such tht it incntivizs th gnts to rch stbl stt. Th prsnt ppr first thorticlly outlins th chrctristics of Booln gms for which stbiliztion cn b chivd by pplying txtion schm. Nxt, srch mthod for n pproprit txtion schm is proposd. Th proposd mthod trnsforms th Booln gm into n Asymmtric Distributd Constrint Optimiztion Problm (ADCOP). ADCOPs r nturl rprsnttion of Booln gms nd nbl ffctiv srch by using xisting lgorithms. A Booln gm tht rprsnts trffic light coordintion gm is usd throughout th ppr s clrifying xmpl. Finlly, n xprimntl vlution of th trffic light xmpl confirms th pplicbility of th proposd srch mthod nd outlins som ttributs of th gm nd th srch procss. Ctgoris nd Subjct Dscriptors I.2.11 [Computing Mthodologis]: Artificil Intllignc Multignt systms Gnrl Trms Algorithms Kywords Booln gms, Txtion, Srch, Trffic light coordintion, ADCOP Supportd by th Lynn nd Willim Frnkl cntr for Computr Scincs nd th Pul Ivnir cntr for Robotics nd Production Mngmnt. Apprs in: Procdings of th 12th Intrntionl Confrnc on Autonomous Agnts nd Multignt Systms (AA- MAS 2013), Ito, Jonkr, Gini, nd Shhory (ds.), My, 6 10, 2013, Sint Pul, Minnsot, USA. Copyright c 2013, Intrntionl Foundtion for Autonomous Agnts nd Multignt Systms (www.ifms.org). All rights rsrvd. 1. INTRODUCTION Booln gms r fmily of gms bsd on propositionl logic [14, 2]. In Booln gm ch prticipnt (gnt) holds distinct st of Booln vribls, nd hs som prsonl gol it ttmpts to stisfy. An gnt s prsonl gol is rprsntd s propositionl logic formul ovr som st of Booln vribls, whr som of ths vribls r not ncssrily hld by th gnt. Th ctions tht n gnt cn tk consist of ssigning vlus to th vribls it holds. In rcntly proposd gnrliztion nd mor xprssiv vrsion of Booln gms, trmd Cooprtiv Booln Gms [7], th ctions vilbl to n gnt r ssumd to hv som cost, nd th gnt s scondry gol is to minimiz its costs. Th prsnt ppr, following othr rcnt studis [9, 11], rfrs to ths cooprtiv Booln gms. A common objctiv in gm thortic rsrch is to rch som sort of stbiliztion. In Booln gms this usully mns finding Pur-strtgy Nsh Equilibrium (PNE), which is stt in which no gnt hs n incntiv to uniltrlly chng its slctd ction. Th problm is tht such stt dos not ncssrily xist in vry Booln gm. In gms with no PNE stt som of th gnts my b mnipultd by som xtrnl principl in ordr to chiv stbiliztion. Two mnipultion schms for Booln gms wr rcntly proposd. In on schm th Booln gms modl is slightly ltrd to includ st of nvironmntl vribls [11]. Th gnts do not hv dirct ccss to ths nvironmntl vribls, so thy only hv blifs rgrding ths vribls vlutions. Th principl my us this disinformtion in ordr to mnipult th gnts into stbl stt by slctivly communicting truthful informtion rgrding th nvironmntl vribls to som of th gnts. A diffrnt typ of mnipultion, which is t th focus of th prsnt ppr, involvs txtion schm tht imposs dditionl costs on th ctions of gnts. By chnging th costs of gnts in this wy, th txtion incntivizs th gnts to chiv som objctiv [9, 10]. In th cs tht svrl txtion schms r pplicbl, on cn ttmpt to find th most fitting schm ccording to som critrion (.g., minimiztion of th ovrll mount of imposd tx). In both mnipultion schms th objctiv my b somthing byond mr stbiliztion, such s rching som socilly dsirbl outcom. 183
Txtion my not lwys nbl th principl to chiv its gol. Th rson for this stms from th fct tht txtion only ffcts th costs of ctions for gnts. Th minimiztion of costs is only th scondry gol for vry gnt prticipting in Booln gm, th primry gol bing thir prsonl gols (whos gin is highr thn ll costs in th gm) [9, 10]. Th first contribution of th prsnt ppr is thorticl chrctriztion of Booln gms for which stbiliztion cn b chivd by pplying txtion schm. Onc such clssifiction is chivd, th rmining tsk involvs th ctul srch for suitbl txtion schm. It hs bn provn tht dciding whthr thr is PNE in Booln gm is p 2 -complt [2]1. Srching for txtion schm is t lst s hrd, sinc for vry potntil txtion schm on must vrify whthr th rsulting Booln gm hs PNE. Sinc finding txtion schm or vn dciding whthr PNE xists r computtionlly hrd qustions, on my pproch th problm by th us of n intllignt srch lgorithm or huristics tht could succssfully prun th srch spc in Booln gm. Howvr, srch lgorithm for txtion schms on Booln gms hs not bn dsignd yt. Th scond contribution of th prsnt ppr is n ffctiv srch lgorithm for txtion schm. In ordr to ccomplish this tsk, th prsnt study proposs th trnsformtion of th Booln gm into distributd constrints problm, for which thr r vrity of srch tchniqus. Sinc Booln gms r both distributd btwn diffrnt gnts nd r gnrlly symmtric (in th sns tht stt of th gm ffcts diffrntly ch of th prticipting gnts), th most nturl formliztion to us for this purpos is Asymmtric Distributd Constrint Optimiztion Problms (ADCOP) [12, 13]. For th purpos of clrity, th trnsformtion into n AD- COP nd th dscription of th srch procss r dmonstrtd through n xmpl gm, which is Booln gm rprsnttion of trffic light coordintion gm (bsd on [16, 6]). Howvr, th trnsformtion into n ADCOP cn b pplid to ny Booln gm nd is dscribd in dtil for th gnrl cs. Th finl contribution of th ppr is n xprimntl vlution of th trffic light xmpl (on rndomly gnrtd grids), which confirms th pplicbility of th proposd srch mthod nd outlins som ttributs of th gm nd of th srch procss. Th pln of th ppr is s follows. First, Booln gms r prsntd in dtil in Sction 2. Two propositions tht chrctriz th xistnc of PNE in Booln gm r prsntd. Th trffic lights xmpl of Booln gm is dscribd in Sction 3. First, th trffic lights problm is dscribd nd its Booln gm rprsnttion, thn n importnt proposition is provn showing tht ll trffic lights problms hv PNE. Sction 4 dscribs th srch procdur for txtion. It strts by dfining th problm nd th frmwork of ADCOPs, procds to dscrib th construction procdur for n ADCOP tht srvs for txtion srch on givn Booln gm, nd thn prsnts th rquird modifictions to n xisting ADCOP lgorithm so it could hndl k-ry constrints. An xtnsiv mpiricl vlution of th proposd srch for txtion is in Sction 5. Sction 6 outlins our conclusions. 1 p 2 = NPNP is th clss of ll th lngugs tht cn b rcognizd in polynomil tim by nondtrministic Turing mchin quippd with NP orcls [17]. 2. BOOLEAN GAMES A Booln gm [14, 2] contins st of gnts A = {1,..., n}, th plyrs of th gm. Ech gnt A i A controls st of Booln vribls (ϕ i is th st of vribls controlld by gnt A i). Controlling vribls mns tht th gnt A i hs uniqu bility within th gm to st th vlus for ch vribl p ϕ i. It is rquird tht ϕ 1,..., ϕ n form prtition of th gm vribls Φ. In othr words vry vribl is controlld by som gnt nd no vribl is controlld by mor thn on gnt (ϕ i ϕ j = for i j nd i A ϕi =Φ). Ech gnt hs prsonl gol, rprsntd by Booln formul. Thus, γ i rprsnts th gol of gnt A i. Evry gol γ i my contin th vribls of gnt A i nd possibly vribls controlld by othr gnts. A choic of gnt A i, dfind by function v i : ϕ i B, is n lloction of truth or flsity to ll of th gnt s vribls, ϕ i. Lt V i dnot th st of ll vilbl choics for gnt A i. Th intuitiv intrprttion of V i is tht it dfins ll ctions or strtgis vilbl to gnt A i. An outcom (v 1,..., v n) V 1... V n is collction of choics, on for ch gnt. It is clr tht vry outcom uniquly dfins vlution for ll vribls in th gm nd w oftn think of outcoms s vlutions. W ssum tht ctions vilbl to gnts hv costs dfind by cost function c :Φ B R,sothtc(p, b) is th cost of ssigning th vlu b B to vribl p Φ[7]. Consquntly, s in [7, 9, 10], Booln Gm is 2n +3 tupl: G =< A,Φ,c,γ 1,..., γ n,ϕ 1,..., ϕ n > whr A = {1,..., n} is th st of gnts, Φ = {p, q, r,...} is finit st of Booln vribls, c :Φ B R is cost function for vilbl ssignmnts, γ 1,..., γ n r th gols of gnts A, ndϕ 1,..., ϕ n is prtition of vribls Φ ovr gnts A. Th primry im of ch gnt A i is to choos n ssignmnt to th vribls ϕ i undr its control, so s to stisfy its prsonl gol γ i. Th min difficulty is tht γ i my contin vribls controlld by othr gnts who r lso trying to choos vlus for thir vribls, so s to gt thir gols stisfid, nd thir gols my b dpndnt on vribls controlld by gnt A i. If n gnt cn chiv its prsonl gol in mor thn on wy, thn it will prfr to minimiz costs. If th gnt cnnot gt its gol chivd it will prfr to choos vlution tht minimizs th costs. 2.1 Txtion schm A txtion schm [9, 10] dfins dditionl costs on ctions, ovr thos givn by th cost function c. W modl txtion schm s function τ :Φ B R,sotht τ(p, b) is th tx tht should b lvid on th gnt controlling vribl p Φincsthvlub B is ssignd. Whil th cost function c is fixd for ny givn Booln gm G, th txtion schm cn b chngd to fit our rquirmnts. Agnts lwys sk to minimiz thir costs, so by ssigning diffrnt txtions w cn incntiviz gnts to prforming som ctions ovr othrs. On importnt ssumption w mk is tht whil txtion schms cn influnc th dcision mking of rtionl gnts, thy cnnot chng th prsonl gol of n gnt. Thrfor, if n gnt hs chnc to chiv its gol, it will tk it, no mttr wht th txtion incntivs r. 184
2.2 Impct of txtion schm on th xistnc of pur-strtgy Nsh quilibrium Lt us considr th following xmpl to illustrt Booln gm with no PNE for ny txtion schm. Throughout th ppr th symbol is usd to dnot th ngtion of (i.., ). This is clrr to th y for formuls tht includ th ngtion of longr litrls. Th xmpl gm consists of two gnts A = {1, 2}, ch of thm controlling singl vribl: A 1 controls vribl (ϕ 1 = {}) nda 2 controls vribl b (ϕ 2 = {b}, soφ= {, b}). Th cost function will rmin undfind for this xmpl du to its irrlvnc. Th prsonl gol of gnt A 1 is γ 1 =( b) ( b), whrs th prsonl gol of gnt A 2 is γ 2 =( b) ( b). Th mtrix form of th gm is dpictd in Figur 1: A 1\A 2 b = F b = T = F γ 1 γ 2 = T γ 2 γ 1 Figur 1: An xmpl of Booln gm with no PNE for ny txtion schm Now w cn nsur tht th givn Booln gm hs no PNE for ny txtion schm. Considr th outcom { = F, b = F }. This pur strtgy is not Nsh quilibrium bcus of gnt A 2. Givn th currnt outcom, th gnt dos not chiv its gol, but if it chngs th ssignmnt of its vribl (b = T ) it will chiv its gol. By dfinition of Booln gm, th primry im of ch gnt is to stisfy its prsonl gol (if possibl), so th givn pur strtgy is not Nsh quilibrium. Anothr possibl outcom is { = F, b = T }. In this cs gnt A 1 dos not chiv its gol, but by chnging th vribl it controls ( = T ) th gnt s gol will b chivd. Thus, th givn strtgy is lso not PNE. Th rmining two outcoms r symmtric to th considrd ons nd r thrfor lso not PNEs. Aftr considring ll possibl pur strtgis w conclud tht th givn Booln gm hs no PNE for ny txtion schm. Dfinition 1. A prtil vlution (PV) for gnt A i is th vlution of ll vribls Φ in th Booln gm xcpt for th vribls controlld by gnt A i. Th prtil vlution of gnt A i is dnotd by v i =(v 1,..., v n) \ v i. Dfinition 2. A spcil prtil vlution (SPV) is prtil vlution v i for gnt A i such tht thr xists t lst on choic v i tht combind with v i stisfis th gol γ i, nd thr xists nothr choic v i tht dos not stisfy th gol γ i. Dfinition 3. A spcil outcom (SO) is n outcom tht t lst on gnt dos not chiv its prsonl gol nd xcluding tht gnt s choic rsults in spcil prtil vlution. Proposition 1. Givn Booln gm, n outcom is not pur-strtgy Nsh quilibrium stt for ny txtion schm if nd only if th outcom is spcil outcom. Proof. In cs th outcom is spcil outcom, thn by dfinition of SO thr is t lst on gnt tht dos not chiv its prsonl gol nd xcluding tht gnt s choic rsults in spcil prtil vlution. Suppos WLOG tht A i is such n gnt, thn by th dfinition of SPV thr xists t lst on choic v i tht stisfis th gnt s prsonl gol. Th gnt dos not chiv its prsonl gol givn th currnt outcom but by chnging its choic it cn chiv th gol. Following th ssumption on txtion schms th gnt will chiv its prsonl gol if it is possibl no mttr wht th txtion schm is. Thus, thr is t lst on gnt tht wnts to chng its choic in th givn outcom. Consquntly, th currnt outcom is not PNE stt for ny txtion schm. In cs th outcom is not spcil outcom, thn ch gnt ithr chivs its prsonl gol or xcluding tht gnt s choic dos not rsult in SPV. In cs n gnt chivs its gol thn thr xists txtion schm tht minimizs th cost of its currnt choic. In cs n gnt (A i) dos not chiv its gol thn thr dos not xist choic v i tht hlps th gnt chiving its gol, sinc this is not SPV. Hnc, thr is txtion schm tht minimizs th cost of its currnt choic. Sinc th txtions on th ctions of ch gnt r indpndnt, thr is clr rul to crting txtion schm so tht no gnt wnts to chng its choic, which mns tht th outcom is PNE stt givn this txtion schm. Proposition 2. A Booln gm hs no pur-strtgy Nsh quilibrium for ny txtion schm if nd only if vry outcom is spcil outcom. Proof. In cs vry outcom is spcil outcom, thn following Proposition 1, ch of th outcoms is not PNE stt for ny txtion schm. Consquntly, th Booln gm hs no PNE for ny txtion schm. In cs thr xists t lst on outcom tht is not spcil outcom, thn following Proposition 1 thr xists txtion schm tht convrts th givn outcom to PNE stt. Thus, th Booln gm hs PNE givn this txtion schm. 3. TRAFFIC LIGHT COORDINATION Th incrs dmnd for mobility in our socity poss chllngs tht hv to b ddrssd by th r of intllignt trnsporttion systms. Among th fforts currntly undr invstigtion or dploymnt, on trditionl (but nonthlss importnt) ffort is rltd to optimiztion mthods nd trffic control by mns of trffic signl controllrs (in short, trffic lights). 3.1 Approchs for trffic light coordintion Signlizd intrsctions r oprtd by trffic lights tht implmnt th signl timing. A signl-timing pln is uniqu st of timing prmtrs comprising th cycl lngth L (th lngth of tim for th complt squnc of th phs chngs), nd th split (th division of th cycl lngth mong th vrious movmnts or phss). Trffic signls cn b oprtd in vrity of mods. For th purpos of this ppr, w r intrstd in th coordintd control. Th gol of coordintd systms (lso clld synchronizd or progrssiv systms) is to synchroniz trffic signls long n rtril in ordr to llow pltoon of vhicls, trvling t givn spd, to cross th rtril without stopping t rd lights. Thus, if pproprit signl plns r slctd to run t djcnt trffic signls, grn wv is 185
built. This is chivd by mns of so-clld offst (tim btwn th bginning of th grn phs of two conscutiv trffic signls) tht is computd bsd on th dsird spd nd on th distnc btwn intrsctions. Anothr importnt concpt is th bndwidth. It is th tim diffrnc btwn th first nd th lst vhicl tht cn pss through without stopping. Wll dsignd synchronizd signl plns cn chiv ccptbl rsults in on flow dirction. Thus on my xpct th othr dirction to hv mor dlys. Although it is thorticlly possibl to st th synchroniztion for mor thn on flow, th bndwidth dcrss in mor constrind problms. For xmpl, whn th synchroniztion is to b st in two dirctions of n rtril, th bndwidth gnrlly dcrss. Th difficulty is tht th gomtry of th rtril is fixd nd with it th spcing btwn djcnt intrsctions. If on wnts to hv long bndwidth in on trffic dirction (.g., bndwidth is qul to th grn tim), this my hv consquncs in othr dirctions. Nonthlss, synchronizd or coordintd trffic lights in rtrils r commonly sn, spcilly in citis with wll bhvd trffic pttrns. Thr r svrl mthods to comput th synchroniztion. Optimiztion of trffic lights in n off-lin wy is th bsis of wll stblishd lgorithms such s TRANSYT [18], which gnrts optiml coordintd plns for fixd-tim oprtion. On drwbck of this mthod is tht plns r computd for sttic sitution, bsd on historicl dt. Altrntivs r SCOOT, SCATS, nd TUC, which r bsd on rl-tim dt. Howvr, ll ths pprochs focus on synchroniztion of trffic lights in n rtril. Th min difficulty to xtnd th synchroniztion to ntwork or to mor dirctions of trffic is th fct tht in som ky intrsctions conflicts my ppr bcus diffrnt dirctions compt for bndwidth. Th convntionl pproch is to lt trffic xprt solv ths conflicts. Altrntivs to such pproch sk to rplc th trditionl rtril grn wv by shortr grn wvs in sgmnts of th ntwork. This cn b don,.g., using ngotition ovr th qustion of which trffic dirction shll b givn mor bndwidth. An pproch bsd on Distributd Constrint Optimiztion Problms (DCOPs) ws proposd in [6]. Th constrints in this problm ris from th fct tht, in ch nod of th grph, trffic signl cnnot coordint to stblish synchroniztion with nighbors loctd in diffrnt dirction t th sm tim. A conflict occurs whn two nighbors wnt to coordint in two diffrnt trffic dirctions. Jungs nd Bzzn [16] hv ltr xtndd this scnrio to biggr ntworks, iming t invstigting computtionl issus rltd to DCOP prformnc, such s tim to rch n grmnt nd numbr of xchngd mssgs. In th prsnt ppr, th gol of th coordintion mong gnts is to synchroniz th trffic lights in djcnt intrsctions in ordr to llow vhicls trvling t givn spd to cross th intrsctions without stopping t rd lights. Th critri for obtining th optimum signl timing t singl intrsction is tht it should ld to th minimum ovrll dly t th intrsction. In gnrl, th mor nighbors tht r synchronizd, th shortr th quus. In our stting w us grid, in which synchroniztion cn b chivd in ithr south-north nd north-south or stwst nd wst-st dirctions. W ssum on-wy trffic to simplify th systm. 3.2 Booln gm rprsnttion of th trffic light coordintion On cn think of trffic light coordintion s Booln gm, in which ch trffic light is n gnt i,j controlling singl Booln vribl p i,j tht indicts th synchroniztion dirction (Tru = SN/NS nd Fls = EW/WE). Th cost function c(p i,j,b) will b th mount of vhicls in th ln incoming to th trffic light tht controls vribl p i,j from th opposit dirction of th vribl s synchroniztion. Th prsonl gol of n gnt i,j is to b synchronizd with two djcnt gnts in th sm dirction to crt mini-grn-wv, i.., γ i,j =(p i 1,j p i,j p i+1,j) (p i,j 1 p i,j p i,j+1). Th prsonl gol of gnts tht rsid on th dgs of th grid r slightly simplifid. Lt us considr simpl xmpl tht illustrts th gnrl stup of th Booln gm rprsnttion. Suppos w hv grid s dpictd in Figur 2: Figur 2: Ntwork of 9 intrsctions W hv 9 gnts A = { 1,1, 1,2,..., 3,3}, ch controlling singl vribl ϕ i,j = {p i,j} (so Φ = {p 1,1,p 1,2,..., p 3,3}). Th prsonl gols r γ 1,1 = (p 1,1 p 2,1) (p 1,1 p 1,2), γ 1,2 = (p 1,2 p 2,2) (p 1,1 p 1,2 p 1,3),..., γ 2,2 = (p 1,2 p 2,2 p 3,2) (p 2,1 p 2,2 p 2,3),.... Lt r i,j k,l b th mount of vhicls in th ln btwn trffic light i, j nd trffic light k, l, thnc(p i,j,tru)=r i 1,j i,j nd c(p i,j,fls)=r i,j 1 i,j. Proposition 3. Evry Booln gm rprsnting trffic light coordintion problm hs t lst on txtion schm tht nsurs pur-strtgy Nsh quilibrium. Proof. Considr n outcom for th Booln gm constructd from th trffic light coordintion problm, in which vry gnt ssigns Tru to its vribl 2. In this prticulr outcom ll th gnts chiv thir gols (by dfinition), so this is not spcil outcom. Following Proposition 2, thr is t lst on txtion schm tht nsurs PNE. 4. SEARCHING FOR TAXATION Givn Booln gm on wnts to find txtion schm tht nsurs th xistnc of PNE. Thr my b mny pproprit txtion schms, so on cn srch for th optiml schm ccording to som critri. Th most common txtion schm critrion is minimizing th ovrll tx in th gm [9]. Th ovrll tx of gm is dfind by th prsnt ppr in th following wy: T (G) = τ(p, b) (1) p Φ,b B 2 This is on of th possibl outcoms, but in th gnrl cs th Booln gm my contin mor such outcoms. 186
On cn think of this pproch s minimizing th dgr of intrvntion in th gm. Altrntivly, thr r mny othr, mor socil, critri for txtion schms. Ths includ th minimx pproch, in which th mximl tx is minimizd, s wll s n pproch tht minimizs th diffrnc in txs [9]. Additionl critri look t th nd rsult of th gm nd not only t th txtion itslf. Ths includ glitrin socil wlfr, which looks t how wll th worstoff gnt is trtd, nd horizontl quity, inwhichthdiffrnc in txs is minimizd sprtly for ch clss of gnts [5]. Th rlvnt clsss of ntitis of Booln gms for th bov socil critrion r th gnts tht chiv thir prsonl gol nd thos tht do not. For simplicity, th prsnt study will focus on minimizing th ovrll tx, lthough ny of th bov mntiond critri could b pplid with som smll djustmnts. Som intuition bout how to find suitbl txtion schm follows from Proposition 1. Givn Booln gm, th srch procss must slct only thos outcoms tht r not spcil outcoms. For ch of ths outcoms th srch procss must find th pproprit txtion schm (Proposition 1 nsurs tht such txtion xists) nd finlly slct th txtion tht minimizs th ovrll tx T (G). It is importnt to not tht in th cs tht th gm lrdy hs PNE th srch procss should rturn n mpty txtion (T (G) =0). A problm riss whn on ttmpts to trnslt th bov intuition into srch procss for Booln gms. To th bst of th uthors knowldg thr r currntly no lgorithms or huristics for srching for txtion or for gnrl purpos srch in Booln gms. Th srch tsk in th prsnt ppr involvs xhustiv srch of th ntir srch spc of th Booln gm. This mns tht for th sk of finding txtion schm on should first go ovr ll possibl outcoms nd thn find th pproprit txtion schm for ch outcom. Evn prforming only th first stg of th srch rsults in th xplortion of n xponntil numbr of stts with no possibility of pruning. Svrl studis im t rducing th computtionl ffort for som Booln gms tsks. Bonzon t l. [1] xploit th dpndncy structur btwn th prsonl gols of th vrious gnts to fcilitt th computtion of PNE, by prtly dcomposing gm into svrl sub-gms tht r only loosly rltd. In nothr study [3] th uthors connct btwn Booln gms nd CP-nts [4]. Suro nd Villt [19] furthr study th dpndncy structur nd propos rduction tht rducs th srch spc whn srching for colitions. Dunn nd Wooldridg [8] study css in which Booln gms my bcom trctbl. On such cs is to srch for n ltrntiv typ of quilibri, nd nothr cs rfrs to spcific Booln gms for which finding quilibri is sir thn in gnrl. As fr s w cn tll, non of th bov pprochs cn ssist on in srching for txtion schm. In th bsnc of n ffctiv srch lgorithm for Booln gms, on nds to turn to n ltrntiv formliztion for which srch tchniqus tht nbl pruning of th srch spc do xist. Booln gms r both distributd btwn diffrnt gnts nd r gnrlly symmtric in th sns tht stt of th gm my diffrntly ffct ch of th prticipting gnts. Consquntly, th most nturl formliztion to us for this purpos is Asymmtric Distributd Constrint Optimiztion Problms (ADCOP) [12, 13]. By trnsforming th Booln gm into n ADCOP, on cn xploit xisting ADCOP srch lgorithms. In th following subsctions short rmindr of th ADCOP modl is prsntd, s wll s th procss of csting th Booln gm s n ADCOP nd prforming th srch procss. 4.1 ADCOP An ADCOP [12, 13] is tupl <A,X,D,R> whr A = {A 1,A 2,..., A n} is finit st of gnts. X = {X 1,X 2,..., X m} is finit st of vribls. Ech vribl is hld by singl gnt (n gnt my hold mor thn on vribl). D = {D 1,D 2,..., D m} is st of domins. Ech domin D i consists of th finit st of vlus tht cn b ssignd to vribl X i. R is th st of rltions (constrints). Ech constrint C R is function C : D i1 D i2... D ik k j=1 R tht dfins nonngtiv cost for vry prticipnt in vry vlu combintion of st of vribls. Th symmtry of constrints in th ADCOP modl stms from th potntilly diffrnt costs for vry prticipnt. An ssignmnt (or lbl) is pir including vribl, nd vlu from tht vribl domin. A prtil ssignmnt (PA) is th st of ssignmnts, in which ch vribl pprs t most onc. vrs(pa) is th st of ll vribls tht ppr in PA, vrs (PA)={X i D i (X i,) PA}. A constrint C R is pplicbl to PA if X i1,x i2,..., X ik vrs(pa). Th cost of prtil ssignmnt PA is th sum of ll pplicbl constrints to PA ovr th ssignmnts in PA. A full ssignmnt is prtil ssignmnt tht includs ll th vribls (vrs(pa)=x). A solution is full ssignmnt of miniml cost. 4.2 Txtion srch using ADCOP Following th intuition prsntd in th bginning of th sction, w dscrib th construction procdur of n AD- COP. Srching this ADCOP should rvl th txtion schm tht nsurs th xistnc of PNE nd imposs th miniml ovrll tx. Givn Booln gm G w dfin th ADCOP s follows: Th st of gnts in th ADCOP is xctly th st of gnts from G. Th vribls of th ADCOP r xctly th vribls from G with th sm vribl lloction. Evry domin D i consists of two vlus (0 rprsnts Fls nd 1 rfrs to Tru). For vry gnt A i construct constrint. This constrint includs vlutions of vribls tht ppr in γ i (in th following th vribls tht ppr in γ i will b dnotd by (v i1,..., v ik )). Th constrints of th ADCOP r not in th form of tbl, but r computd during srch from formul tht tks constnt computtion tim. Th dtild dscription of th constrints is blow. On wnts to slct only thos outcoms tht r not SO, so for vry spcil outcom th cost should b lrgr thn th mximl possibl ovrll tx. Txtion vlus my of cours b infinitly lrg, but sinc w r srching for txtion 187
schm with th miniml ovrll tx, w lwys rfr to th miniml ndd vlu. Thus, th mximl possibl ovrll tx M cn b computd using th following qution: M = c(p, T ru) c(p, F ls) (2) p Φ Out of ll th outcoms tht r not SO, w wnt to find th on tht minimizs th ovrll tx. Thrfor, th cost of constrint should b th ndd (miniml) tx. For ll such outcoms th tx is ndd only in th cs whr th chivmnt of th prsonl gol is indpndnt of th gnt s choic v i. This outcom is PNE only in cs th following qution holds for vry gnt A i nd vry vlution v i V i of th gnt: c(p, v i(p)) c(p, v i(p)) (3) p ϕ i p ϕ i whr v i(p) mns th Booln vlu of vribl p ccording to th choic v i (of gnt A i). Thus, txtion must b pllid in cs th qution dos not hold. In such cs on cn rwrit Eqution 3 by dding txtion in th following mnnr: c(p, v i(p)) c(p, v i(p)) + τ(p, v i(p)) (4) p ϕ i p ϕ i Th txtion cn b found by solving systm of linr qutions tht is constructd following Eqution 4. In th spcific cs of Booln gm tht rprsnts trffic light coordintion problm, vry gnt A i owns xctly on vribl p i ϕ i. Thus, Equtions 3 nd 4 cn b simplifid to: τ(p, v i(p)) = { c(p, vi(p)) c(p, v i(p)) if c(p, v i(p)) >c(p, v i(p)) 0 othrwis (5) whr v i(p) rfrs to th ngtion of v i(p). Consquntly, th bov qution dscribs th miniml txtion to singl vribl p tht nsurs tht th gnt A i holding vribl p will not hv ny incntiv to chng p s vlution. This incntiv xists whnvr th currnt cost c(p, v i(p)) is highr thn th cost of th ngtion c(p, v i(p)). So in such css th miniml ndd tx is vlu tht mks both rlvnt costs qul, i.., th tx quls th diffrnc btwn th two costs. Finlly, th cost function is dfind s follows: C Ai (v i1,..., v ik )= { M + ɛ if (vi1,..., v ik )isprt of SO p ϕ i,b B τ(p, b) othrwis (6) for ny ɛ>0. Th trm prt of SO rlts to vlution of γ i tht stisfis Dfinition 3. Eqution 6 nsurs tht no spcil outcoms would b slctd if thr is t lst on outcom which is not spcil. Not tht th constrints dfind bov r symmtric bcus ch constrint incurs cost on singl gnt. Th ADCOP s solution is full ssignmnt (v 1,v 2,..., v n) tht rprsnts PNE stt whn th pproprit txtion schm is usd. Th txtion schm is clcultd during th srch procss nd cn b stord long with its mtching ssignmnt. Not tht th txtion schm only dds costs to th ngtiv vlutions. Th rson for this is tht th solution is potntil PNE (unlss thr is no txtion schm tht chivs stbiliztion for this Booln gm), so th txtion schm only nds to mk sur tht th ngtiv vlutions r txd ccordingly, so s to rmov th gnts incntivs to mov wy from this stt. In cs th Booln gm lrdy hs t lst on PNE (without txtion) thn th ADCOP s solution will b PNE stt. Th corrctnss of this pproch stms from Proposition 1. Th ADCOP will chck ll th outcoms tht cn hv PNE stt with som txtion schm. In cs thr r no such outcoms thn th cost of th solution will b highr thn M, so th cost must b vrifid in ordr to know if thr is n pproprit txtion schm or not. According to Proposition 3, this vrifiction is not ndd in th spcil cs of Booln gm rprsnttion of th trffic light coordintion problm. Th corrctnss of th ovrll tx s minimlity coms dirctly from th corrctnss of th chosn ADCOP lgorithm. Endriss t l. [9] suggst tht txtion schms my lso b usd to incntiviz gnts to rch som socilly dsirbl outcom. For instnc, in th trffic lights coordintion problm such socil outcom my b th formtion of long grn wv (longr thn th mini-grn-wv of siz 3 tht srvs s n gnt s prsonl gol). In ordr to fcilitt such socil outcom, th only chng to th ADCOP is th ddition of nothr constrint to on of th gnts. Th cost of this (globl) constrint is M + ɛ (for ny ɛ>0) for vlutions in which th socil outcom is not chivd (nd 0 othrwis). 4.3 k-ry SyncABB-1ph Grubshtin t l. [12] introducd svrl complt AD- COP lgorithms. Th most simpl, yt ffctiv, complt ADCOP lgorithm is SyncABB-1ph, which is n symmtric vrsion of th fmous Synchronous Brnch & Bound (SyncBB) lgorithm [15]. Aftr ch stp of th lgorithm, whn n gnt dds n ssignmnt to th Currnt Prtil Assignmnt (CPA) nd updts on dirction of th bound, th CPA is snt bck to th ssignd gnts to updt its bound by th costs of ll bckwrds dirctd constrints (bck-chcking). This is don by rplcing th CPA MSG mssg snt ftr ch vlu ssignmnt to th nxt gnt with CPA BACK MSG mssg to th prcding gnt. Th problm with ll th prsntd ADCOP lgorithms, including SyncABB-1ph, is tht thy wr dvlopd to hndl binry constrints, whrs th constrints in th bov constructd ADCOP r k-ry. Thus, w slightly djust th SyncABB-1ph lgorithm to hndl k-ry constrints: As in th originl psudo-cod [12], A i rfrs to th gnt tht currntly holds th CPA, A j rprsnts th gnt tht inititd th currnt bck-chcking, A n is th lst gnt in th ordr, nd B dnots th currnt bound. Th modifictions w md r to nsur tht th CPA hs ll th ssignmnts ndd for clculting th constrint cost nd tht th lst ssignmnt ws ddd by nighbor of th currnt gnt (lin 2). Th scond condition ssurs tht vry constrint cost is vlutd only onc for vry full ssignmnt. Morovr, th constrint cost is clcultd using th ntir CPA rthr thn with singl gnt (lin 3). In cs A i is th lst gnt in this prticulr constrint thn th cost is ddd s in th rgulr SyncBB lgorithm (with th slight chng of th constrint bing k-ry). 188
Algorithm 1 k-ry SyncABB-1ph: bck-chcking whn rcivd CPA BACK MSG, CPA,cost do 1: j CPA.lstId 2: if ll vribls from A i s constrint r in th CPA nd A j is nighbor of A i thn 3: f cost of th constrint with th CPA 4: ls 5: f 0 6: if cost + f B thn 7: snd CPA MSG, CPA to A j 8: ls if A i A 1 thn 9: snd CPA BACK MSG, CPA,cost+ f to A i 1 10: ls if A j = A n thn 11: B cost + f 12: brodcst NEW SOLUTION, CPA,B 13: snd CPA MSG, CPA to A n 14: ls 15: CPA.cost cost + f 16: snd CPA MSG, CPA to A j+1 5. EXPERIMENTAL EVALUATION Th Booln gms usd in th following xprimnts rprsnt trffic lights coordintion problms of diffrnt grid sizs 3x3, 4x4, 5x5, nd 6x6. Problms wr rndomly gnrtd nd th rportd rsults r vrgs ovr 100 diffrnt xprimnts for ch stting. 5.1 Problm gnrtion For ch xprimnt rndom problm ws gnrtd. First, trffic lights coordintion problm ws gnrtd by rndomly slcting th numbr (in th rng [0,mx cost)) of vhicls in th ln btwn vry two djcnt trffic lights. Nxt, th pproprit Booln gm rprsnting th gnrtd trffic lights coordintion problm ws constructd ccording to th ruls dscribd in Sction 3.2. Thn, n ADCOP problm ws gnrtd from th Booln gm using th procdur dscribd in Sction 4.2. Finlly, th problm ws solvd using th k-ry SyncABB-1ph lgorithm tht ws prsntd in Sction 4.3. 5.2 Exprimntl rsults Th first prt of th xprimntl vlution is imd to hlp on undrstnd th proprtis of th rndomly gnrtd gms. For this purpos two msurs r considrd. E 60 N P h 50 it w s 40 m g 30 f o g 20 t n 10 r c P 0 3x3 4x4 5x5 6x6 Grid siz mx-cost = 1000 mx-cost = 100 mx-cost = 10 Figur 3: Prcntg of gms tht hv PNE without ny txtion schm Figur 3 prsnts th first msur th prcntg of gms tht originlly hv PNE stt (without txtion). It is sy to s tht th probbility for PNE drops whn th problms bcom lrgr nd whn th rng of costs (siz of mx-cost) is widr. Th costs ffct th xistnc of PNE bcus two vlutions of th sm vribl hv diffrnt costs, which cnnot b blncd for n quilibrium. Th fct tht lrgr rng of costs incrss th probbility tht th costs will b diffrnt, xplins th ffct of mxcost. Th ffct of th grid siz cn lso b xplind, sinc it is clrly hrdr to find stbl stts whn thr r mor plyrs prticipting in th gm. Th scond msur is th siz of th ovrll tx with rspct to th originl costs of th gm. Th prcntg of th ovrll tx is clcultd s follows: p Φ,b B τ(p, b) 100% (7) c(p, b) x t l r v o f o g t n c r P 8 7 6 5 4 3 2 1 0 p Φ,b B 3x3 4x4 5x5 6x6 Grid siz mx-cost = 1000 mx-cost = 100 mx-cost = 10 Figur 4: Prcntg of ovrll tx Whil it ws shown tht both th dimnsions of th problm nd th rng of costs ffct th xistnc of PNE, th ndd tx lod to chiv stbiliztion dos not sm to b ffctd by ths prmtrs. Figur 4 shows tht th ovrll ndd tx is only bout 4% in ll of th problm sttings in th vlution. 1000000 900000 800000 700000 600000 s C500000 C C 400000 N 300000 200000 100000 0 3x3 4x4 5x5 6x6 Grid siz mx-cost = 1000 mx-cost = 100 mx-cost = 10 Figur 5: Mn numbr of NCCCs In ordr to vlut th lgorithm, w considr th mn numbr of Non-Concurrnt Constrint Chcks (NCCCs), which is commonly usd msur for th runtim prformnc of distributd constrints lgorithms [20]. Th xponntil growth of th computtionl lod with rspct to th problm siz is clrly sn in Figur 5 nd is of no surpris s ADCOPs r NP-Hrd problms. In contrst, th rng of mx-cost dos not sm to hv ny ffct on th prformnc. 189
In ordr to undrstnd th impct of th pruning tht ws nbld by th trnsformtion into n ADCOP, on cn compr to niv lgorithm tht xhustivly trvrss th ntir srch spc of th Booln gm. Th niv lgorithm must run through vry possibl outcom, chck if it is spcil outcom, nd in cs it is not n SO clcult th ovrll tx. Th pproximt complxity of this niv pproch is n 2 n (thr r 2 n possibl outcoms nd th lgorithm must trvrs ll th vribls in ordr to clcult th ovrll tx). Grid siz k-ry SyncABB-1ph Niv pproch 3x3 550 4, 608 4x4 6, 468 1, 048, 576 5x5 63, 841 838, 860, 800 6x6 893, 863 2, 473, 901, 162, 496 Tbl 1: Runtim prformnc of k-ry SyncABB- 1ph vs. niv pproch Tbl 1 comprs th NCCC rsults of th k-ry SyncABB- 1ph lgorithm with th pproximt numbr of oprtions prformd by th niv pproch (problms with mx-cost = 1000). Although th computtionl tim of n NCCC my b somwht diffrnt thn tht of th niv pproch s oprtion, th diffrnc in ordrs of mgnitud btwn th two ltrntivs stblishs th grt impct of th pruning tht is chivd whn th ADCOP rprsnttion is usd. 6. CONCLUSIONS Txtion schms tht impos dditionl costs on th ctions of gnts in Booln gm my in som css incntiviz th gnts to rch stbl stt. Th prsnt ppr thorticlly outlins th chrctristics of Booln gms for which stbiliztion cn b chivd by pplying txtion schm. Whn Booln gm is on tht mts th thorticl critri, on must srch for th most pproprit txtion schm. Th prsnt ppr proposs mthod tht ffctivly srchs for th txtion schm. Th proposd mthod trnsforms th Booln gm into n Asymmtric Distributd Constrint Optimiztion Problm (ADCOP). Th rsulting ADCOP nbls n ffctiv srch by using n xisting lgorithm with som minor djustmnts. Th mthod is vlutd on Booln gms tht rprsnt trffic light coordintion gm of diffrnt grid sizs. Th runtim prformnc of th proposd mthod ws shown to b bttr by svrl ordrs of mgnitud thn niv pproch for finding txtion, tht xhustivly gos ovr ll th possibl outcoms of th Booln gm. Th substntil dvntg of th proposd mthod stms from th ffctiv pruning of th srch spc tht is inhrnt to th usd ADCOP lgorithm. 7. REFERENCES [1] E. Bonzon, M.-C. Lgsqui-Schix, nd J. Lng. Dpndncis btwn plyrs in Booln gms. Int. J. Approx. Rsoning, 50(6):899 914, Jun 2009. [2] E. Bonzon, M.-C. Lgsqui-Schix, J. Lng, nd B. Znuttini. Booln gms rvisitd. In ECAI, pgs 265 269, 2006. [3] E. Bonzon, M.-C. Lgsqui-Schix, J. Lng, nd B. Znuttini. Compct prfrnc rprsnttion nd Booln gms. Autonomous Agnts nd Multi-Agnt Systms, 18:1 35, 2009. 10.1007/s10458-008-9040-2. [4] C. Boutilir, R. I. Brfmn, H. H. Hoos, nd D. Pool. CP-nts: A tool for rprsnting nd rsoning with conditionl Ctris Pribus prfrnc sttmnts. Journl of Artificil Intllignc Rsrch, 21:2004, 2003. [5] J. J. Cords. Horizontl quity. In Th Encyclopdi of Txtion nd Tx Policy. Urbn Institut Prss, 1999. [6] D. d Olivir, A. L. C. Bzzn, nd V. R. Lssr. Using cooprtiv mdition to coordint trffic lights: cs study. In AAMAS, pgs 463 470, 2005. [7] P. E. Dunn, W. vn dr Hok, S. Krus, nd M. Wooldridg. Cooprtiv Booln gms. In AAMAS (2), pgs 1015 1022, 2008. [8] P. E. Dunn nd M. Wooldridg. Towrds trctbl Booln gms. In AAMAS, pgs 939 946, 2012. [9] U. Endriss, S. Krus, J. Lng, nd M. Wooldridg. Dsigning incntivs for Booln gms. In AAMAS, pgs 79 86, 2011. [10] U. Endriss, S. Krus, J. Lng, nd M. Wooldridg. Incntiv nginring for Booln gms. In IJCAI, pgs 2602 2607, 2011. [11] J. Grnt, S. Krus, M. Wooldridg, nd I. Zuckrmn. Mnipulting Booln gms through communiction. In IJCAI, pgs 210 215, 2011. [12] A. Grubshtin, T. Grinshpoun, A. Misls, nd R. Zivn. Asymmtric distributd constrint optimiztion. In Proc. of th 11th intrntionl workshop on Distributd Constrint Rsoning t IJCAI 09, Psdn CA, Unitd Stts, July 2009. [13] A. Grubshtin, R. Zivn, T. Grinshpoun, nd A. Misls. Locl srch for distributd symmtric optimiztion. In AAMAS, pgs 1015 1022, 2010. [14] P. Hrrnstin, W. vn dr Hok, J.-J. Myr, nd C. Wittvn. Booln gms. In Procdings of th Eighth Confrnc on Thorticl Aspcts of Rtionlity nd Knowldg, pgs 287 298, Sn Mto, CA, 2001. Morgn Kufmnn Publishrs. [15] K. Hirym nd M. Yokoo. Distributd prtil constrint stisfction problm. In Procdings of th Third Intrntionl Confrnc on Principls nd Prctic of Constrint Progrmming (CP-97), pgs 222 236, 1997. [16] R. Jungs nd A. L. C. Bzzn. Evluting th prformnc of DCOP lgorithms in rl world, dynmic problm. In AAMAS (2), pgs 599 606, 2008. [17] C. H. Ppdimitriou. Computtionl complxity. Addison-Wsly, 1994. [18] Robrtson. TRANSYT: A trffic ntwork study tool. Rp. LR 253, Rod Rs. Lb., London, 1969. [19] L. Suro nd S. Villt. Dpndncy in cooprtiv Booln gms. Journl of Logic nd Computtion, 2011. [20] R. Zivn nd A. Misls. Mssg dly nd DisCSP srch lgorithms. Annls of Mthmtics nd Artificil Intllignc (AMAI), 46:415 439, 2006. 190