Applying Retrograde Analysis to Nine Men's Morris. 1. Introduction

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1 Applyin Rtror Anlysis to Nin Mn's Morris Rlph Gssr Inormtik ETH 809 Zürih Switzrln Astrt This ppr prsnts rsults in throuh rtror nlysis o Nin Mn's Morris. First, I sri som o th mthos mploy to lrt th lultions, or xmpl runlnth noin o th tss. Son, th rsults thmslvs r xmin. Whil mny humn plyrs intuitivly rr Nin Mn's Morris s rthr simpl, rwn m, my rsults sm to sust othrwis. Som o th optiml ply is lrly yon humn ility, somtims rquirin ovr 0 plis o sminly rnom movs or win is rh.. Introution Work on this sujt oriinlly volv rom my Nin Mn's Morris prorm [Gssr 90]. Du to th irnt phss o th m, it ws vint tht th mim vlution untion lon ws insuiint. It ws prtiulrly unsuit urin th nm. Inst o imprtly tunin th vlution untion to ths ss, I i to solv th prolm y xhustivly nlyzin rtin nm positions. Rtror nlysis is thniqu whih hs om quit populr rntly or solvin suh xhustiv nlysis prolms. It is muh ppli to hss, whr som intrstin n unsuspt rsults hv turn up. For summry s [Hrik 8]. Som o ths rsults hv or FIDE, th Intrntionl Chss Frtion, to hn thir ruls [Hrik 89]. Althouh mny popl hv lry work on rtror nlysis, it sms to m tht th mploy lorithms still hv th potntil or improvmnt. Som improvmnts will rthr prolm spii, whil othrs my hv ror pplition. Alon ths lins, rtror nlysis or Nin Mn's Morris ws implmnt. I will xplin how run-lnth omprssion ws us to sp up th pross n isuss othr mor prolm pnnt onsirtions. Atrwrs, th rsults in or Nin Mn's Morris r prsnt. Althouh ths rsults o not yt hv ny inlun on th qustion o whthr th omplt m o Nin Mn's Morris is rw or not, thy o sust tht Nin Mn's Morris is muh rihr in possiilitis thn most humn plyrs suspt. Prtiulrly th sum whr oth plyrs hv only stons on th or is hrly vr (0.%) rw, ut or humns, inin win is xtrmly iiult.

2 . Nin Mn's Morris Nin Mn's Morris is ply on or with points whr stons my pl (Fi.). Initilly th or is mpty n h o th two plyrs rivs nin stons. Th plyr with th whit stons strts. Th m onsists o thr phss: opnin phs mim phs nm phs Durin th opnin phs, plyrs ltrntly pl thir stons on ny vnt point. Atr ll stons hv n pl, th or miht look s in Fi.. Now ply pros to th mim phs. Hr plyr my sli ny on o his stons to n jnt vnt point. I t ny tim urin th m, plyr sus in rrnin thr o his stons in row, lso ll "losin mill", h must rmov n opponnt's ston. Any ston whih is not likwis prt o mill my rmov. I ll th opponnt's stons r prt o mills, ny ston my rmov. In Fi., Whit hs just ply to (opnin phs) n n now rmov Blk's ston on, ut not th on on. Fi. Th mpty or Fi. Atr th opnin phs As soon s t lst on plyr hs only thr stons lt, w pro to th nm phs. Whn it is his turn, th plyr with thr stons my tk on o his stons n jump to ny vnt point on th or. I h loss mill, h my, s usul, rmov n opponnt's ston. Th m ns in th ollowin wys: Th irst plyr who hs lss thn thr stons loss. Th irst plyr who nnot mk ll mov loss (Fi. ). I rptition o position ours, th m is rw. Fi. Whit movs to Eithr or is rmov Fi. Blk to mov No ll movs vill

3 .. Wht is rtror nlysis?. Rtror Anlysis Th xprssion "rtror nlysis" is us xtnsivly in omputr hss irls, whr it stns or mtho o lultion whih ins th optiml ply or ll possil or positions in spii nm. In omputr sin, th somwht nloous trm "ynmi prormmin" is otn us. Th hrtristi o this mtho is tht kwr lultion is prorm. On strts with ll positions tht r immit wins or losss. From ths, ll won or lost positions in ply n lult n thn with ll ths positions, th positions tht r won or lost in plis t. Thr r two min rsons or pplyin this mtho to Nin Mn's Morris. First, it sms possil tht nw insihts into th mtho miht in tht r not s sily visil usin hss. Son, Nin Mn's Morris is onsirly lss omplx thn hss. This smllr omin ttrt m, us in th lon run thr is hn o solvin th m ompltly... Stnr rtror nlysis Rtror nlysis is omplish y th ollowin thr prours: Initiliztion Loss kup Win kup Th initiliztion loop os throuh ll ors n intiis trminl positions, whr on plyr hs ithr won or lost. In Nin Mn's Morris ths trminl position r ithr positions whr th plyr to mov is lok n thror loss (Fi. ) or positions whr th plyr to mov n los mill n thry ru his opponnt to lss thn thr stons. Th loss kup prour srhs or ll positions in whih th plyr to mov hs lost in spii numr o plis. For ths positions ll ll prssor positions r nrt. All o ths prssor positions r wins. Whit to mov Nwly intii won position Blk to mov Lost position

4 Th win kup prour is th most prossin intnsiv. First, th won positions or spii ply-pth r intii, n thir prssor positions nrt. For prssor position to loss, ll o its possil sussor positions must wins. Blk to mov Nwly intii lost position Whit to mov Won position Whit to mov Won position Rtror nlysis is omplish y rptly pplyin th loss n win kup prours to ll or positions. Th lorithm trmints whn no nw wins or losss r intii.. Possil Improvmnts This stion ls with possil improvmnts to th stnr mtho sri ov. Thy r not ll qully pplil to othr ms or othr hrwr oniurtions, ut wr spiilly tilor to Nin Mn's Morris n th Appl Mintosh's oprtin systm.... Initiliztion In th lst stion, w sw tht th irst stp in rtror nlysis is to initiliz ll th immit won/lost position. As irst pproh this n solv y lookin t vry possil or position n trminin i it is won or lost. Th ost o oin this initiliztion n hih prnt o th totl run-tim o rtror nlysis, spilly i most positions r rws (i.. th win n loss kup loops on't rquir muh tim). An xmpl o suh sitution ours in th - sum o Nin Mn's Morris whr only 88 o th ''9 positions r non-rws. To in n ltrntiv, lt's tk look t spii st o our inst our positions. Fi. shows possil oniurtion o our whit stons. Thr r 8 possil wys o plin th our lk stons. I it is Whit's turn to mov w woul normlly lult ll 8 possil positions n or h o thm tst i Whit is lok. Thr is str wy howvr. Fi. shows tht w n t lst 9 lk stons to rnr ll th whit stons immoil, thror non o th lk ston oniurtions n lok th our whit stons. A similr pross n us whn it is Blk's turn to mov. In Fi. w s how th our whit stons prtition th or into rs. Th rs ontin, n mpty squrs. It is thror possil to pl,,,,8,9 or 0 lk stons so tht thy hv no jnt n vnt squrs, i.. thy r lok. But thr is no wy or Whit's stons to lok our lk stons. So on in, w n rpily trmin tht non o th 8 lk oniurtions rsult in Blk in lok.

5 Fi. Whit ston oniurtion Fi. Minimum mount o lk stons n to lok Whit Fi. Whit nnot lok lk stons This mtho o lookin t multitu o similr positions simultnously is mittly rthr prolm pnnt. But th min i n proly otn us in on orm or othr. It is rtinly lso pplil to hss. For instn, tr plin ll pis o on olor, on n trmin sust o th opponnt's possil kin positions, so tht mt n only our y plin th kin on position in this sust.... Win kup In th stnr win kup prour, w o throuh ll ors intiyin th positions whih win in spii numr o plis. For h o ths, w nrt thir possil prssor positions. For vry prssor position whih hs not yt n intii s win or loss w nrt its sussor positions. I ll th sussor positions r wins, w hv nwly intii loss. Th omplxity o this lultion is : O ( #Wins * #AvPr * #AvSu ) whr # Wins : numr o winnin positions in spii numr o plis # AvPr : vr numr o prssor positions # AvSu : vr numr o sussor positions Thr is nothr wy o hivin th sm rsult. Inst o strtin with th won positions, kin thm up n hkin i thy r so r unintii, w n simply o throuh th il srhin or unintii positions. For ll ths positions w thn prorm th sm hk with thir sussor positions. Doin th win kup this wy rsults in th ollowin omplxity: O ( #Unintii * #AvSu ) Rouhly spkin, this susts tht th son mtho is str i #Win * #AvPr > #Unintii. In Nin Mn's Morris suh situtions n our. For xmpl in th - sum w hv som 0 million positions n th numr o wins in plis is out 900'000. Sin #AvPr is out 0 it is justii to intiy th losss in plis usin th son mtho.... Run-lnth noin My lultions wr on on n Appl Mintosh, whih ws lso us or othr tsks, so tht only 0-0 MByts o hr isk stor pity ws vill. Thror lultin tss with 0 million positions (with yt pr position to irntit twn plis) wsn't possil without som sort o t omprssion. Run-lnth omprssion sms irly wll suit to Nin Mn's Morris, us thr r mny rwn positions whih n stronly omprss. I lso tri to trnslt 'similr' or positions to jnt il positions. Hr 'similr' ors r thos whih rquir n intil numr o plis until th m ns. I this n on, run-lnth omprssion is mor tiv.

6 Stnr run-lnth noin trnsorms squn o intil vlus into only on opy o th vlu plus th lnth o th squn. It nrts lr svins or lon squns, ut or short squns (or xmpl squn lnth o on) it my tully us mor mmory. Exminin th Nin Mn's Morris tss show tht thr wr mny irly short squns n lso tht rtin vlus (thos orrsponin to low ply numrs) wr mor rqunt. Cominin run-lnth noin with vril lnth noin miht thror oo i. Ths is l to th oin shm isply low. Vlu Ino Lnth Ino X Vlu 0 Lnth X Vlu - 0 Lnth Lnth This is silly run-lnth noin, ut sin most positions will ithr rwn or won/lost in smll numr o plis, ths mor rqunt vlus r stor usin only on yt. Svn its r us to stor rws n wins/losss rom 0 to plis. For up to 8 plis son yt is us. Sin thr r mny runs o lnth on, it in th vlu ino (it X) ws us to init run-lnth o on. I th run-lnth is rtr thn on, on yt is us or run-lnths rom 0-. Two yts or 0-'8 n our yts or vn lonr runs. This o hs rtin runny, ut hs prov suiint. Blow th rsultin omprssion tors r list or th lult sums. A sum is hrtriz y th numr o stons on th or, i.. th - sum hs stons o on olor n o th othr. Sum Unomprss Fil Siz Comprss Fil Siz Comprssion tor - 0'0 89'80. - ''90 9' ''.8 - '8'0 8'. - '98'0 ' '9'8 9'.0 - '90'0 9'.8 - '9'880 ''8. Run-lnth noin hs prviously n sust or omprssin th inl tss [Mrris 89]. As r s I m wr, no prvious ttmpts wr m to prorm rtror nlysis usin omprss t. Any tim urin rtror nlysis whn w prorm squntil srh throuh th il, or xmpl whn w n ll or positions with ivn vlu, run-lnth omprssion sps our srh us th il is smllr. A prolm only ours whn w n th vlu o spii or position. Whn usin unomprss tss, w know xtly t whih position th vlu n oun. Usin run-lnth omprssion, w must silly srh throuh th whol il until w rh th rquir position. By usin n inx whih points into th omprss il t numrous pls w n sp this up. With this inx, w n rstrit th squntil srh to smllr prt o th il. I th omprssion tor is lr nouh, this squntil srh n still str thn usin unomprss t, us lrr prt o th il n hl in th mmory h, limintin slow isk sss. Anothr t to kp in min is tht th omprss il siz is smll t th innin o rtror nlysis n rows lrr th mor nw positions w intiy. This mns tht or smll ply-pths, whn most nw positions will oun, w still hv smll il i.. st ss. Whn th omprss il rows lrr with inrsin ply-pth, th numr o positions tht w hv to hk usully rows smllr, so tht st ss rows lss ritil.

7 . Rsults Th ollowin rsults wr ll otin usin Mintosh IIx with 8 MByts o mmory. Th run-tims vri rom w minuts (- sum) to wks (- sum). Mor xt timin is not vill us this prorm ws only run whn th omputr ws not othrwis us. For h ts, som ri sttistis lon with n xmpl o on o th most iiult positions r ivn. Whn thr is mor thn on optiml mov, th ltrntivs r innt. Th trminoloy is s ollows: A W-B sum inlus ll or positions whr thr r W whit stons n B lk stons i.. th - sum ompriss ll positions with whit stons n lk stons. Th positions with lk n whit stons n trnsorm to - positions y swithin ston olors. A position is ll 'won' i th plyr to mov n win, likwis or losss. Th 'Mx Win Ply' n 'Mx Loss Ply' olumns iv th lonst winnin or losin squn or position in th sum. Plyin this squn n rsult in losin mills n ntrin irnt sums. 'Mx Win Ply' n 'Mx Loss Ply' msur th numr o plis until win/loss is inlly rh n not th numr o plis until th sum hns. Ths rsults isprov mny sttmnts oun in [Müllr 8]. It sms tht humns r not spilly oo t ths omintoril ms... - Sum This ws th irst sum to xmin n to t it is lso th most intrstin, i.. th prnt o rws is xtrmly low (out 0.%). This ontrits humn intuition, whih susts tht th - sum is lmost lwys ti. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov ' 9 9'9 An xmpl o win position in plis is shown low lon with possil winnin squn: Whit to mov win in plis.. - Sum This sum is somwht lss intrstin thn -, ut w n still in som nw insihts rom this ts. First, osrv tht th plyr with stons hs mximum win ply o i.. h n only win i it's his turn n h hs n opn mill, rthr ptholoil s whih proly won't our urin tul m plyin. Son, i it's Blk's ( stons) turn, h n't

8 los! Oviously th itionl moility in y hvin th ility to jump (i.. stons lt) is quit vlul. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov '9 8'90 '09 Blk To Mov 0'8 8' 0 - An xmpl o win position in plis is shown low lon with possil winnin squn: Whit to mov win in plis.. - Sum This is rthr orin sum. Thr r hrly ny winnin positions (out 0.00%) n vn wr losin positions. But th wins tht n rh r o irnt ntur thn or. Wins r not hiv y rmovin n opponnt's ston, ut rthr y lokin his pis. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov 9 '' Th ollowin xmpl shows tht vn th lonst winnin squns r sily roniz y humn.

9 Whit to mov win in 9 plis.. - Sum Th - sum is similr to th - sum. Th plyr with stons hs mny winnin positions, ut sin thy r t suh shllow pths, it sms unlikly tht thy will rh in tul ply. Th plyr with stons hs intrstin winnin squns, ut thr r so w o thm tht this sum is ssntilly ti. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov 80'0 '999'0 0 - Blk To Mov '0 '' 9' Th ollowin xmpl shows tht to win this sum, th opponnt must irst ru to stons. This susts mtho o winnin rwn position inst n imprt opponnt: th plyr with stons shoul try to ru his opponnt to stons, us orrt - (or -) ply is mor iiult thn orrt - ply. Whit to mov win in plis

10 .. - Sum Th - sum is lso silly ti. All possil wins r rh y lokin th opponnt. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov 9'889 0'00' Blk To Mov 0'08'9 '0 8 Whit to mov win in 9 plis.. - Sum Th - sum is similr to th - sum. But thnks to th itionl ston, Whit nnot los. Th possil wins r t shllow pths, ut strnly nouh humns sm to in optiml ply rthr iiult. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov '' '' Blk To Mov 0 '8'0 9'000 - Whit to mov win in plis

11 .. - Sum All wins r hiv y lokin th opponnt. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov 8'89 0'88'800 '9 Whit to mov win in plis.8. - Sum This is th sum tht to t hs th lonst winnin squn, nmly plis. Th tsts I hv prorm init tht inin winnin squn is lrly yon humn ility. # Wins # Drws # Losss Mx Win Ply Mx Loss Ply Whit To Mov '98'9 9'80'9 Blk To Mov ''98 '9'9

12 Whit to mov win in plis Unortuntly, th winnin squn is muh too lort to shown hr, ut n otin rom th uthor.. Conlusion On purpos o this work ws to ttr unrstn th ilitis n limittions o rtror nlysis. Usin th improvmnts sri rlir n othr Mintosh spii justmnts, th sp o th rtror lultions ws inrs y mor thn tor 0. Furthr work will ous on nrlizin th xprin in in Nin Mn's Morris to othr pplitions o rtror nlysis. A son, irly lon trm, ol o this work is th omplt nlysis o Nin Mn's Morris. Whthr or not this n hiv in th nr utur pns rt l on th m itsl. I Nin Mn's Morris is rw, thn mny or vn most positions will rws n th runlnth noin shm will prov tiv in oth spin lultions n in ruin mmory sp rquirmnts. At prsnt I sm to pprohin th limits impos y th Mintosh. In th utur I will ithr hv to port my o to str mhin or sp up my lultions. A urthr spup my hiv y not irntitin twn th numr o plis rquir to win, ut rthr to lssiy positions only s wins,losss or rws.. Rrns [Gssr 90] [Hrik 8] [Hrik 89] [Mrris 89] Gssr, R., "Huristi Srh n Rtror Anlysis: thir pplition to Nin Mn's Morris", Diplom thsis, ETH Zürih, Frury 990. vn n Hrik H.J. n Hrshr I.S.,"A t s on t ss", ICCA Journl 9(), p. 9 (Mrh 98). vn n Hrik H.J. n Hrshr I.S.,"Th 0-mov rul rvisit", ICCA Journl (), p. 9 (Sptmr 989). Mrris, C.A.,"Comprssin hss-nm ts", ICCA Journl (), p. (Mrh 989). [Müllr 8] Müllr, R.F., "Mühl", ECON Tshnuh Vrl, Otor 98.