Seprle isrete funtions: reognition n suffiient onitions Enre Boros Onřej Čepek Vlimir Gurvih Novemer 21, 217 rxiv:1711.6772v1 [mth.co] 17 Nov 217 Astrt A isrete funtion of n vriles is mpping g : X 1... X n A, where X 1,...,X n, n A re ritrry finite sets. Funtion g is lle seprle if there exist n funtions g i : X i A for i = 1,...,n, suh tht for every input x 1,...,x n the funtion g(x 1,...,x n ) tkes one of the vlues g 1 (x 1 ),...,g n (x n ). Given isrete funtion g, it is n interesting prolem to sk whether g is seprle or not. Although this seems to e very si prolem onerning isrete funtions, the omplexity of reognition of seprle isrete funtions of n vriles is known only for n = 2. In this pper we will show tht slightly more generl reognition prolem, when g is not fully ut only prtilly efine, is NP-omplete for n 3. We will then use this result to show tht the reognition of fully efine seprle isrete funtions is NP-omplete for n 4. The generl reognition prolem ontins the ove mentione speil se for n = 2. This se is well-stuie in the ontext of gme theory, where (seprle) isrete funtions of n vriles re referre to s (ssignle) n-person gme forms. There is known suffiient onition for ssignility (seprility) of two-person gme forms (isrete funtions of two vriles) lle (wek) totl tightness of gme form. This property n e teste in polynomil time, n n e esily generlize oth to higher imension n to prtilly efine funtions. We will prove in this pper tht wek totl tightness implies seprility for (prtilly efine) isrete funtions of n vriles for ny n, thus generlizing the ove result known for n = 2. Our proof is onstrutive. Using grph-se isrete lgorithm we show how for given wekly totlly tight (prtilly efine) isrete funtion g of n vriles one n onstrut seprting funtions g 1,...,g n in polynomil time with respet to the size of the input funtion. Keywors: seprle isrete funtions, totlly tight n ssignle gme forms MSIS Dep. of RBS n RUTCOR, Rutgers University, 1 Rokfeller Ro, Pistwy, NJ 8854-854, USA. (enre.oros@rutgers.eu) Deprtment of Theoretil Informtis n Mthemtil Logi, Chrles University, Mlostrnské nám. 25, 118 Prh 1, Czeh Repuli. (onrej.epek@mff.uni.z) MSIS Dep. of RBS n RUTCOR, Rutgers University, 1 Rokfeller Ro, Pistwy, NJ 8854-854, USA; Ntionl Reserh University Higher Shool of Eonomis, Mosow Russi. (vlimir.gurvih@rutgers.eu) 1
1 Introution A isrete funtion of n vriles is mpping g : X 1... X n A, where X 1,...,X n, n A re ritrry finite sets. Disrete funtion g is lle seprle if there exist n seprting funtions of one vrile eh, g i : X i A for i = 1,...,n, suh tht g(x) = g 1 (x 1 ) or... or g(x) = g n (x n ) for every input x = (x 1,...,x n ) X 1 X n. To see some simple exmples for seprle n non-seprle isrete funtions onsier the following isrete funtions of two vriles. For n = 2 we n interpret isrete funtion s n rry, where the rows re inexe y X 1, the olumns y X 2, n the entries of the rry re the g(x 1,x 2 ) A vlues for x 1 X 1 n x 2 X 2. [ ] [ ] [ ] (1) The first two exmples re lerly seprle (we leve to the reer the esy tsk of efining the two seprting funtions, i.e. of ssigning the orret vlues to rows n olumns) while the lst three exmples re not (whih is esy to prove y piking n entry in the rry, ssigning this vlue first to the orresponing row n then to the orresponing olumn, n following in oth ses the sequene of fore ssignments to ontrition). The onept of seprility n e nturlly extene to prtilly efine isrete funtions of n vriles y requiring the ove property only for those inputs for whih the funtion is efine. Note tht the seprting funtions n e lwys ssume to e fully efine - inee if set of n prtilly efine seprting funtions fulfills the ove onition, then so oes ny ompletion of this set to fully efine funtions. Severl onepts of eomposility for prtilly efine Boolen funtions ( speil sulss of isrete funtions) re surveye, for exmple, in [3]. Although isrete funtions re generl onept foun in mny res of isrete mthemtis, our motivtion n interest in stuying them me from gme theory, n so we will in the rest of this pper swith to the stnr gme theoretil terminology. We enote y I = [n] = {1,...,n} the set of plyers, y X i the set of strtegies of plyer i I, n y A the set of possile outomes. A isrete funtion g : X 1... X n A of n vriles is lle n n-person gme form. In other wors, one ll plyers hose prtiulr strtegy, sy x i X i for i I, then g(x 1,...,x n ) enotes the orresponing outome of the gme. The vetor of the hosen prtiulr strtegies, x = (x 1,...,x n ) X 1 X n is lle strtegy profile. In se g is seprle, the seprting funtion g i ssigns n outome to every strtegy of plyer i, n so in this ontext the term ssignility (or property AS in short) is use inste of seprility. The prolem stuie in this pper n e formulte s follows: given n n-person(prtilly efine) gme form (n n-vrite isrete funtion) eie whether it is ssignle (seprle). We shll see in Setion 3 tht this prolem is solvle in polynomil time for n = 2 n tht it is omputtionlly hr for n 3 for the prtilly efine se, n for n 4 for the fully efine se. In those situtions when reognizing ssignility is iffiult, it mkes sense to look for onitions tht (i) re suffiient for ssignility n (ii) n e teste in polynomil time. Suh onitions will e stuie in Setion 2. First, let us rell results relte to the se of two plyers. This se hs een most stuie n est unerstoo so fr. 2
1.1 The two-person se Sine ssignility is oviously hereitry property (ny suform of n ssignle gme form, inue y susets of the strtegy sets of the plyers, is oviously lso ssignle), nturl question rises, whether ssignle gme forms mit hrteriztion y finite set of forien suforms. Interestingly, this is not the se lrey for two-person gme forms, s the following infinite fmily (from [2]) of non-ssignle gme forms emonstrtes it: (2) It is not hr to see, tht eh suh gme form is not ssignle, while removl of ny row or olumn mkes it ssignle, i.e. the presente gme forms re miniml non-ssignle. On the other hn, it n e shown [2], tht if ssignility is me more restritive y requiring tht every strtegy profile is overe y extly one of the plyers (either g(x 1,x 2 ) = g 1 (x 1 ) or g(x 1,x 2 ) = g 2 (x 2 ) ut not oth), then suh strong ssignility n e hrterize y finite set of miniml non-ssignle gme forms. Note lso, tht ny two person gme form with t most two outomes is lerly ssignle (ssign one outome to ll rows n the other outome to ll olumns), n hene the gme forms in (2) use the miniml numer of three outomes neee for non-ssignility. This oservtion n e generlize to the se of n plyers: ny n-person gme form with t most n outomes is lerly ssignle, n it is esy to onstrut non-ssignle n-person gme forms with n+1 outomes. Suh onstrution proees s follows. Tke n m m m gme form, n for eh of the first n outomes selet m strtegy profiles (thus we nee nm < m n to hve enough strtegy profiles) suh tht no two of them shre ny oorinte vlue. In the two plyer se these re two permuttion sumtries, in (2) the min igonl with outome n the igonl ove it plus the ottom left orner with outome. Eh of these m-tuples of strtegy profiles requires m ifferent plyer strtegies to over ll m outomes. For eh outome k we nee g i (x j ) = k for m istint pirs (i,j)), n so ltogether they use ll nm ville strtegies g i (x j ). Hene, putting the lst outome n+1 to ny still vnt strtegy profile uses the gme form to e non-ssignle. Let us finlly remrk, tht this onstrution works for ny n n m suh tht nm < m n, ut to mke suh gme form miniml non-ssignle is proly very triky tsk (we o not see ny esy wy how to generlize the onstrution of miniml non-ssignle gme forms to ny n > 2). Severl other properties of two-person gme forms whih re onnete to ssignility were stuie intheliterture. Animportnt propertyis tightness. Wesy tht plyer n gurntee susetb A of outomes, if he/she hs strtegy suh tht no mtter wht strtegy the other plyer hooses, the orresponing outome elongs to B. A two-person gme form g : X 1 X 2 A is lle tight (hs property T, in short) if for ny suset B A either one plyer n gurntee B or the other plyer n gurntee B = A\B. Note tht oth of the ove nnot hppen, while there my e suset B A in non-tight gme form suh tht neither plyer 1 n gurntee B, nor plyer 2 n gurntee B. The importne of tightness stems prtly from the ft tht it is in the two plyer se equivlent to Nsh-solvility of gme form [7, 8, 9], whih is pivotl onept in gme theory. No polynomil time lgorithm for verifying tightness is known, however qusi-polynomil one ws suggeste y Fremn n Khhiyn in [5]. Another property of gme forms relte to tightness is totl tightness. A two-person gme form g : X 1 X 2 A is lle totlly tight (TT) if every 2 2 suform of g (whih is two-imensionl rry in this se) is tight, or equivlently, if it ontins onstnt line (i.e. row or olumn). More preisely, 3
let us ll g : X 1 X 2 A to e 2 2 restrition of g if X 1 = {x 1,x 1 } X 1 n X 2 = {x 2,x 2 } X 2 re 2-element susets of X 1 n X 2. Then g is TT if for every 2 2 restrition g of g we hve g (x 1,x 2 ) = g (x 1,x 2), or g (x 1,x 2 ) = g (x 1,x 2 ), or g (x 1,x 2) = g (x 1,x 2 ), or g (x 1,x 2) = g (x 1,x 2). It ws shown in [4] tht totlly tight two-person gme forms re oth tight n ssignle. It is esy to oserve, tht heking whether two-person gme form is TT n e one in polynomil time, s there is only polynomil numer of 2 2 suforms to hek (this esy ft ws oserve e.g. in [1] ). Thus property T T provies simple suffiient onition for the ssignility of two-person gme forms. 1.2 The n-person se for n 3 As we shll rell in Setion 3 tht ssignility of two-person gme forms n e teste in polynomil time. A nturl ie is to try to reue the ssignility of higher imensionl gme forms to the two-imensionl se y onsiering projetions. Let g : X 1... X n A e n-person gme form. An i-th two imensionl projetion g i of g is two person gme form where the set of strtegies of the first plyer onsists of the strtegies of plyer i n the set of strtegies of the seon plyer is the iret prout of ll the strtegies of the remining n 1 plyers. Is it true tht g is ssignle if n only if g i is ssignle for every i? Unfortuntely not. Both implitions fil lrey for n = 3. First we shll show n exmple of 3 3 3 three-person gme form whih is not ssignle, ut eh of its three two-imensionl 3 9 projetions is ssignle. The gme form is given in Figure 1. 7 4 1 5 2 3 8 9 6 Figure 1: A non-ssignle 3D exmple. Clerly, this three person gme form is not ssignle s there re 1 outomes ut only 9 vlues to e ssigne. On the other hn, eh of the three two-imensionl projetions is 9 3 two-person gme form whih is ssignle y ssigning to ll three olumns n ssigning outomes 1 to 9 to the rows (note tht eh row ontins extly two s n extly one other outome in eh of the three projetions). Seon we shll show n exmple of 3 3 3 three-person gme form whih is ssignle, ut none of its three two-imensionl projetions is ssignle. The gme form is given in Figure 2. In this exmple the symol stns for n ritrry outome from the set {,,}. Sine there re only three outomes, the three-person gme form is trivilly ssignle y ssigning outome to plnes orthogonl to iretion 1, outome to plnes orthogonl to iretion 2, n outome to 4
Figure 2: Assignle 3D exmple tht hs no 2D ssignle projetion, where symol n e ny of, or. plnes orthogonl to iretion 3. On the other hn, eh of the three 9 3 two-imensionl projetions is gme form tht ontins 2 3 suform [ ] whih is not ssignle, n hene none of the projetions is ssignle (note tht this 2 3 suform is the mile exmple in (1)). Therefore, to otin suffiient onitions for ssignility for n 3 we nee to look t other properties of gme forms, e.g. generliztions of the properties stuie in the two-person se. The onepts of tightness n totl tightness n e extene to the generl n-person se in the following wy. Given n n-person gme form g : X 1... X n A n prtition I = K K of the plyers into two omplementry non-empty olitions, the two-person gme form g K : X K X K A is efine s follows. The strtegies of the first n seon plyers re the elements of the iret prouts X K = i K X i n X K = i K X i, respetively. For x X K n x X K we efine g K (x,x) = g(y) where y origintes from x n x y ontenting them n reorering the oorintes oring to the X 1,...,X n orer. An n-person gme form g is lle tight (respetively, totlly tight) if g K is tight (respetively, totlly tight) for ll non-empty K I. Similrly, we ll g wekly tight (respetively, wekly totlly tight if g K is tight (respetively, totlly tight) for ll K I suh tht K = 1. Note, tht in this se we onsier extly the two-imensionl projetions efine ove in the first prgrph of this susetion. We shll enote these onepts y WT n WTT, respetively. Let us remrk tht, y the ove efinition, T = WT n TT = WTT for n 3. Inee, inthisse fornynon-trivil prtition oneofthetwo olitions ontins onlyoneplyer. Let us lso remrk, tht the ove efine onepts (T, WT, TT, n WTT) n e strightforwrly extene to prtilly efine gme forms y repling ll unefine vlues of the gme form y single extr outome, n y requiring the orresponing property for the resulting fully efine gme form. Let us oserve tht testing whether three-person gme form g is (wekly) totlly tight n e one in polynomil time. In ft, s we shll show in Setion 3, property WTT n e teste in polynomil time (with respet to the size of g) for ny n. Therefore, property WTT my e goo nite for 5
property tht we re looking for, property whih n e teste in polynomil time n whih implies ssignility. This les to nother results of this pper, nmely tht wek totl tightness implies ssignility of n-person gme forms (oth prtilly n fully efine) for ll n. The struture of the pper is s follows. In Setion 2 we prove tht WTT AS for every n, tht is, tht property W T T implies ssignility for every n. In Setion 3 we prove tht eiing ssignility of prtilly efine gme forms is NP-omplete for n 3, n tht eiing ssignility of fully efine gme forms is NP-omplete for n 4. We lose the pper y proviing further onnetions to gme theory in Setion 4, n y listing some open prolems in Setion 5. 2 Wek totl tightness implies ssignility When eling with gme forms it is sometimes onvenient to think of the n-person gme form s of n n-imensionl rry n use geometri interprettion for surrys. A line in iretion i is set of strtegy profiles ( 1-imensionl surry) where ll oorintes re fixe n only oorinte i is use s running inex. In gme theoreti terms line in iretion i is 1-imensionl suform otine y fixing the strtegies of ll plyers exept of plyer i. A hyperplne perpeniulr to iretion i is set of strtegy profiles (n (n 1)-imensionl surry) where ll oorintes re use s running inies n only oorinte i is fixe. In gme theoreti terms hyperplne perpeniulr to iretion i is n (n 1)-imensionl suform otine y fixing the strtegy of plyer i. Definition 1 Given gme form g, set S X of strtegy profiles will e lle onstnt region if ll strtegy profiles in S get the sme outome, i.e. if there exists n outome A) suh tht for ll strtegy profiles x S we hve g(x) =. Remrk 2 We will ssume in the reminer of this pper tht the gme form we re eling with ontins no onstnt hyperplne ( hyperplne whih is onstnt region) n no pir of uplite prllel hyperplnes. These ssumptions n e me without loss of generlity s suh hyperplnes oviously influene neither totl tightness nor ssignility of the onsiere gme forms. Let us lso note tht for WTT gme form g the entries in ny 2 2 surry of g {i} (rell tht g {i} is two plyer gme form where plyer i onstitutes one person olition n ll other plyers onstitute the omplementry olition) n e geometrilly thought of s the four intersetions of two ritrry istint lines in iretion i with two ritrry istint (n 1)-imensionl hyperplnes perpeniulr to iretion i. Let us strt with simple lemm esriing forien sustruture for WTT gme forms. Lemm 3 Let g e n n-person gme form, i n ritrry iretion (plyer), n H j,h k e two istint prllel hyperplnes perpeniulr to iretion i. Furthermore let l i,1 i 4 e four lines (not neessrily ll istint) in iretion i interseting H j in strtegy profiles x i j,1 i 4 n interseting H k in strtegy profiles x i k,1 i 4, suh tht (1) g(x 1 j ) g(x2 j ), (2) g(x 3 k ) g(x4 k ), (3) g(x i j ) g(xi k ),1 i 4. Then g is not WTT gme form. 6
H j H k g(x 1 j) g(x 1 k) l 1 g(x 2 j) g(x 2 k) l 2 g(x 3 j) g(x 3 k) l 3 g(x 4 j) g(x 4 k) l 4 Figure 3: A forien onfigurtion in WTT gme forms s in Lemm 3. Proof. Using the inequlities (1) n (3) for the quruple (x 1 j,x2 j,x1 k,x2 k ) we either get iret ontrition to the WTT property of g or we get g(x 1 k ) = g(x2 k ) = k for some outome k A. Similrly using (2) n (3) for the quruple (x 3 j,x4 j,x3 k,x4 k ) we either get iret ontrition to the WTT property of g or we get g(x 3 j ) = g(x4 j ) = j for some outome j A. So let us ssume the ltter in oth ses. Now using (1) we get tht one of g(x 1 j ),g(x2 j ) must iffer from j, so let us enote the iffering strtegy profile y x u j, for u {1,2}. Similrly, using (2) we get g(x3 k ) k or g(x 4 k ) k n let us enote the iffering strtegy profile y x v k, for v {3,4}. This ltogether implies tht the quruple (x u j,xv j,xu k,xv k ) ontrits the WTT property of g, see Figure 3. Note tht if some of the lines l i,1 i 4 oinie (of ourse y the ssumptions l 1 must iffer from l 2 n l 3 must iffer from l 4 ), the proof eomes even simpler. If l 1 = l 3 n l 2 = l 4 (or l 1 = l 4 n l 2 = l 3 ) then the four intersetions immeitely give ontrition to the WTT property. If l 1 = l 3 n l 2 l 4 then the proof ove goes through for u = 2 n v = 4. Symmetrilly, if l 1 l 3 n l 2 = l 4 then the proof ove goes through for u = 1 n v = 3. Let us now efine nottion for speil hyperplne prtitions n stte n prove key property of these prtitions. Definition 4 Let g e n n-person gme form n i n ritrry iretion (plyer). Let H j n H k e two istint prllel hyperplnes perpeniulr to iretion i. For n ritrry line l in iretion i let us enote y x l j n xl k the strtegy profiles t the intersetions of line l with hyperplnes H j n H k respetively. We efine prtition of H j into H j = (k) n H j (k) s follows: H j = (k) = {xl j g(xl j ) = g(xl k )} n H j (k) = {xl j g(xl j ) g(xl k )} Lemm 5 Let g e n n-person WTT gme form, i n ritrry iretion (plyer), n H j,h k e two istint prllel hyperplnes perpeniulr to iretion i. Then H j (k) is onstnt region or H k (j) is onstnt region (or oth re onstnt regions). 7
Proof. Assume y ontrition tht neither H j (k) nor H k (j) is onstnt region. This mens tht there exist four lines l i,1 i 4 (not neessrily ll istint) in iretion i interseting H j (k) in strtegy profiles x i j,1 i 4 n interseting H k (j) in strtegy profiles xi k,1 i 4 suh tht (1) g(x 1 j ) g(x2 j (2) g(x 3 k ) g(x4 k (3) g(x i j ) g(xi k ) (euse H j (k) ontins two istint outomes), ) (euse H k (j) ontins two istint outomes), ),1 i 4 (y the efinition of H j (k) n H k (j)). The four lines l i,1 i 4 n their intersetions with hyperplnes H j,h k oviously fulfil the ssumptions of Lemm 3 n hene g is not WTT whih is ontrition. Definition 6 Let g e n n-person WTT gme form, i n ritrry iretion (plyer), n H j,h k e two istint prllel hyperplnes perpeniulr to iretion i. If H j (k) is onstnt region for some outome, then we sy tht H j omintes H k y n enote this ft y H j H k. If H j H k n there exists no outome suh tht H k H j then we sy tht H j stritly omintes H k y n write H j = H k. Note tht H j = H k implies tht H k (j) is not onstnt region. Using the just efine nottion, Lemm 5 n the ft tht we hve no two ientil prllel hyperplnes y Remrk 2 implies the following esy orollry. Corollry 7 Let g e n n-person WTT gme form, i n ritrry iretion (plyer), n H j,h k e two istint prllel hyperplnes perpeniulr to iretion i. Then extly one of the following three onitions is true 1. H j = H k for some outome n there exist two istint outomes in H k (j) (whih re oth ifferent from ) 2. H k = H j for some outome n there exist two istint outomes in H j (k) (whih re oth ifferent from ) 3. H j H k n H k H j for some outomes. Remrk 8 It shoul e note tht Corollry 7 gives omplete hrteriztion of WTT gme forms. Nmely, gme form is WTT if n only if ny pir of prllel hyperplnes fulfills extly one of the three properties speifie in Corollry 7. The left to right implitions is prove in Lemm 5 while the reverse implition is trivil. Now we shll show tht hyperplne nnot stritly ominte two other hyperplnes y two ifferent outomes. Lemm 9 Let g e n n-person WTT gme form, i n ritrry iretion (plyer), n H l,h j,h k three istint prllel hyperplnes perpeniulr to iretion i suh tht H l = H j n H l = H k for some outomes n. Then =. 8
Proof. Assume y ontrition tht. Then, hyperplne H l n e prtitione into three isjoint regions, nmely the onstnt region H l (j), onstnt region H l (k), n region H= l (j) H= l (k) where the outomes re the sme in ll three hyperplnes for ny perpeniulr line in iretion i. This in prtiulr implies tht H l (j) H= l (k) n H l (k) H= l (j), see Figure 4. Now H l = H j implies tht there exist two istint lines l 1,l 2 in iretion i interseting H l (j) (n thus lso H l = (k)) in profiles x1 l,x2 l for whih g(x1 l ) = g(x2 l ) =, interseting H j (l) in profiles x 1 j,x2 j for whih g(x1 j ) = 1 2 = g(x 2 j ) (here we use the strit omintion), n interseting H= k (l) in profiles x 1 k,x2 k for whih g(x1 k ) = g(x2 k ) =. Note tht the outomes 1, 2, re pirwise istint. Similrly, H l = H k implies tht thereexist two istint lines l 3,l 4 in iretion i interseting H l (k) (n thus lso H l = (j)) in profiles x3 l,x4 l for whih g(x3 l ) = g(x4 l ) =, interseting H= j (l) in profiles x3 j,x4 j for whih g(x 3 j ) = g(x4 j ) =, n interseting H k (l) in profiles x3 k,x4 k for whih g(x3 k ) = 1, g(x 3 k ) = 2 for 1, 2, pirwise istint. H j = H l = H k 1 2 = = l 1 l 2 H l (j) H= l (k) = = 1 2 l 3 l 4 H l (k) H= l (j) Figure 4: Strit ominne y two ifferent outomes les to the forien onfigurtion s in Lemm 3. It is esy to hek tht the four pirwise istint lines l i,1 i 4 n their intersetions with hyperplnes H j,h k fulfil the ssumptions of Lemm 3 n hene g is not WTT whih is ontrition. Lemm 9 llows us to ssoite unique outome to every hyperplne tht stritly omintes t lest one other hyperplne. Definition 1 Let g e n n-person WTT gme form, i n ritrry iretion (plyer), n H j e hyperplne perpeniulr to iretion i. If there exists hyperplne H k prllel to H j n n output suh tht H j = H k then we ll the proper outome of H j. Note tht hyperplne H j tht oes not stritly ominte ny other hyperplne must hve the property (y Corollry 7), tht it is ominte (non-stritly or stritly) y every other hyperplne prllel to H j. We shll ll suh hyperplnefor whih noproperoutome is efinesink hyperplne. 9
Definition 11 Let g e n n-person WTT gme form, i n ritrry iretion (plyer), n H j e hyperplne perpeniulr to iretion i. We shll ll H j sink hyperplne if for every hyperplne H k, k j, perpeniulr to iretion i there exists n outome k suh tht H k k Hj. If there exist no sink hyperplne in ny iretion then g is lle no-sink gme form. Definitions 1 n 11 llow us to formulte the following simple orollry. Corollry 12 Let g e n n-person WTT gme form, i n ritrry iretion (plyer), n H j e hyperplne perpeniulr to iretion i. Then either H j hs proper outome or it is sink hyperplne. We will now stuy no-sink WTT gme forms. In suh gme form every hyperplne stritly omintes t lest one of its prllel hyperplnes, whih in turn implies tht there must e yle (or severl yles) of strong ominne reltions mong ll hyperplnes in every iretion. Note tht suh gme forms exist, onsier for instne the following 2-person gme form (3) in whih R 1 = R 3, R 3 = R 2, n R 2 = R 1, where R i is the i-th row of the gme form. The sme yle of strit ominne hols mong the olumns C 1, C 2, n C 3 ue to the symmetry w.r.t. the min igonl. Generting n exmple for three plyers is muh more iffiult to o y hn, ut two suh gme forms were omputer generte using oe of V.Oulov. Eh of them is 3 3 3 rry isplye elow s set of three 2-imensionl 3 3 surrys (hyperplnes). The first exmple is shown in Figure 5. = H 1 H 3 = H 2 = Figure 5: First no-sink 3D exmple. Clerly H 1 = H 3, H 3 = H 2, n H 2 = H 1. Note tht H 3 is extly the two imensionl gme form from (3) n the reltions from there rry over to the row hyperplnes R i, 1 i 3 (in 1
prtiulr R 1 = R 3, R 3 = R 2, n R 2 = R 1 ) n ue to symmetry lso to the olumn hyperplnes C i, 1 i 3. The seon exmple we hve is shown in Figure 6. Note tht only H 3 is ifferent. We leve the etetion of the three strong ominne yles in ll three iretions to the reer. We onjeture, tht no-sink gme forms exist for ny n, not just for n = 2 n n = 3 s isplye ove. Before we strt to stuy the properties of no-sink WTT gme forms let us introue two efinitions. = H 1 H 3 = H 2 = Figure 6: Seon no-sink 3D exmple. Definition 13 Let g e n n-person WTT gme form, x = (x 1,...,x n ) strtegy profile, n i n ritrry iretion (plyer). Then H x i enotes the hyperplne perpeniulr to iretion i ontining the profile x. Tht is, H x i onsists of ll strtegy profiles in g whih hve the i-th oorinte equl to x i. Definition 14 Let g e n n-person no-sink WTT gme form. We sy tht g ontins k-ox if there exist two strtegy profiles x = (x 1,...,x n ) n y = (y 1,...,y n ) suh tht: 1. g(x) g(y), 2. x n y iffer in extly k oorintes i 1,...,i k (so x n y spn k-imensionl sugme form of 2 k strtegy profiles), n 3. for every 1 j k it hols tht g(x) is not the proper outome of H x i j n g(y) is not the proper outome of H y i j. Now we re rey to stte severl properties of no-sink WTT gme forms. Lemm 15 Let g e n n-person no-sink WTT gme form whih ontins k-ox for some 1 k n. Then g ontins (k 1)-ox or 1-ox. 11
Proof. Let us ssume without loss of generlity tht profiles x n y spnning the k-ox iffer in the first k oorintes (if not we n permute the oorintes), i.e. x = (x 1,...,x n ) n y = (y 1,...,y n ) where x i y i for 1 i k n x i = y i for k+1 i n. Now onsier iretion 1 n strtegy profiles x = (y 1,x 2...,x n ) n y = (x 1,y 2...,y n ). Notie tht the pirs x,x n y,y lie on lines in iretion 1 while the pirs x,y n x,y elong to two hyperplnes perpeniulr to iretion 1 (nmely H x 1 n H y 1 ). Thus the retngle x,x,y,y must ontin onstnt line ue to the WTT property. Now we hve four possiilities: 1. if g(x ) = g(y) then x,x spn 1-ox, 2. if g(x ) = g(x) then y,x spn (k 1)-ox, 3. if g(y ) = g(y) then x,y spn (k 1)-ox, n 4. if g(y ) = g(x) then y,y spn 1-ox. In ll four ses the three properties efining the speifie 1-ox or (k 1)-ox follow esily from the properties of the k-ox. In prtiulr, in the first se g(x) g(x ) = g(y), x n x iffer only in the first oorinte, n pir of prllel hyperplnes H1 x n Hx 1 = Hy 1 fulfills the thir property. In the seon se g(y) g(x ) = g(x), y n x iffer in extly (k 1) oorintes (nmely 2,...,k), n (k 1) pirs of prllel hyperplnes H y i n Hi x = Hi x for 2 i k fulfill the thir property. The thir n fourth se re symmetri. Lemm 16 Let g e n n-person no-sink WTT gme form. Then g ontins no 1-ox. Proof. By ontrition let x n y e two profiles spnning 1-ox, i.e. suh tht 1. g(x) g(y), 2. x n y iffer in extly 1 oorinte, i.e. lie on line l in some iretion i, n 3. g(x) is not the proper outome of H x i n g(y) is not the proper outome of H y i. Let us enote the proper outome of H 1 = Hi x y g(x) n the proper outome of H 2 = H y i y g(y). Sine g(x) g(y) we n hve neither H 1 = H 2 nor H 2 = H 1. Consequently, H 1 nnot stritly ominte H 2 y Lemm 9, sine we ssume to e its proper outome. Similrly H 2 g(x) g(y) nnot stritly ominte H 1. Therefore, y Corollry 7 we must hve oth H 1 H 2 n H 2 H 1, see Figure 7. Hene, there must exist thir hyperplne H 3 perpeniulr to iretion i n istint from H 1,H 2 suh tht H 1 = H 3. This implies tht line l intersets H 3 in some strtegy profile z with g(z) = g(x), euse x elongs to H 1 = (3). Moreover, there must exist two istint lines l n l in iretion i interseting H 1 (3) in profiles x n x with g(x ) = g(x ) = n interseting H 3 (1) in profiles z n z where g(z ) =, g(z ) = with,, pirwise istint. One of, must e ifferent from g(x) so let us ssume g(x) = g(z). Let l n l interset H 2 in profiles y n y g(x). Beuse H 1 H 2, lines l n l interset H 1 = (2) n so g(y ) = g(y ) =. Now onsier the retngle y,z,z,y on lines l,l n in hyperplnes H 2,H 3. We hve g(y) g(z) = g(x), g(z) g(z ) =, n = g(z ) g(y ) =. Therefore, WTT property implies g(y) = g(y ) =. However, now the quruple y,y,z,z with g(y ) = g(y ) = g(y) =, g(z ) =, g(z g(y) ) = with,, pirwise istint implies H 2 = H 3 whih ontrits the ft tht the proper outome of H 2 is g(y). Lemm 15 n Lemm 16 of ourse hve n ovious orollry. 12
g(x) H 3 = H 1 g(y) H 2 g(z) = g(x) = g(x) g(y) l = g(z ) = g(x ) = = l g(z ) = g(x ) = = l Figure 7: Configurtion showing tht no 1-ox n exists in WTT gme form. Corollry 17 Let g e n n-person no-sink WTT gme form n let k e ritrry, 1 k n. Then g ontins no k-ox. Proof. Let us ssume y ontrition tht g ontins k-ox for some 1 k n. Using Lemm 15 we get tht g ontins (k 1)-ox or 1-ox, the ltter eing impossile ue to Lemm 16. Iterting the rgument we susequently get tht g ontins (k 2)-ox, (k 3)-ox n so on, until we finlly get tht g ontins 1-ox, whih ontrits Lemm 16. Let us introue more terminology onnete to ssignle gme forms. Let g e n ssignle n-person gme form n g i, 1 i n, funtions gurnteeing the ssignility of g. Let x i X i e strtegy of plyer i. If g i (x i ) = for some outome, we sy tht hyperplne H efine y fixing the strtegy of plyer i to x i is ssigne n outome. If hyperplne H is ssigne outome then we sy tht strtegy profile x H is overe y H if g(x) = n not overe y H if g(x). Now let us formulte the finl sttement out no-sink WTT gme forms. Lemm 18 Let g e n n-person no-sink WTT gme form. Then g is ssignle. Proof. Let us ssign to every hyperplne perpeniulr to iretion i, 1 i (n 1), its proper outome (i.e. ll hyperplnes exept of those perpeniulr to iretion n re now ssigne). Let H e hyperplneperpeniulrtoiretion nnlet x,y etwostrtegy profilesinh whihrenotovere y ny hyperplne orthogonl to H, i.e. g(x) is not the proper outome of H x i for ny 1 i (n 1), n g(y) is not the proper outome of H y i for ny 1 i (n 1). If g(x) g(y) then g ontins k-ox for some 1 k (n 1) where k is the numer of oorintes in whih x n y iffer (they nnot iffer in ll n oorintes sine they re oth in H). However, this is impossile in WTT gme 13
form ue to Lemm 17, n therefore g(x) = g(y) = for some must hol. Sine x,y were selete s ritrry two not overe strtegy profiles, it follows tht ll not overe profiles in H hve the sme outome n thus n e overe y ssigning to H. The sme n e one for every hyperplne perpeniulr to iretion n n hene g is ssignle. It seems quite intuitive, tht lso every hyperplne perpeniulr to iretion n, whih is ssigne the ommon outome of ll unovere profiles, is in ft ssigne its proper outome. We onjeture tht it is inee the se whih woul mke the sttement of the lgorithm prouing fesile ssignment muh simpler: ssign to eh hyperplne (in ny iretion) its proper outome. However, we urrently hve neither proof of this ft nor ounterexmple, so we leve this s n open reserh question. Now we re finlly rey to stte n prove the min result of this pper. Theorem 19 Let g e n n-person WTT gme form. Then g is ssignle. Proof. We shll proee y inution on n. The se se n = 1 is trivil. In this se g is just single line whih is trivilly WTT (there re no 2 2 sumtries to onsier) n whih is lso esily ssignle (eh strtegy of the single plyer is ssigne the only outome elonging to tht strtegy). Let us ssume for the inution step tht the sttement of the theorem is true for ll (n 1)-person WTT gme forms n let g e n n-person WTT gme form. Now there re two possiilities. Either g hs no sink hyperplne in ny iretion n then it is ssignle y Lemm 18, or there exists iretion i n hyperplne H j perpeniulr to i suh tht H j is sink hyperplne. In the ltter se there exists n outome k suh tht H k k Hj for every hyperplne H k perpeniulr to iretion i, k j. We ssign k to H k for every k j whih overs ll profiles in H k (j) regions of the hyperplnes H k, k j (the omintion H k k Hj implies g(x) = k for every x H k (j)). It remins to over profiles in the H k = (j) regions of the hyperplnes H k, k j, n in hyperplne H j. However, H j is n (n 1)-person WTT gme form whih is ssignle y the inution hypothesis. Moreover, if we exten the ssignment of ll (n 2)-imensionl hyperplnes insie of H j given y the hypothesis to e the sme for the (n 1)-imensionl hyperplnes originting from the (n 2)-imensionl hyperplnes y ing the oorinte i s running inex (extening the (n 2)- imensionl hyperplnes long the lines in iretion i) then this extene ssignment lerly overs ll profiles in the H k = (j) regions of ll hyperplnes H k, k j. Thus ll strtegy profiles in g re overe, whih finishes the proof. Note tht the proof of Theorem 19 gives reursive lgorithm onstruting fesile ssignment for n ritrry n-person WTT gme form. Tht hs n impt on the omplexity of reognizing the WTT property n susequently onstruting fesile ssignment for WTT gme form, s we shll see in the next setion. 3 Complexity of reognition of WTT n AS gme forms First let us relize tht WTT gme forms n e reognize in polynomil time with respet to their size, i.e. with respet to the totl numer of strtegy profiles. Moreover, let us note tht for WTT gme form fesile ssignment n e onstrute in polynomil time s well using the lgorithm impliitly present in the proof of Theorem 19. Theorem 2 Let g e n n-person gme form of size s 1 s 2 s n. Let us enote s = n i=1 s i the sum of sizes in ll iretions n p = n i=1 s i the prout of sizes in ll iretions, i.e. let p e the totl numer of ll strtegy profiles in g. Then it n e teste in O(np 2 ) time whether g is WTT n in the ffirmtive se fesile ssignment for g n e onstrute in O(nsp) time. 14
Proof. To test the WTT property, it suffies to test for eh iretion i ll 2 2 surrys efine y hoie of two of the p/s i istint lines in iretion i n two of the s i istint hyperplnes perpeniulr to iretion i. There re O((p/s i ) 2 ) pirs of suh lines n O((s i ) 2 ) pirs of suh hyperplnes. Thus there re O(p 2 ) 2 2 surrys to hek for iretion i n thus ltogether O(np 2 ) surrys for ll iretions (heking eh 2 2 surry tkes of ourse just onstnt time). Now let us ssume tht g is WTT. Given two hyperplnes perpeniulr to iretion i (eh ontining p/s i strtegy profiles) it tkes O(p/s i ) time to etet the ominne reltion etween them. There re O(s 2 i ) suh pirs of hyperplnes n so it tkes O(s ip) time to uil the ominne grph for plyer i. Thus it tkes O(sp) time to uil the ominne grphs for ll plyers. Inseg isno-sinkgmeformthen, followingtheproofoflemm18, ll hyperplnesperpeniulr to iretions other thn n re ssigne their proper outomes (these outomes re ontine in the ominne grphs s ege lels). Given hyperplne H perpeniulr to iretion n n profile x H it tkes O(n) time to hek whether it is overe y one of the n 1 lrey ssigne hyperplnes going through x. Therefore it tkes O(np) time to hek ll profiles n ssign outomes lso to hyperplnes perpeniulr to iretion n whih is ominte y O(sp) time neessry to uil the ominne grphs. In se g ontins sink hyperplne then, following the proof of Theorem 19, reursion is invoke. This reursion hs epth t most n n t eh level the time neee to uil the ominne grphs is of ourse ominte y O(sp). Thus the totl time neee to get fesile ssignment is O(nsp). Oviously, the ove theorem is vli lso for prtilly efine gme forms. Rell tht the WTT property in this se mens tht ll unefine vlues re ll reple y single extr outome n WTT property of this fully efine gme form is require. Thus the omplexity of the reognition prolem is equivlent to the fully efine se. When onstruting fesile ssignment for prtilly efine WTT gme form using the proeure for the fully efine se, some hyperplnes my e ssigne the extr outome. This just mens tht suh hyperplnes re not neee to over the efine outomes in the prtilly efine gme form, n hene eh of these hyperplnes my e ssigne n ritrry outome (inste of the extr outome) in fesile ssignment of the prtilly efine gme form. We hve seen ove tht WTT gme forms (fully or prtilly efine) n e reognize in polynomil time. Wht is the omplexity of reognition for ssignle gme forms? As we showe in the previous setion, the set of ssignle gme forms is superset of WTT gme forms. In ft, it is strit superset of the WTT ones even in the two imensionl se n = 2. See exmples in [4], where the implition AS TT is isprove, or the first exmple in (1) in the introution. The next three susetions show how iffiult it is to reognize this strit superset uner ifferent itionl onitions. 3.1 Complexity of reognition of ssignle gme forms for n=2 One wy to ttk the reognition prolem is to formulte the ssignility of gme form g (oth for the fully efine n prtilly efine ses) s CNF stisfiility prolem. If we introue Boolen vriles y k ij for 1 i n, j X i, n k A, where y k ij = 1 mens g i(j) = k, then the esire CNF onsists of two types of luses. The first type gurntees for every strtegy profile x = (x 1,...,x n ) X 1 X n where g(x) = k tht it is overe y one of the seprting funtions: (y k 1x 1 y k 2x 2... y k nx n ) Note tht the size of eh suh luse (the numer of literls in it) is given y the imension of g (y the numer of plyers), n the numer of suh luses is equl to the numer of profiles for whih g is efine. The seon type of luses then gurntees tht t most one outome from A is ssigne to 15
every g i (j), 1 i n, j X i : kl A (y k ij y l ij) These luses re ll qurti (two literls per luse). It is not neessry to require tht extly one outome is ssigne to every g i (j) (requiring t most one outome suffies), euse ny prtilly efine fesile ssignment n e of ourse ritrrily omplete to fully efine one. Note, tht for n = 2 the ove formultion yiels 2-SAT instne (ll luses re qurti), whih immeitely implies tht the ssignility of two-person gme form (prtilly or fully efine) n e reognize (n fesile ssignment onstrute, if it exists) in polynomil time with respet to the size of the gme form. On the other hn, given fully efine n-person gme form g with n 3, the omplexity of reognizing whether g is ssignle is not known. We shll ress this prolem in the rest of this setion. First we shll show tht for prtilly efine gme forms the reognition prolem is NP-omplete for n 3. Then we will moify this proof to show tht for fully efine gme forms the reognition prolem is NP-omplete for n 4, leving the se n = 3 open. 3.2 Complexity of reognition of prtilly efine ssignle gme forms for n 3 In this susetion we show tht reognizing ssignle prtilly efine n-person gme forms is NPomplete lrey for n = 3. This prolem is oviously in NP (for ny n) s heking fesiility of given ssignment n e esily one in polynomil time with respet to the size of the gme form. The hrness prt is prove in the following theorem. Theorem 21 It is NP-hr to reognize, whether given prtilly efine 3-person gme form is ssignle or not. Proof. We will proee y onstruting reution from the known NP-hr prolem 3-SAT, stisfiility of CNFs with extly three literls per luse, where we lso ssume without ny loss of generlity tht no luse ontins two literls of the sme vrile. Let m m Φ = C i = (u i v i w i ) i=1 i=1 e n instne of 3-SAT, i.e. 3-CNF on vriles x 1,...,x n where eh u i, v i, n w i is positive or negtive ourrene of some vrile. We ssoite outomes 1,..., m with the luses C 1,...,C m of Φ n efine prtilly efine 3-person gme form g Φ. It onsists of n m m m ox B, where g Φ (i,i,i) = i, i.e. ox B ontins the outomes 1,..., m on its min igonl n it is unefine everywhere else. Let us enote H 1 1,...,H1 m the hyperplnes perpeniulr to iretion 1, n similrly for iretions 2 n 3. Box B serves s seletor. The ft tht hyperplne H 1 i is ssigne outome i mens tht luse C i is stisfie y literl u i (the literl u i gets vlue true), n similrly for H 2 i n v i n lso H 3 i n w i. Clerly, B y itself is ssignle in mny wys - eh strtegy profile on the min igonl of B n e overe y ny of the three hyperplnes inient with it. However, not every suh ssignment orrespons to truth ssignment to vriles x 1,...,x n s it isregrs the ft tht two istint literls my shre the sme vrile. To estlish one-to-one orresponene etween fesile ssignments of g Φ n stisfying truth ssignments of Φ we will ggets to the ox B whih will gurntee tht: 1. If u i = u j for i j, i.e. oth literls re two ourrenes of the sme vrile with the sme polrity, then either H 1 i is ssigne i n simultneously H 1 j is ssigne j (oth u i n u j re true), or neither H 1 i is ssigne i nor H 1 j is ssigne j (oth u i n u j re flse). Similrly for 16
v i = v j n w i = w j. In eh of these three ses we nee to fore the ssigne outomes in the ove esrie wy for pir of prllel hyperplnes. 2. If u i = v j for i j, i.e. oth literls re two ourrenes of the sme vrile with the sme polrity, then either H 1 i is ssigne i n simultneously H 2 j is ssigne j (oth u i n u j re true), or neither H 1 i is ssigne i nor H 2 j is ssigne j (oth u i n u j re flse). Similrly for u i = w j n v i = w j. In eh of these three ses we nee to fore the ssigne outomes in the ove esrie wy for pir of perpeniulr hyperplnes. 3. If u i = u j for i j, i.e. oth literls re two ourrenes of the sme vrile with omplementry polrities, then either H 1 i is ssigne i or H 1 j is ssigne j ut not oth (extly one of u i n u j is true n extly one flse). Similrly for v i = v j n w i = w j. In eh of these three ses we nee to fore the ssigne outomes in the ove esrie wy for pir of prllel hyperplnes. 4. If u i = v j for i j, i.e. oth literls re two ourrenes of the sme vrile with omplementry polrities, then either H 1 i is ssigne i or H 2 j is ssigne j ut not oth (extly one of u i n v j is true n extly one flse). Similrly for u i = w j n v i = w j. In eh of these three ses we nee to fore the ssigne outomes in the ove esrie wy for pir of perpeniulr hyperplnes. Eh suh gget will onstnt numer of eite hyperplnes outsie of ox B, where eite mens tht no two ggets shre ommon e hyperplne (of ourse they my shre hyperplnes inient to ox B). Sine the numer of the ove pir-wise reltions is t most qurti in the size of Φ, the onstrute gme form g Φ hs polynomil size with respet to the size of Φ. Before onstruting the four types of ggets with ove esrie properties, let us onstrut ommon foring omponents of suh ggets (lle foring ues). These re 2 2 2 rrys with six istint outomes,,,,e,f rrnge in the eight orners (strtegy profiles) of the rry in one of the two possile wys shown in Figure 8. e e f f Figure 8: Two foring ues: Here we ssume tht,,, e, n f re six istint outomes. Sine they re overe y six plnes, oth opies of must e overe y the front plne, n oth opies of must e overe y the k plne. Consequently, we must hve ssigne to the front n to the k plnes in ll fesile ssignments. In oth foring ues there re six hyperplnes to over ll eight orners with six istint outomes. Thus, oth outomes must e overe y the sme hyperplne n so o oth outomes. Hene, in oth foring ues the front hyperplne is fore to e ssigne n the k hyperplne is fore to e ssigne in ny fesile ssignment. Let us now onstrut the four types of require ggets. 17
1. Let u i = u j. Let us two hyperplnes Hl 2 n H2 l+1 perpeniulr to iretion 2 n two hyperplnes Hk 3 n H3 k+1 perpeniulr to iretion 3 for some k,l > m. Let us onsier four istint outomes,,,e not ontine mong 1,..., m, n the 2 2 2 rry efine y the intersetions of the four e hyperplnes with Hi 1 n Hi 2 s in Figure 9. H 1 i H 1 j H 2 l j e H 3 k+1 H 2 l+1 i H 3 k Figure 9: Gget 1: This is n instne of one of the foring ues, thus ssignments Hk 3 n Hk+1 3 re implie in ny fesile ssignment. These leve only two possile yli fesile ssignments for the remining four plnes. Either i Hi 1, H2 l, j Hj 1 n e H2 l+1, or H1 i, j Hl 2, e Hj 1 n i Hl+1 2. Consequently, in ll fesile ssignments we hve either oth i Hi 1 n j Hj 1 or we hve neither one, simultneously. This gget is foring ue whih fores Hk 3 to e ssigne n H3 k+1 to e ssigne. Now there re only two wys how the remining four hyperplnes n over the four istint outomes i, j,,e. If Hi 1 is ssigne i, it fores Hl 2 to essigne, whih in turn fores H1 j to e ssigne j, whih finlly fores Hl+1 2 to e ssigne e. On the other hn, if H1 i is ssigne, it fores Hl+1 2 to e ssigne i, whih in turn fores Hj 1 to e ssigne e, whih finlly fores H2 l to e ssigne j. Thus the onstrute gget fulfills extly the require properties. Note lso, tht the hyperplnes Hi 1,H1 j whih re inient to the seletor ox B re fore to e ssigne one of the outomes i, j,,e in every fesile ssignment. 2. Let u i = v j. Let us one hyperplne Hl 1 perpeniulr to iretion 1, one hyperplne H2 p perpeniulr to iretion 2, n two hyperplnes Hk 3 n H3 k+1 perpeniulr to iretion 3 for some k,l,p > m. Let us onsier four istint outomes,,,e not ontine mong 1,..., m n the 2 2 2 rry efine y the intersetions of the four e hyperplnes with Hi 1 n Hj 2. We ssign the outomes to this 2 2 2 rry s in Figure 1. This is gin foring ue whih fores Hk 3 to e ssigne n H3 k+1 to e ssigne. Clerly, either Hi 1 is ssigne i n Hj 2 is ssigne j or lterntively Hi 1 is ssigne j n Hj 2 is ssigne i (n the outomes n e re similrly overe y Hl 1 n H2 p in one of the two possile wys). Thus the onstrute gget fulfills extly the require properties. Note lso, tht the hyperplnes Hi 1,H2 j whih re inient to the seletor ox B re fore to e ssigne one of the outomes i, j in every fesile ssignment. 3. Let u i = u j. Let us two hyperplnes Hl 1 n H1 l+1 perpeniulr to iretion 1, two hyperplneshp 2 nhp+1 2 perpeniulrtoiretion2, ntwohyperplnesh3 k nh3 k+1 perpeniulr to iretion 3 for some k,l,p > m. Let us ssign six istint outomes,,,,e,f not ontine 18
H 1 i H 1 l j H 2 j i e H 3 k+1 H 2 p H 3 k Figure 1: Gget 2: This is n instne of one of the foring ues, thus ssignments Hk 3 n Hk+1 3 re implie in ny fesile ssignment. These leve only two possile fesile ssignments for Hi 1 n Hj 2. We hve either i Hi 1 n j Hj 2, or j Hi 1 n i Hj 2. Consequently, in ll fesile ssignments we hve either oth i Hi 1 n j Hj 2 or we hve neither one, simultneously. mong 1,..., m to the 2 2 2 rry efine y the intersetions of the six e hyperplnes in suh wy tht we get foring ue whih fores Hk 3 to e ssigne outome. Let us two more hyperplnes Hp+2 2 n H2 p+3 perpeniulr to iretion 2 n onsier the intersetions of Hk 3 with H1 i, H1 j, H2 p+2, n H2 p+3. Let us ssign outomes to this 2 2 surry s in Figure 11. Sine Hk 3 is fore to e ssigne outome, there re just two wys how to over the four outomes in this 2 2 surry. Either Hi 1 is ssigne i, Hp+3 2 is ssigne e, H1 j is ssigne f, n Hp+2 2 is ssigne j, or lterntively Hi 1 is ssigne e, Hp+2 2 is ssigne i, Hj 1 is ssigne j, n Hp+3 2 is ssigne f. Thus the onstrute gget fulfills extly the require properties. Note lso, tht the hyperplnes Hi 1,H1 j whih re inient to the seletor ox B re fore to e ssigne one of the outomes i, j,e,f in every fesile ssignment. 4. Let u i = v j. Let us two hyperplnes Hl 1 n H1 l+1 perpeniulr to iretion 1, two hyperplneshp 2 nh2 p+1 perpeniulrtoiretion2, ntwohyperplnesh3 k nh3 k+1 perpeniulr to iretion 3 for some k,l,p > m. Let us ssign six istint outomes,,,,e,f not ontine mong 1,..., m to the 2 2 2 rry efine y the intersetions of the six e hyperplnes in suh wy, tht we get foring ue whih fores Hk 3 to e ssigne outome. Let us one more hyperplne Hl+2 1 perpeniulr to iretion 1 n one more hyperplne H2 p+2 perpeniulr to iretion 2 n onsier the intersetions of Hk 3 with H1 i, H2 j, H1 l+2, n H2 p+2. Let us ssign outomes to this 2 2 surry s in Figure 12. Sine Hk 3 is fore to e ssigne outome, there re just two wys how to over the four outomes in this 2 2 surry. Either Hi 1 is ssigne i, Hj 2 is ssigne e, H1 l+2 is ssigne j, n Hp+2 2 is ssigne f, or lterntively H1 i is ssigne e, Hp+2 2 is ssigne i, Hl+2 1 is ssigne f, n Hj 2 is ssigne j. Thus the onstrute gget fulfills extly the require properties. Note lso, tht the hyperplnes Hi 1,H2 j whih re inient to the seletor ox B re fore to e ssigne one of the outomes i, j,e,f in every fesile ssignment. It follows from the ove onstrutions tht if there exists fesile ssignment to gme form g Φ then Φ hs stisfying ssignment. Inee, the fesiility of the ssignment for the strtegy profiles on 19
H 1 i H 1 j H 2 p+3 i j H 1 l H 1 l+1 H 2 p+2 e f f H 2 p+1 e H 3 k+1 H 2 p H 3 k Figure 11: Gget 3: In this gget we hve foring ue isjoint from the hyperplnes inient with the seletor ox, implying tht is ssigne to Hk 3 in ll fesile ssignments. The intersetion of this hyperplne with four others, s in the piture ove, reues the possile fesile ssignments to the ses when either i is ssigne to Hi 1 or j is ssigne to Hj 1, ut not oth. the min igonl of the seletor ox implies tht eh luse hs stisfying literl, n the fesiility of the ssignment in the strtegy profiles of the ggets imply tht these truth vlues re onsistent. Conversely, if we hve stisfying truth ssignment to Φ, then we n erive fesile ssignment to ll hyperplnes Hi 1, H2 j n H3 k whih over the strtegy profiles long the igonl of the seletor ox, n exten these to over ll other strtegy profiles y the proven properties of the ggets. 3.3 Complexity of reognition of fully efine ssignle gme forms for n 4 In this susetion we will moify the proof of Theorem 21 to show tht the reognition prolem is NP-omplete lso for fully efine gme forms, this time for n 4, leving the se n = 3 open. Theorem 22 It is NP-hr to reognize, whether given fully efine 4-person gme form is ssignle or not. Proof. Let us repet the onstrution from the proof of Theorem 21 with these hnges: All strtegy profiles whih were unefine in the onstrution now get new itionl outome, whih proues fully efine gme form. If the onstrution proue 3-person gme form of size s 1 s 2 s 3 we shll onsier it now s 4-person gme form of size s 1 s 2 s 3 1, n enote the single hyperplne perpeniulr to the e iretion H 4 1. 2
H 1 i H 1 l+2 H 2 j e j H 1 l+1 H 1 l H 2 p+2 i f f H 2 p+1 e H 3 k+1 H 2 p H 3 k Figure 12: Gget 4: In this gget we hve foring ue isjoint from the hyperplnes inient with the seletor ox, implying tht is ssigne to Hk 3 in ll fesile ssignments. The intersetion of this hyperplne with four others, s in the piture ove, reues the possile fesile ssignments to the ses when either i is ssigne to Hi 1 or j is ssigne to Hj 1, ut not oth. We shll ssume tht the input 3-CNF Φ stisfies the following itionl property: if we elete ny two luses from Φ, the remining 3-CNF ontins some luse C i with non-trivil literl in the first position, luse C j with non-trivil literl in the seon position, n luse C k with non-trivil literl in the thir position, where trivil literl is literl whih represents the only ourrene of its vrile in the entire formul. This ssumption n e me without losing the NP-hrness of the 3-SAT prolem restrite to suh inputs. Let us first note tht we n ssume tht every vrile ppers t lest twie in two ifferent luses of the input 3-CNF. Otherwise, we n fix the vlue of the unique pperne without hnging stisfiility of the input. We lim tht 3-CNF tht oes not stisfy the property lime ove, n tht hs every vrile ppering t lest twie, nnot hve more thn 1 luses. To see this let us ssume tht y eleting the first two luses we hve trivil literl in eh of the remining luses. All of the vriles of these trivil literls must hve then their seon pperne in the elete luses, tht is we nnot hve more thn 6 suh trivil literls. Therefore, if we hve t lest 11 luses, then we must hve three suh tht they o not involve trivil literls in ny positions. Now it is ler tht ny fesile ssignment of outomes to hyperplnes in the proof of Theorem 21 n e extene to fesile ssignment of outomes to hyperplnes for the fully efine gme form y ssigning outome to H1 4. Now we shll show the other iretion, i.e. prove, tht the ssignment of outome to H1 4 is fore, i.e. there is no fesile ssignment of outomes to hyperplnes of the 4-person gme form in whih H1 4 is ssigne something else. This will in turn prove, tht ny fesile ssignment of the fully efine 4-person gme form efines fesile ssignment for the prtilly efine 3-person gme form whih is otine y eleting ll outomes n onsiering the three imensionl 21