Optimal football strategies: AC Milan versus FC Barcelona

Transcription

1 Optml footll strteges: C Mln ersus C Brcelon Chrstos Pphrstodoulou strct In recent UE Chmpons Legue gme etween C Mln nd C Brcelon, plyed n Itly fnl score -, the collected mtch sttstcs, clssfed nto four offense nd two defense strteges, were n four of C Brcelon y ersus ponts. The m of ths pper s to exmne to wht extent the optml gme strteges dered from some determnstc, posslstc, stochstc nd fuy LP models would mproe the pyoff of C Mln t the cost of C Brcelon. Keywords: footll gme, mxed strteges, fuy, stochstc, Nsh equlr Deprtment of Industrl Economcs, Mälrdlen Unersty, Västerås, Sweden chrstos.pphrstodoulou@mdh.se Tel +

2 . Introducton The mn oecte of the tems mngers s to fnd ther optml strteges to wn the mtch. Thus, t should e pproprte to use gme theory to nlye footll mtch. Moreoer, s lwys wth gme pplctons, the ccess of ccurte dt to estmte the pyoffs of the selected strteges s ery dffcult. ccordng to Crlton & Perloff, only few mxed strtegy models he een estmted n Industrl Economcs. In ddton to tht, contrry to professonl usness mngers who he sold mngerl, mthemtc or economc educton, tem mngers lck the necessry forml knowledge to use the gme theoretc methods. ootll mngers, when they decde ther most pproprte tctcl moe or strtegy, rely more on ther -pror elefs, ntuton, tttude towrds rsk nd experence. In footll gme f we exclude fortune nd smple mstkes, y plyers nd referees s well, gols scored or conceed re often the results of good offense nd/or d defense tctcs nd strteges. There re ryng numer of strteges nd tctcs. s s well known, tctcs re the mens to chee the oectes, whle strtegy s set of decsons formulted efore the gme strts or durng the hlf-tme rke, specfyng the tctcl moes the tem wll follow durng the mtch, dependng upon rous crcumstnces. or nstnce, the sc elements of tem s tctcs re: whch plyers wll ply the gme, whch tsks they wll perform, where they wll e postoned, nd how the tem wll e formed nd reformed n the ptch. Smlrly, tem s strteges mght e to ply short pssng gme wth hgh ll possesson, ttckng wth the ll mong quckly nd pressng hgh up ts compettors, whle nother tem s strtegy mght e to defend wth onl or mn-to-mn system, nd usng the speed of ts fullcks to ttck, or plyng long psses nd crosses s counter ttckng see Consequently, oth mngers need somehow to guess correctly how the opponents wll ply n order to e successful. ll decsons mde y humns re ulnerle to ny cognte ses nd re not perfect when they try to mke true predctons. Not only the numer of tctcs nd strteges n footll mtch s lrge, ther mesures re ery hrd ndeed. How cn one defne nd mesure correctly counter ttcks, hgh pressure, ttckng gme, long psses, runs etc? The exstng dt on mtch-sttstcs coer reltely esy rles, lke: ll possesson, shots on trget, fouls commtted, corners, offsde nd yellow or red crds, see for nstnce UE s offcl ste If one wnts to mesure the pproprte tems strteges or tctcs, one hs to collect such mesures, whch s oously n extremely tme-consumng tsk, especlly f sttstclly lrge smple of mtches, where the sme tems re noled, s requred. In ths cse-study, I he collected detled sttstcs from ust one mtch, UE Chmpons Legue group mtch, etween C Mln CM nd C Brcelon, held n Mln on Noemer,, where defeted CM y -. Despte the fct tht oth tems were prctclly qulfed efore the gme, the gme hd more prestgous chrcter nd would determne to lrge extent, whch tem would e the wnner of the group. Gen the fct tht I he concentrted on sx strteges per tem, four offenses, nd two defense, nd tht wns oer CM n more strtegy prs, the m of ths pper s ndeed to exmne to wht extent the optml gme strteges dered from some determnstc, posslstc nd fuy LP models would mproe the pyoff of CM. Oously, there re two shortges wth the use of such mtch sttstcs. rst, we cn t lme the tems or ther mngers for not usng ther optml pure or mxed strteges, f the pyoffs from the selected strteges were not known n dnce, ut were osered when the gme ws eng plyed. Second, t s unfr to lme the mnger of CM the looser, f hs plyers dd not follow the correct strteges suggested y hm. It s lso unfr to ge credts to the mnger of the wnner, f hs plyers dd not follow the possly ncorrect strteges suggested y hm. Thus, we modfy the purpose nd try to fnd out the optml strteges, ssumng tht the mngers ntcpted the pyoffs nd the plyers dd wht they he een sked to do. On the other hnd, the merts of ths cse study re to tret footll mtch not s trl ero-sum gme, ut s non-constnt sum gme, or -mtrx gme, wth mny strteges. It s not the gol scored tself tht s

3 nlyed, ut merely under whch mxed offense nd defense strteges the tems nd especlly CM could he done etter nd collected more pyoffs. s s known, n such gmes, t s rther dffcult to fnd soluton tht s smultneously optml for oth tems, unless one ssumes tht oth tems wll he Nsh elefs out ech other. Gen the uncertnty n mesures of some or ll selected strteges, posslstc nd fuy formultons re lso presented. The structure of the pper conssts of fe sectons: In secton we dscuss the selected strteges nd how we mesured them. In secton, usng the pyoffs from secton, we formulte the followng models: clsscl optmton mxmum of mnmum pyoffs LP wth complementry constrnts Nsh Chnce Constrned LP Posslstc LP uy LP. In secton we present nd comment on the results from ll models nd secton concludes the pper.. Selected strteges nd Dt nd CM re two worldwde tems who ply ery ttrcte footll. They use lmost smlr tem formtons, the -- system four defenders, three mdfelders, nd three ttckers. ll footll fns know tht s stndrd strtegy s to ply n excellent pssng gme, wth hghll possesson, nd quck moements when t ttcks. ccordng to offcl mtch sttstcs, hd % ll possesson, een f lrge prt of the ll ws kept wy from CM s defense re. ll mngers who fce expect tht to hppen, nd knowng tht hs the world s est plyer, Mess, they must decde n dnce some defense tctcs to neutrle hm. Snce the offcl mtch sttstcs re not pproprte for our selected strteges, I recorded the gme nd plyed t ck seerl tmes n order to mesure ll nterestng prs of pyoffs. Both tems re ssumed to ply the followng sx strteges: shots on gol, counter-ttcks, ttckng psses, drles, tcklng nd one mrkng. The frst four reflect offense strteges nd the lst two defense strteges. Most of these rles re hrd to osere nd mesure. It s ssumed tht the pyoffs from ll these strteges re eqully worth. One cn of course put dfferent weghts. shots on gol SG Tems wth mny SG, re expected to score more gols. In preous study Pphrstodoulou,, sed on UE CL mtches, t ws estmted tht tems need, on erge, out SG to score gol. In ths pper ll SG count, rrespectely f they sed y the golkeeper or the defenders, s long s they re drected towrds the trget, nd rrespectely of the dstnce, the power of the shot nd the ngle they were kcked. SG from fouls, corners, nd hed-ncks re lso ncluded. ccordng to the offcl mtch sttstcs, hd SG nd corners. ccordng to my own defnton, hd SG. The defenders of CM locked of them ncludng the sngs y the gol-keeper. X turned one of the shots nto gol. On the other hnd, the other two gols scored do not count s SG, ecuse the frst ws y penlty Mess nd the other y own gol n Bommel. Smlrly, ccordng to the offcl mtch sttstcs, CM hd SG nd corners, whle n my mesures CM hd SG. locked of them ncludng good sng y ts golkeeper, nd two of them turned nto gols y Irhmoc nd Boteng. counter-ttcks C The de wth C s to eneft from the other tem s desperton to score, despte ts offense gme. The defendnt tem s wthdrwn nto ts own hlf ut keep mn or two further up the ptch. If mny opponent plyers ttck nd loose the ll, they wll e out of poston nd the defendnt tem hs more spce to deler Snce gme theoretc termnology s ppled, we use the term strtegy n the entre pper, een f we refer to tctcs. Pollrd nd Reep estmted tht the scorng prolty s % hgher for eery yrd nerer gol nd the scorng prolty doules when plyer mnges to e oer yrd from n opponent when shootng the ll.

4 long-ll for the own strkers, or own plyers cn run reltely free to the compettors defense re nd proly score. Ths tctc s rther rsky, ut t wll work f the defendnt tem hs relnt nd sold defense, nd excellent runners nd/or ll kckers. In ths study C he een defned s those whch he strted from the own defense re nd contnued ll the wy to the other tem s penlty re. On the other hnd, slow pce wth psses nd/or the exstence of more defenders thn ttckers n ther correct poston do not count. ccordng to tht defnton, hd C nd CM. ttckng psses P The golden rule n footll s to pss nd moe quckly. There re not mny tems whch hndle to pply t successfully though. mnly, nd CM to less extent, re two tems whch re known to ply n entertnng gme wth ery lrge numer of successful psses. In recent pper Pphrstodoulou, t ws estmted tht CM, n n erge mtch, could chee out successful psses nd he ll possesson of more thn %. or ll Itln tems see for nstnce, Smlrly n preous study Pphrstodoulou,, cheed een hgher ll possesson. Moreoer, ery often, the plyers choose the esest possle pss, nd mny tmes one oseres defenders pssng the ll long the defense lne. There s smple logc ehnd ths pprently ttrcte strtegy. By keepng hold of the ll wth psses, the opponents get frustrted, try to chse ll oer the ptch, e tred nd dsposed nd consequently lee open spces for the opponent quck ttckers to score. Gen the fct tht the numer of psses s ery lrge, compred to the other osertons, the pyoff gme mtrx wll e extremely unlnced nd oth tems would smply ply ther domnnt P strtegy. To mke the gme less trl, I he used ery restrcte defnton of P, ssumng the followng crter re fulflled: Only successful psses nd hed-ncks, whch strt t most pproxmtely meters outsde the defendnt tem s penlty re, count. The psses nd hed-ncks should e drected forwrd to the trgeted tem plyer who must e runnng forwrd too.e. psses to sttc plyers re excluded. Bckwrd psses count s long s they tke plce wthn the penlty re only. Nether long crosses, nor psses from free kcks nd corners count. Consequently, hd successful P nd CM hd ones. CM mnged to defend successfully tmes whle defended successfully eery thrd pss tht CM ttempted. drles D Drlng,.e. the cton to pss the ll round one or more defenders through short skllful tps or kcks, cn tke plce nywhere n the ptch. Moreoer, snce D n ths pper s treted s offense strtegy, only the offense ones re of nterest. The cton wll e mesured f t strts no more thn meters outsde the defendnt tem s penlty re nd the plyer must moe forwrd. Drlng counts een f the plyer turns ckwrd, s long s he remns wthn the penlty re. If the offense plyer mnges to drle more thn one plyer ut wth dfferent ctons susequently, the numer of D ncreses nlogclly. ccordng to tht defnton, ech tem hd D. tcklng T

5 stndrd defense strtegy s to tckle the opponents n order to stop them from gnng ground towrds gol, or stop ther SG, P nd ther D. Tcklng s defned when the defender uses ether hs left or rght leg ut not oth legs to wrest possesson from hs opponent. Een sldng n on the grss to knock the ll wy s treted s T. The tckle must lwys e t the ll, otherwse t my e llegl nd often punshed y the referee, especlly f the plyer mkes contct wth hs opponent efore the ll, or mkes unfr contct wth the plyer fter plyng the ll. Very often, tems, whch use T frequently, ply mn-to-mn mrkng,.e. when certn defenders who re responsle to gurd prtculr opponent re forced nto tht cton, ecuse they re dspossessed or re slower thn the opponents re. Mn-to-mn mrkng s prtculrly effecte when the tem hs sweeper who hs free role nd ports hs temmtes who re dspossessed or hng prolems wth the opponents. Only T t less thn pproxmtely meters outsde the defendnt tem s penlty re s counted. Tcklng nd hed-ncks s well from free kcks nd corners re lso counted, ecuse n these cses, the defenders ply the mn-to-mn tctc. On the other hnd, SG, C, P nd D stopped y unust T nd punshed y the referee, does not count. ccordng to these crter, defenders hd successful T gnst SG, gnst C, gnst P nd gnst D. Smlrly, CM hd,,, nd successful T respectely. one mrkng ZM In ZM eery defender nd the defense mdfelders too, re responsle to coer prtculr one on the ptch to hnder the opponent plyers from SG, P, D or C nto ther re. In perfect ZM, there re two lnes of defenders, usully wth four plyers n the frst nd t lest three n the second lne, coerng roughly the onehlf of the ptch. successful ZM requres tht eery defender fulflls hs dutes, communctes wth hs temmtes, coers ll empty spces, nd synchrones hs moement. In tht cse, the defense lne cn explot the offsde rules nd preent the success of long-lls, C, P, D nd SG. Bd communcton from the defenders though cn e ery decse, especlly f the opponents he ery quck ttckers who cn drle, pss, nd shot eqully well. Snce mesurng ZM s ery dffcult, the followng condtons re ppled to smplfy tht tctc. The two lnes of defenders should e plced t out less thn nd meters respectely, outsde the defendnt tem s penlty re,.e. ZM ner the mddle of the ptch does not count. Normlly, ZM ner the mddle of the ptch s osered when the tem controls the ll through psses or when t ttcks. To dfferentte the ZM from the T, the own defenders should e t lest - meters wy from ther offense plyers when he they ntercepted the ll. Despte the fct tht offsde postons re the result of good ZM, do not count. Precsely s n T, unust ctons y ZM do not count. ccordng to these condtons, defenders hd successful ZM gnst SG, gnst C, gnst P nd gnst D. Smlrly, CM hd,,, nd successful ZM respectely. The pyoff of the gme for ll sx strteges s depcted n the Tle elow. Notce tht some entres re empty ecuse oth tems cn t ply smultneously offense or defense. When one tem ttcks defends the other tem wll defend ttck. The frst entry refers to nd the second entry to CM. Consequently, snce the pyoff from tem s offense strtegy s not equl to the negte pyoff from the other tem s defense strtegy, the gme s non-ero sum nd the pyoff mtrx s -mtrx. There seem to e some doutful prs, where the defense lues re hgher thn the offense ones, such s,. How cn D e defended y ZM? Smply, some D whch counts ws defended occsonlly y ZM

6 whch lso counts the ll s then lost to the offense plyer who tred to drle gn, ut fled. Consequently, the new D ttempt does not count whle the new ZM does. Tle : The pyoff mtrx B = C Mln CM Offense Defense = C Brcelon Notce lso tht there re no pure domnnt strteges. Howeer, despte the fct tht there re no pure domnnt strteges, gets more ponts thn CM from the mtch. or nstnce, hd P,, n comprson wth CM, whch hd only,. s whole, ets CM n sx offense-defense prs y totl of ponts, s eten y CM n fe prs, y ponts, whle n fe prs there s te. The hghest dfferences n for of re n,,.e. when plys ts P nd CM does not succeed wth ts defense T, nd n,, when CM tres wth ts D ut defends successfully wth ts T.. Models In ths secton, I wll present four determnstc models, one chnce constrned, one posslstc nd one fuy LP. e of them re formulted seprtely for ech tem nd two smultneously for oth tems.. Clsscl Optmton Let nd B represent nd CM respectely, ther respecte sx strteges nd, wth, ounds. Ech tem mxmes seprtely the sum of ts pyoffs tmes the product of nd of the relent strtegy prs. Consequently, the oecte functons gen elow, re non-lner. Two models he een formulted: unrestrcted,.e. the sum of ll sx strteges s equl to unt restrcted,.e. oth offense nd defense strteges must e plyed. Consequently, n model the two condtons,, Offense,,,,,,,, Defense,,,,,,,, CM = SG = C = P = D = T = ZM = SG = C = P = D = T = ZM re modfed nto the four:,,, Model mx s. t.,,,,..,,, mx B s. t.,,,,..,,,

7 Model : Ths formulton ensures tht tem for nstnce, wll recee ts respecte pyoffs from ts offense strtegy, f tem B wll ply ts nd/or ts. In fct, when tem or B mxme, oth strteges nd re decded smultneously. Oously, wthout the strteges of the other tem, the oecte functon would e trl or een erroneous snce the hghest pyoff strtegy would not e ensured.. Mx-mn Let e the mnml lue from ll four offense strteges nd s the mnml lue from oth defense strteges for. Smlrly, let nd, e the respecte mnml lues for CM. Ech tem mxmes seprtely the sum of these mnml respecte lues. gn, the model s non-lner ecuse ech one of the offense defense strteges of one tem s multpled y the defense offense strteges of the other tem. Model,,..,,,,,.. mx s t,,..,,,,,.. mx s t B. LP formulton wth complementry condtons Whle the frst two models ssume tht tems optme seprtely, we turn now to smultneously optml decsons. Normlly, for mtrx gme wth mny strteges, t s rther dffcult to fnd soluton tht s smultneously optml for oth tems. We cn defne n equlrum stle set of strteges though,.e. the wellknown Nsh equlrum. In the followng two sectons, I wll formulte two models to fnd the Nsh equlrum. s s known, the mx-mn strtegy s defned s:, mn mx rg,.., pyoff, mn mx rg,.., pyoff B stndrd model to fnd mx-mn to oth tems s to use smultneous LP, wth complementry condtons. The complementry condtons re to set the product of ech one of the sx respecte slck, tmes the sx In order to se spce, model wll e excluded n ll susequent formultons.

8 respecte strteges, equl to ero. ccordng to ths formulton, oth tems ehe symmetrclly, snce they mxme ther own mnml pyoffs otned from ther own selected strteges. Compred to the preous models, ech tem selects now only ts own strteges. Notce lso the two extr constrnts, whch ensure tht oth tems cn t ply entrely offensely or defensely. or nstnce, the upper ound for ll offense strteges s set rtrrly equl to. nd the lower ound for the defense strteges s set rtrrly equl to.. Model,,..,,,..,..,,,..,,,.. mx sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl sl s t B. Nsh strteges s s known n the Nsh equlrum, ech tem selects ts prolty mxture of strteges or pure strtegy to mxme ts pyoff, condtonl on the other tem s selected prolty mxture or pure. The prolty mxture of tem s the est response to the other tem s prolty mxture. Consequently, the,..,,,.., s Nsh equlrum f nd only f t stsfes the followng condtons:,..,,,,..,,,..,,,.., pyoff pyoff,..,,,,..,,,..,,,.., pyoff pyoff B B It s lso known tht, f mn-mx nd Nsh equlr concde, the gme hs sddle pont. Such sddle ponts re rther frequent n ero-sum gmes ut not n -mtrx non-ero sum gmes. Wthout these ddtonl constrnts, oth tems plyed offensely plys.% SG nd.% P, whle CM plys.% P nd.% D.

9 I ppled the pckge y Dckhut & Kpln progrmmed n Mthemtc, to fnd the Nsh equlr. In model the entre pyoff mtrx ws used. In model I used two su-mtrces when CM ws plyng offensely nd CM defensely.. Chnce-Constrned Progrmmng CCP When tems re uncertn out compettors ctons or out the pyoff mtrx, gmes ecome ery complex. ccordng to Crlton & Perloff much of the current reserch n gme theory s undertken on gmes wth uncertnty. I moe now to some more plusle models nd modfy the determnstc prmeters nd constrnts. In CCP the prmeters of the constrnts re rndom rles nd the constrnts re ld wth some mnmum prolty. Let us ssume tht the determnstc prmeters re expected lues, ndependent nd normlly dstruted rndom rles wth the mens s preously, nd rnces gen n Tle. The frst entry depcts the rnce for nd the second for CM. Tle : The rnce of the pyoff mtrx = C Brcelon Offense Defense B = C Mln CM Offense Defense,,,,,,,,,,,,,,,, Moreoer, n CCP, when we mxme for one tem, we ssume tht the other tem s lues re determnstc nd dsregrd ther rnce. We lso ssume tht, Josep Gurdol, the mnger of, mght expect tht the prolty of the expected lue of hs tem s defense strteges nd s t lest %, whle the prolty of ll four expected lues of offense strteges,,, nd s t lest %. The frst stochstc constrnt s now formulted s: P, where, s the cumulte densty functon of the stndrd norml dstruton. If K s the stndrd norml lue such tht K = -, then the oe constrnt reduces to: Gen., the constrnt s smplfed to: K The rnces of the pyoffs re oously ery suecte nd re gen ust to show the formulton of the model. Moreoer, sed on my numerous plyng ck of the mtch, the rnces reflect rther well the uncertnty of the respecte pyoffs.

10 . Smlrly, gen., the frst defense constrnt s modfed to:. So, the CCP model for s: mx s. t.,....,,,..,,.., smlr formulton pples for CM, ssumng tht ts mnger Mssmlno llegr expects tht the prolty of the expected lue of hs tem s defense strteges nd s lso t lest %, whle the prolty of ll four offense strteges,,, nd s t lest %. llegr lso trets Brcelon s lues s determnstc nd therefore the prolem s formulted smlrly.. Posslstc LP PLP model No mtter how well one hs defned nd mesured the sx rles, the osered pyoffs re stll rther mguous. The mguty of mesured lues cn e restrcted y symmetrc trngulr fuy numer, determned y center, nd spred w, respectely, nd c c w whch s represented s:, w c, respectely B, w c. or nstnce, the estmte of C for, when tems ply,, cn e restrcted y fuy x numer, wth the followng memershp functon: x mx,. Thus, the center s.e., the ntl lue, ts upper lue s nd ts lower lue s. Consequently, tht fuy C rle s expressed s:,., In ddton to tht, we cn use posslty mesures n order to mesure to wht extent t s possle tht the posslstc lues, restrcted y the posslty dstruton re t lest equl to some certn lues. I wll follow Inuguch & Rmk, who used posslty nd/or necessty mesures to de-fufy fuy LP. Gen two fuy sets, nd Z, nd posslty dstruton of posslstc rle, the posslty mesure s defned s:,

11 Z mn x, x r Z If Z, g],.e. Z s determnstc non-fuy set of rel numers not lrger thn g, the posslty ndex s defned s: Pos g g If Z g, Pos, x x g, the respecte posslty ndex s defned s: g g, x r g The necessty mesures mesure to wht extent t s certn tht the posslstc lues, restrcted y the posslty dstruton re t lest or t most some certn lues. The necessty mesures nd the necessty ndex re smlrly defned s: N Nes Nes Z nf mx x, x r g N g g N g Z, x x g, x x g In my estmtes, I ssume spred equl to for the most fuy mesures, C, D nd ZM, equl to for P nd equl to, for the less fuy lue, SG. Thus, I use the followng fuy sets: B B,,,,,,,,,,, B,,,, B,,,,,,,, B,,,, B,,,,,,,, B,,,, B,,,,,,,, B,,,, B,,,,, B,,,,,,, B,,,,, B,,,,,,, B,,,,, B,,,,,,, B,,,,, I wll lso mke the rght-hnd sde prmeters mguous nd use only possle mesures. I ssume tht the certnty degrees of oth defense strteges eng t lest equl to., s not less thn %. Smlrly, I ssume tht the certnty degrees of ll four offense strteges eng t lest equl to, s not less thn %. These ounds pply to oth tems nd re ery moderte compred to the determnstc estmtes from the preous models. Gen the symmetrc trngulr fuy lues, nd the ssumptons oe, the PLP model for s: mx s. t.,,......,,..,,....,

12 smlr formulton pples for CM.. Vn Hop s uy LP model Let us fnlly mke oth left-nd rght-hnd sde prmeters fuy. Vn Hop formulted uy LP model, usng erorty nd nferorty mesures. ~ ~ where, u, = centrl lues nd,,c,d R, ~ ~ Z Gen two fuy numers, u,,, Z, c, d.e. the left nd rght spreds respectely, nd f, the erorty of Z ~ oer ~ s defned s: Sup Z ~, ~ d u, nd the nferorty of ~ to Z ~ e defned s: Inf ~, Z ~ c u. Smlrly, gen two trngulr fuy rndom rles ~ B ~, the erorty of B ~ oer ~ s defned s: d w w B ~ ~ Sup, w u w, nd the nferorty of ~ to B ~ e defned s: w c w ~,B ~ Inf w u w. Let us now ssume the followng symmetrc trngulr type, fuy rndom prmeters. The four offense fuy prmeters for re: ~ ~ ~ ~ ~, ~, ~, ~, ~, ~, ˆ, ~, ~. ~ ~ ~ ~ ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, w, w,, ~, wth p w., p w. ~, w, w Notce tht the frst row s dentcl to the respecte determnstc lues frst entres of Tle nd hs ~, s prolty of %. In order to e consstent wth the PLP model preously, we ssume tht the fuy the expected lue oe the mnmum lue. The second row conssts of the respecte fuy rles nd hs lower prolty. Smlrly, the two defense fuy prmeters gn for only re: ~ ~, ~ ~ ~ ~, w, w, w, w, ~, ~, ~, ~, w, w, w, w ~, ~, ~, ~, ~. ~, ~, ~, ~, ~. ~ ~, ~, ~, ~, ~. ~ ~, ~, ~, ~, ~., wth p w., p w. Notce tht n ths mtrx, the frst nd thrd rows re the respecte determnstc lues from Tle, whle the second nd fourth rows re the true fuy ones. In order to e consstent wth the symmetrc trngulr fuy lues n the PLP model preously, we keep the sme spreds. Thus, we he the followng fuy numers:

13 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, nlly, sed on the fuy numers oe, we construct n erge fuy numer for the respecte offense nd defense constrnts, such s:,,, =,,, = +++/ =. nd, =, = +/ =. ollowng Vn Hop, the correspondng LP model for s:,..,,,,,..,,,..,,,,, mx nf nf nf nf nf nf nf nf nf nf m k s t m k m k m m k k m m k k smlr formulton pples for CM.. Results The unrestrcted offense nd defense strteges, model, re presented n Tle nd the restrcted ones, model, re presented n Tle. The mxmng tem s n ld nd the other tem n tlcs. In LP wth complementry constrnts nd n the Nsh, oth tems mxme nd re n ld. In clsscl optmton model, oth tems ply pure strteges, recees ponts nd CM. It s rther surprsng ecuse plys defensely nd CM plys offensely, no mtter whch tem mxmes. In oth cses, plys ZM nd CM plys D, = =.

14 In model, the strteges chnge. When mxmes, t plys the pure strteges P nd ZM = = nd recees, under the condton tht CM plys lso ts pure strteges T nd D, = =. When CM mxmes, t recees, y plyng P nd T, = =, gen tht plys T nd C, = =. In the mxmton of the mnmum pyoffs, model, oth tems use mxed offense strteges when they mxme seprtely. When mxmes, t gets. ponts, f t plys offensely.% P nd.% SG nd CM plys defensely.% ZM nd.% T. Smlrly, when CM mxmes, t gets. ponts when t lso plys offensely.% P nd.% D nd plys defensely.% ZM nd.% T. In model, when mxmes, t contnues wth the sme offense gme ut t plys % T s well. Gen the fct tht CM contnues wth the sme defense gme nd lmost eqully lnced wth ll the offense strteges, gets. +. =. ponts. When CM mxmes, t contnues wth lmost the sme weghts n P nd D, nd lso plys lmost % ZM nd % T. Snce contnues wth the sme mxture n defense, nd lso wth ll four offense ones, wth P ust oe %, CM gets. +. =. ponts. In the LP wth complementry constrnts model, plys mnly defensely lmost % T nd CM lmost offensely.% SG nd.% C, wth two poste slcks sl =., sl =., gng CM more ponts thn! In model, mxes two offense strteges.% SG nd.% P nd plys lso % T. CM shfts strteges y plyng % SG nd % C, nd lso mxng ts defense strteges, wth more weghts n ZM. gets more ponts from ts offense strteges =., =, whle CM gets slghtly more ponts from ts defense strteges =., =.. Notce though tht n ths cse there re fe poste slcks, sl =, sl =, sl =., sl =., sl =.. In Nsh, model, Mthemtc found seen equlr, wth three of them eng pure strteges nd four mxed ones. The three pure strteges nd one of the four mxed ones re dentcl n model s well. Notce lso tht the pure strteges Nsh equlrum = = s dentcl wth the soluton from the clsscl optmton when mxmes nd s the only one where CM plys offensely. prt from the Nsh pyoff,, n ll other equlr gets more ponts thn CM, wth the lrgest dfference, when plys % P nd CM defends wth % T. In nother Nsh equlrum,.,., the dfference s pproxmtely % n for of. Tht equlrum s found f plys % ts nd % ts, whle CM plys.% ts nd.% ts. In tht cse the product for s: =.. Smlrly the product for CM s: =.. In CCP model, when mxmes, t plys % C f CM defends y.% wth T nd.% y ZM, gng. ponts. On the other hnd, when CM mxmes, t mxes four strteges, wth D domntng y.%, gen tht defends y out / T nd / ZM, nd gng CM. ponts,.e. rther lnced gme. In model, oth tems shft strteges. When mxmes, t plys % P nd % ZM, whle CM plys ll sx strteges wth chnges n defense weghts. When CM mxmes, t shfts to two pure strteges, % C nd % T, whle plys ll sx strteges too nd chnges ts defense weghts. In ths model, the offense strteges ge. ponts to nd. ponts to CM. On the other hnd, oth tems get lmost the sme ponts. ersus. from ther defense strteges. Notce tht CM would recee ponts too f t ccepted the soluton n whch mxmes,.e. = = nd = =.

15 Tle : Unrestrcted offense nd defense strteges Model Tem Offense strteges Defense SG C P D T ZM Clssc - Opt. CM CM Mx-mn CM - - of pyoffs CM LP & Compl. Constr. Nsh CCP PLP uy.. CM - - CM CM... CM.... CM CM CM... CM. - - CM CM CM CM CM CM In PLP model, the results re rther smlr s n CCP. Both tems, when they mxme, mx ther offense strteges, wth most weghts n C. Both tems mx lso ther defense strteges wth lmost dentcl weghts when the other tem mxmes. gets. nd CM. ponts. Notce though the two negte lues n the defense strteges, = -. nd = -., ndctng tht the certnty degree of defense strteges eng t lest equl to., should not e less thn %, s olted. or CM, the ddtonl -. s explned y the fct tht =.. On the other hnd, the certnty degrees of ll four offense strteges eng t lest equl to, should not e less thn %, s ld.

16 Tle : Restrcted offense nd defense strteges Model Clssc Opt. Mx-mn of pyoffs LP & Compl. Constr. NSH : offense CM: defense CM: off BC: def CCP PLP uy Tem CM CM Offense strteges Defense strteges SG C P D T ZM CM CM CM CM CM CM CM CM CM CM CM CM CM In PLP, model, oth certnty degrees re stsfed. Both tems, when they mxme, ply % ZM, plys lso % C, whle CM mxes ts C wth P. When one tem mxmes, the other tem mxes ll sx strteges, wth roughly smlr weghts. gets. +. =. ponts, whle CM gets. +. =. ponts, gn rther lnced gme. nlly, n uy model, oth tems, ply smlr strteges when they mxme, % T for CM nd lmost % for, nd mx ll four offense strteges, wth lmost smlr weghts, when the other tem mxmes. They get the sme ponts from ther offense strteges, ut CM gets more ponts thn BC from ts pure defense strtegy T.

17 In uy, model, whle the strteges from model remn unchnged, plys.% SG,.% C nd CM mxes out / C nd / D. Both tems mx lso ther defense strteges when the other tem mxmes. gn, CM gets. more ponts from ts T, whle gets. more ponts from ts offense strteges, ledng to lmost lnced gme. Notce tht n ll fuy models ll nferor lmds re poste, ryng etween. nd.. In generl, the erge strteges from ll models re: plys out % offensely,.e. out % P, % SG nd % C. Its hghest weght though s n defense, ZM wth out %. CM plys more offensely,.%, mnly through % D, nd y out % C. CM lnces ts defense strteges, y out % ZM nd % T. The closest to these erges s the Nsh equlrum tht ges. ponts nd CM. ponts. Smlrly, the erge strteges from ll models re: plys out.% P,.% SG nd.% C offensely nd.% T nd.% ZM defensely CM plys.% C,.% D,.% P nd.% SG offensely nd.% T nd.% ZM defensely. Conclusons numer of determnstc nd stochstc models where used to fnd out the optml offense nd defense strteges tht nd CM could he ppled durng ther UE CL mtch, sed on the selected mtch sttstcs. Snce won the mtch, the queston posed ws f CM could he done etter y followng etter strteges. Despte the fct tht the optml strteges ry wth the selected model, there re ndeed four dfferent strteges tht CM could he selected ccordng to the followng models: n fuy model CM gets % more ponts thn, f t plys purely defense,.e. % T n LP wth complementry constrnts model CM gets. % more ponts, f t plys offensely,.e. % SG nd % C n one Nsh equlrum where oth tems get ponts, f t plys % ZM n order to defend the % SG plyed y n fuy model where CM gets % less ponts thn, f t plys % T n defense nd lso.% C nd.% D n offense. On the other hnd, CM does t dly wth the followng strteges: n PLP model, f t reles mnly to C y % wth lmost no defense n one Nsh equlrum when plys % P nd CM % T n nother Nsh equlrum when plys % P nd % ZM whle CM plys.% D nd.% T n clsscl mxmton model, f oth tems ply % P, nd lso plys % ZM whle CM plys % T. Therefore, the fnl suggeston to CM s: When oth tems mxme smultneously, ply offensely, strt wth counter-ttcks wth fnsh wth shots on gol. would try wth tcklng, ut not wth hgh success. On the other hnd, excesse C f SG does not follow t should e oded snce cn mx ts two defense strteges successfully. If plys mnly ts mng pssng gme or ts SG, defend wth ZM nsted of T. I lee t open to the reders, the funs nd the mngers to conclude f CM s defet cn e explned ecuse t dd not followed the suggested strteges oe.

18 References Crlton, D.W. & J.M. Perloff: Modern Industrl Orgnton, th ed., Person ddson Wesley,. Dckhut J. & T. Kpln: Progrm for ndng Nsh Equlr, n H.R. Vrn, ed. Economc nd nncl Modelng wth Mthemtc, Telos, Sprnger-Verlg. Inuguch, M. nd J. Rmík: Posslstc lner progrmmng: ref reew of fuy mthemtcl progrmmng nd comprson wth stochstc progrmmng n portfolo selecton prolem, uy Sets nd Systems,, -. Luhndul, M.K.: Optmton under hyrd uncertnty, uy Sets nd Systems, -. Luhndul, M.K.: uness nd rndomness n n optmton frmework, uy Sets nd Systems, -. Pphrstodoulou, C.: The optml lyout of footll plyers: cse study for C Mln, Workng Pper, Munch Personl RePec rche,, downlodle t: Pphrstodoulou, C.: n nlyss of UE Chmpons Legue Mtch Sttstcs, Interntonl Journl of ppled Sports Scences, -. Pollrd, R. nd C. Reep: Mesurng the effecteness of plyng strteges t soccer, The Sttstcn,,, -. Th, H..: Opertons reserch: n Introducton, th ed., Person Prentce Hll, Upper Sddle Rer, NJ,. Vn Hop, N.: Solng fuy stochstc lner progrmmng prolems usng erorty nd nferorty mesures, Informton Scences,, -. Wolfrm Reserch, Mthemtc: uy Logc