results of playing an evasion game

8/14/2019 RESULTS OF PLAYING AN EVASION GAME

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NPS55 78 3N V L POSTGR U TE S OOL

Mnt y CaI r

RESULTS OF PLAYING N EVASION G M

byAlan R Washburn

January 1978

Approved fo r pub l ic re lease d i s t r i bu t i on un l imi ted


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N V L POSTGR DU TE S HOOLMonterey California

Rear Admiral Isham LinderSuperintendent

Reproduction of l l or part of this report s authorized.

Jack R BorstingProvost

This report was prepared by:

Reviewed by:

l f ~ t ~ ~Michael Sovereign hairmanDepartment of Operations Research

Released by:

William M TollesActing Dean of Research


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UN L SSIFIEDSECURITY CLASStFICATION OF THIS PAGE When Data Entered)

REPORT DO UMENT TION P GE READ INSTRUCTIONSBEFORE COMPLETING FORMI. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT S CATAL.OG NUMBERNPS55-78-34. TITLE and Subtitle 5. TYPE OF REPORT at PERIOD COVERE

Results of Playing an Evasion Game Technical6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(a) 8. CONTRACT OR GRANT NUMBER(a)Alan R. Washburn

9. PERFORMING O R N I ~ T I O N NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASAREA at WORK UNIT NUMBERSNaval Postgraduate SchoolMonterey California 9394011. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Naval Postgraduate School January 1978Monterey California 93940 13. NUMBER OF PAGES1014. MONITORING AGENCY NAME a AODRESS(II dllterent lrotn ControlUn, Olllce) 15. SECURITY CLASS. o t thla report)Unclassified

ISa. DECL A S FICATIo...1DOWNGRADINGSCHEDULE16. DISTRIBUTION STATEMENT o t thla Report)

Approved fo r public release; distr ibution unlimited

17. DISTRIBUTION STATEMENT o t the abetract entered In Block 20 dllterent lrOtZl Report)

18. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on rever.e aide neceea.,.,. and Identity by block number)GameEvasion

20. ABSTRACT (Continue on reveree aide neceea.,.,. and Identi ty by b lock number)A Searcher whose speed s vS attempts to come within distance R of anEvader whose speed s vE knowing that the Evader must stay within a squarof side L. This game has been played repeatedly t the Naval PostgraduateSchool under each of the folloWing three assumptions: Game 1 The Evader sgiven no information about the position of the Pursuer. Game 2 The Evaderknows t l l times the bearing to the Pursuer. Game 3 The Evader knows t l l times the range and bearing to the Pursuer. For L/v = 15.42 secondsp

FORM1 JAN 73 47 EDITION OF 1 lOV 6S IS OBSOLETESIN 0102-014- 6601 I SECURITY CLASSIFICATION OF THIS PAGE When Data IInt


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:: li: CURlTY CLASSIFICATION OF THIS PAGE WhenD Entered20 Continued.v v = .2, and R =


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R SULTS OF PL YING N V SION G M

byAlan R Washburn

Naval Postgraduate SchoolMonterey l i forn i

January 9 8


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RESULTS OF PLAYING N EVASION G M

byAlan R. WashburnNaval Postgradua te SchoolMonterey Cal i forn ia

SUMM RYA Searcher whose speed Vs a t tempts to come within

dis tance R o f an Evader whose speed vE knowing t h a t theEvader must s tay within a square o f s ide L. This game has beenplayed repeatedly a t the Naval Postgraduate School under each o fthe fol lowing th ree assumptions:

Game 1 The Evader given no informat ion about th e pos i t ionof the Pursuer .

Game 2 The Evader knows a t a l l t imes th e bear ing to the PursuerGame 3 The Evader knows a t a l l t imes the range and bear ing

to th e P ursu er

For L/Vp = 15.42 seconds vE/vp = 2 and R/L = .0286 themean t imes to de tec t ion were 265 367 and 407 seconds in theth re e c as es i s concluded t h a t bear ing alone i s almost asgood as range and bear ing fo r purposes o f evas ion.

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INTRODU TIONModern W o f t e n i n v o l v e s a s e a r c h f o r a t a rg e t t h a t i s

n o t ig no ra nt o f th e p o s i t i o n o f t h e s e a r c h e r . Assuming t h a tb oth S earc her and Evader must f ol low con ti nuou s t r a c k s t h e r ei s c u r r e n t l y no t h e o r y capable o f computing o r even u s e f u l l yapproximat ing th e mean t ime t o d e t e c t i o n i n such c i r c u m s t a n c e s .

I n o r d e r t o g a i n some unders tanding o f t h e s e n s i t i v i t y o f t h i st ime t o p r e c i s e l y what i nf or m at io n t he Evader has about t h es e a r c h e r t h e experiment d e s c r i b e d t h e Summary has been c a r r i e do u t u s i n g o f f i c e r s t u d e n t s a s p l a y e r s .

Experimental s e t u pEach p l a y e r saw t h e p l a y in g a r e a a s a square on h i s

own ca thode r a y t u b e w i t h i n which he could c o n t r o l h i s ownp o s i t i o n up t o h i s maximum speed u s i n g a two degree o f freedomj o y s t i c k . The p u rs u er s p o si t i on was r e p r e s e n t e d a s a d o t

surrounded py a c i r c l e o f r a d i u s R t h e c a p t u r e d i s t a n c e . InGame 1 t h e Evader saw o n l y a d o t r e p r e s e n t i n g h i s own p o s i t i o n .In Game he a l s o saw a s tr ob e p o in ti n g toward t h e p u rs u e rt h a t emanated from the d o t . In Game 3 th e P u r s u e r s p o s i t i o nt o g e t h e r w ith t h e surrounding c a p t u r e c i r c l e was shown i n s t e a do f th e s t r o b e . I n a l l c a s e s the game cont inued u n t i l e i t h e rthe Evader was inc luded w i t h i n t h e c a p t u r e c i r c l e o r u n t i l t h eEvader touched t h e boundary whichever came f i r s t . In 45t r i a l s t h e r e were t h r e e boundary t e r m i n a t i o n s . There werea l s o some i n s t a n c e s o f t e r m i n a t i o n due t o such t h i n g s a s machinef a i l u r e o r having t o go to c l a s s ; i n which case t h e t r u nc at e d


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t ime was added to the t ime fo r the next p lay count ing thesum o f the two t imes as one t r i a l .

In a l l exper iments th e pos i t ions o f Pursuer and Evaderwere i n i t i a l l y chosen a t random with in the squ are .

Subjec tsThe sUbjec t s were o f f i c e r s tuden ts enro l l ed in courses

t augh t by th e au th or . Each sub jec t played Pursuer and Evadersevera l t imes . Although an a t tempt was made to discover whichsub jec t s were the b e t t e r p layers and al though t he re was def i n i t eevidence o f le arn in g the da ta w i l l be t r e a t e d as coming froma s ing le popula t ion in t h i s r e por t .

Quant i t a t ive Resul t sThe experimental cumula t ive d i s t r i bu t i ons o f the t imes in

the th ree exper iments a re shown in Figures 1 3 in each casetoge. ther with an exponen t i a l curve having the same mean. Thehypothes is o f exponent ia l t imes i s no t re j ec ted a t the .5 l e v e lin any o f the exper iments by a Kolmogorov Smirnov t e s t and theexponent ia l curve provides a s t r i k i ng l y good v i su a l t inFigures and 3 . The f a c t t h a t the th ree exper iments invo lved i f f e r e n t numbers o f t r i a l s i s because one c l a s s o f s tu de ntsplayed Game 2 whereas another c l a s s played Game o r 3 a t theprefe rence o f the p layers . Game 3 was usual ly chosen becauseGame i s r a the r bor ing .

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Avai lable TheoryWhen the E vader s pos i t ion s randomly chosen bu t

s t a t i ona ry , the theory o f random search [1] pred ic t s t h a t thet ime to cap tu re wi l l be exponen t ia l with mean .5 L/Vp)/ R/L)= 270 seconds . A cor rec t ion s av ailab le fo r dynamic enhance-

ment th e even t the Evader moves about randomly. The cor r e c -t ion has th e e f f e c t o f d ec re asin g th e average t ime to cap tu re ;the e f f e c t would be smal l when vE/vp = .2 . The theory predic t sthe r e su l t s o f the f i r s t exper iment qu i t e accura te ly , bu t f a i l sin the experiments where the Evader maneuvers i n t e l l i g en t l y .

Observat ions on p layFor a s ta t ionary Evader, the Searcher should perform

an e xh au sti ve s ea rc h o f some so r t a r a s t e r - scan o r l addersearch , fo r example) , and the t ime to cap tu re should be uniformover th e i n t e rva l [0, 270 sec] with a mean of 135 seconds .Most Searchers in f a c t at tempted to search exhaust ive ly inExperiment 1 , bu t the r e su l t i ng search t imes a re ce r t a i n ly no tuniform over the spec i f ied i n t e r va l . There a re four reasonsfo r t h i s , l i s t ed order o f what the au tho r f ee l s to be in -creas ing importance:1) The Searcher was no t i n i t i a l l y placed in a corner , thus

requ i r ing him to spend a few seconds to ge t to the c loses t one .2 The pers i s tence o f th e scope face was only a few seconds ,

so t h a t was d i f f i c u l t fo r th e Searcher to place segmentn+l r i gh t next to segment n .

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3 The theory does no t al low fo r the double coverage t h a t i sunavoidable in ac tua l l y c arry in g o ut an exhaus t ive sea rch .

4 The Evader was no t qu i t e s t a t iona ry . His speed, w hilesmal l an d ran dom ly d i r e c t ed was su f f i c i en t to t u rn whatwas planned to be an exhaus t ive search i n to what tu rned

ou t to be a random sea rch .

In Garnes and 3, Searchers who t r i ed to search exhaus-t i ve ly quickly figured o ut t h a t the r epe t i t i ve nature o f th er e su l t i ng motion made t easy fo r the Evader to avoid i nde f in i t e ly .The word go t around no a t tempt was made to preserve t r i a l i nde-pendence by forbidding communication , and soon a l l Searcherswere employing unpredic table t a c t i c s . There seemed to be twod i f f e ren t phi losophies about how to do t h i s :1 Employ broad , tw i s t ing sweeps o f about the same rad ius o f

curvature as th e l a r ge s t c i r c l e t h a t would t in th esc reen with occas ional v i s i t s to the four corners . Theidea was t h a t sooner o r l a t e r the Evader would guess thedi rec t ion o f th e c urv atu re wrong and ge t caught .

Selec t a smal l re gio n w ith in which the Evader could bet rapped cond i t iona l on h is being there in th e f i r s t p lace .Af te r s ea rc hin g the smal l region se l ec t another one, andcont inue un t i l f i na l ly the co r r e c t region i s s e l e c t ed .

There were l ikewise two t a c t i c s fo r the Evader t h a t were commonlyused:

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1 Attempt to ge t behind the Searcher on the grounds t h a t heprobably would no t re turn to where he has j u s t been.

2 ~ v t n th e cons t ra i n t s o f the r eg ion , t r y to ge t away fromthe Searcher as f a s t as pos s ib l e .

i s i n t e r e s t ing t h a t a mixture o f the Evader s 1 and hasthe po t en t i a l fo r being opt imal in th e sense o f game theory;in o rder to counter 1 th e Searcher must employ re ve rsa ls o rf r equen t t i gh t tu rns t h a t reduce h is speed o f advance, thusmaking him vulnerable to 2 ) .

There were no s ign i f i can t di f fe rences in s tra teg y ev iden tbetween Games and 3. The d i f fe rence in mean t ime to cap tu rei s a t t r i bu ted to th ose occas iona l inc idents in Game when theEvader would mistakenly i n t e r p r e t a s lowly changing bear ing tomean t h a t th e Searcher was fa r away. Usual ly , however, th eEvader was ab le to do h is pass ive ranging wel l enough to movein the c o r re c t d i re c ti o n.

REFERENCE

[ ] Koopman, B.O. , The Theory o f Search , P t . I . KinematicBases , Operat ions Research 4, 324-346 1956

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results of playing an evasion game

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