2006 CCRTS. The State of the Art and the State of the Practice. A Game Theoretic Approach to Threat Intent Inference

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1 006 CCTS The Se of he Ar he Se of he Prcce A Ge Theorec Aroch o Thre Ie Iferece Age Trc: C Moelg Sulo Age Per Nuber: C-076 D She* The Oho Se Uvery 05 Dree Lborory 05 Nel Ave Colubu OH 40 Ph: El: he.00@ou.eu Gehe Che Prcl o of coc Iellge Auoo Ic Clhou Drve Sue 400 ocvlle MD 0874 Tel: rec Fx: El: gche@--.co Joe. Cruz Jr. The Oho Se Uvery 05 Dree Lborory 05 Nel Ave Colubu OH 40 Ph: El: cruz.@ou.eu Ch Kw Iellge Auoo Ic Clhou Drve Sue 400 ocvlle MD 0874 Tel: rec Fx: El: cw@--.co

2 ASTACT A ge heorec roch o hre e ferece I he verrl lry evroe or o effcely roly rec he eey ccl e fro he lower level l eorl foro. I h er we rooe ecerlze Mrov ge MG heorec roch o ee he belef of ech oble eey COA ECOA whch ulze o oel he verry e. I h he followg vge: Decerlze. Ech cluer or e e eco oly be o he locl foro. e u ore uooe ech grou llowg for ore flexble; Mrov Deco Proce MDP c effecvely oel he ucere he oy lry evroe; Ge oel wh hree lyer: re force eee blue force frely force whe force eurl obec; 4 Deceo. h he coero h yerc eey y ule he foro vlble o frely force; we egre he eceo coce our ge roch o oel he co of uroely reerg rl foro o cree he yoff of he eey. A ulo ofwre cge h bee eveloe wh coecvy o he oeg OEP Oe Exerel Plfor o eore he erforce of our rooe lgorh. Sulo hve verfe h our rooe lgorh re clble ble fcory erforce.

3 . Irouco Ge heory rove frewor for oelg lyzg vrou erco bewee ellge rol eco er or lyer coflc uo whch every vul eco er o colee corol of oher eco u eerg o he evroe. The e of he ge c be rce bc o he bylo Tlu whch he colo of ce lw ro e ow urg he fr fve ceure A.D.. However w o ul 944 h he hecl heory of ge w vee by Joh vo Neu Or Morgeer []. Mheclly Ge heory ue o uy regc uo where lyer chooe ffere co e o xze her reur whch ee uo he choce of oher vul. To e ol ovee ul-ge ye rege of oher ge houl be e o ccou herefore eel o be ble o oel he behvor of he ooe. I he verrl lry evroe or o effcely roly rec he eey or verry ccl e fro he lower level l err eorl foro. Sr AI ool for olvg eco-g roble colex uo uch yc eco ewor fluece gr re o lcble o hee of uo. Ge heory o he oher h rove hecl frewor ege for he ly of ge erco uer he uo of roly where oe re o efy he ge equlbr ooe o rol uly xzo rcle. A ge cooe ul-ge ecog hu ue roly ool o rec he behvor of he oher ge [-5]. I h er he focu o he lco of Mrov Ge [] he ul-ge exeo of Mrov Deco Procee MDP o he eo of eey coure of co COA [] whch roxely oel he e of rge. y uccefully eg oble fuure hre fro he verre he eco er c e ore effecve rgeg eco l frely COA ge he c of uexece verry co rec eg ye o beer oberve verry behvor. e hve cheve he followg or reul. Fr we eveloe hghly ovve frewor for hre e reco urb wrfre eg be o Mrov Sochc ge heory. I co of hree cloely coule cve: he roceg egro of foro fro re ource o rouce egre obec e; he reog groug he cooerve obec whch erfor coo ; recg he eo CoA of yerc hre. Seco we hve leee verry Mrov ge oel wh hree lyer: re force eee blue force frely force whe force eurl obec. Th gfcly exeo of exg ge heorec ool for oelg corol of lry r oero whch oe o exlcly coer he eurl force or cvl ellge lyer []. Ihere foro erfeco coere leee wo eho: ecerlze eco g chee; eceo wh boue roly. Thr ofwre rooye h bee eveloe wh coecvy o he MICA Mxe Ive Corol of Auo-Te Oe

4 Exerel Plfor [4] oology-be vrul blece o eore he erforce of our rooe roch. I h verfe h our rooe lgorh re clble ble erfor fcorly ccorg o he uo wree erforce erc. The er orgze follow. I eco we wll urze he echcl roch whch clue bc e of Mrov ge hre e ferece frewor ovg horzo corol roch for ge oluo. Seco ecrbe he exerel reul. Seco 4 coclue he er.. Mrov Ge Frewor e rooe Mrov ge MG heorec roch o ee he belef of ech oble eey COA ECOA becue Decerlze. Ech cluer or e e eco oly be o he locl foro. e u ore uooe ech grou llowg for ore flexble; Mrov Deco Proce MDP c effecvely oel he ucere he oy lry evroe; Ge oel wh hree lyer: re force eee blue force frely force whe force eurl obec. Ge frewor effecve el oel o cure he ure of lry coflc: he eero of oe e rege ghly coule o h of he oher e rege vce ver. h he coero h yerc hre error y c le eurl or whe obec cvl we lo oel he co of whe u our ge frewor; 4 Deceo [0]. h he coero h yerc eey y ule he foro vlble o frely force; we egre he eceo coce our ge roch o oel he co of uroely reerg rl foro o cree he yoff of he eey. Fg. Sochc ge oel for e ferece The ge e ferece frewor how Fg.. I coruce fro he l e evolve ccorg o he ro rule. A ech e blue re co re

5 ece ccorg o he vrou eor rule of eggee roly err foro curre e. Thee co ece he ue robbly rbuo over he e ce ccorg o he ro rule whch lo e err foro oe of he u. Oly oe e elece he ex e. I he rel ble he e choe by he ure whle our oel rw be o he robble of ll oble e. The bgger he robbly he eer wll be rw. y efo Mrov ochc ge gve by fe e of lyer N ; fe e of e S ; for every lyer N fe e of vlble co A we eoe he overll co ce A N A ; v ro rule q : S A S where S he ce of ll robbly rbuo over S ; v yoff fuco N r : S A. For he e ferece roble coo obvouly fe. Coo hol becue we ue h rule of eggee err foro re ow ech lyer h le e of COA gve by he ocre err foro ec. For exle rver wll l he co of grou force. v v re ege ccorg o he ecfc uo clug err foro. For our hre reco roble we ob he followg cree e ecerlze Mrov ge: Plyer Deco Mer --- Alhough our rbue ecerlze Mrov ge oel ech grou cluer e e eco here re hree lyer: eey frely force whe obec. All cluer of eey frely force or whe obec c be coere gle lyer ce hey hve coo obecve. Se Sce --- All he oble COA for eey frely force co of he e ce. A elee S hu le of eey frely force COA cooe of e of rle eource Aco Verb Obecve. A exle eey COA gh be: he re e eource c Aco Verb he blue e Obecve. I he coex of h reor ue h for he eey COA he eource lwy verry ey whle he Obecve frely e. Slrly for he frely force COA eource frely e Obecve verry ey. S S S S where S he COA of lue frely force { r o r A o O } A O re he e of he reource co obecve of blue force reecvely. O he oher h S he COA of e eey force { r o r A o O }. S he COA of he force eurl obec. Aco Sce --- A every e e ech blue grou chooe l of rge wh oce co cofece robbly rbuo over he l of rge.e. he u of he cofece houl be equl o be o locl ble fel foro uch he u ye oo of oble rge fro level fuo. Le efe D eoe he co ce of he h blue e. Ech elee of D

6 { A O 0 < } where he robbly of he co-rge coule whch efe he co o rge. Therefore he co ce of blue e A D. A exle for he blue ll weo UAV blue e co gh be {c re fgher 0. fly o re fgher 0.5 vo re fgher 0.}. Slrly ech re cluer obe fro he level fuo eere robbly rbuo over ll oble co-rge cobo. Le D eoe he co ce of he h re cluer. Ech elee of D efe { A O 0 < } where he robbly of co o rge. Therefore he co ce of re force A D. A oble co for re lfor re fgher {c ll weo UAV 0.6 ove o blue oler 0. vo blue oler 0.}. er: Aco Aco Verb re ffere coce. Aco e of rle wh oce robble whle Aco Verb u cooe of rle cooe of eource Aco Verb Obecve. All Aco re clue A for lyer lue force A for lyer e force. All Aco Verb re euere A for lyer lue force A for lyer e force. The co of whe obec re relvely le. The co ye he ovee. Le D eoe he co ce of he h whe u. Ech elee of D efe { A O 0 < } where he robbly of co o rge Tro rule --- Due o he ucery roere of lry evroe we ue h he e of he Mrov ge hve er o h he eco er hve ore chce uru of he obecve of he revou co. e efe er fcor T vecor for lyer where he uber of he e or cluer of lyer 0 h h. So for he e of he lyer here robbly of o ee he curre co-rge coule robbly of - o ue he ew co cooe of co-rge coule. There re wo e o clcule he robbly rbuo over he e ce S where + re e of e e + reecvely re he eco of lyer blue force or frely force lyer re force or eey ly whe force reecvely e e. Se : wh he coero of er fcor vecor we cobe he curre e wh eco of boh lyer o ob fue robbly rbuo over.

7 ll oble co-rge coule for re blue force. To o h we fr ecooe he curre e o he co-rge coule for ech e of ech lyer re force or blue force. Le eoe he reulg co-rge h coule rele o he e of he h lyer. For exle f here oe rle of blue e c re fgher he curre e he he co-rge coule for blue e he fr e of blue force c re fgher h. Secoly for ech ecfe e y he cluer of lyer eey force we fue co-rge coule v ofyg he robbly of ech oble co-rge coule be o he followg forul There re four ce Eq 6: The co-rge coule oly occur h he curre co of cluer of lyer o he curre e whch c be heclly rereee by { }. The we ow he robbly of curre e 0 robbly of curre co. So ccorg o he efo of er he fue robbly of he co-rge coule +0. h The co-rge coule he boh he curre co of cluer of lyer he curre e. The we ow he robbly of curre e robbly of curre co. So ccorg o he efo of er he fue robbly of he co-rge coule + +. The co-rge coule oly occur he curre e he we ow he robbly of curre e robbly of curre co 0. So ccorg o he efo of er he fue robbly of he co-rge coule The co-rge coule occur h eher he curre e or he curre co of cluer of lyer he we ow he robbly of curre e 0 robbly of curre co 0. So ccorg o he efo of er he fue robbly of he co-rge coule Slrly he ew robbly rbuo for he h e of lyer blue force

8 + } { 0 } { } { } { 5 The ew robbly rbuo for h e of lyer whe force + } { 0 } { } { } { 6 Se : we eere he robbly rbuo over he ll oble oucoe of e + + q 7 whe } { } { } { r r r +. Oherwe 0 + q. where he uber of he e or cluer of lyer blue force he uber of he e or grou of lyer re force he uber of he u of lyer whe force. } { r he e of he ll oble wh ove robbly rle for he h e of lyer blue. Therefore } { r co ll he oble wh ove robbly rle for he blue force. Fro he e we ow h he fue robbly of ech ecfe efe equo 5. h he uo h ll e of blue force re eee we ob he overll robbly of blue force. Slrly re he overll robble of he eey whe force reecvely. So he robbly rbuo over he ll oble oucoe of e + cooe of ll oble ub-e of blue force re force c be clcule v equo 7. Pyoff Fuco --- I our rooe ecerlze Mrov ge oel here re wo level of yoff fuco for ech lyer eey frely force or whe force. The lower level yoff fuco re ue by ech e cluer or u o eere he e co be o he locl foro. For he h e of blue force he yoff fuco efe ~ f where ~ he locl foro obe by he e he wegh for ll oble co-rge coule of blue force ouce o ll blue e eere ccorg he o level yoff fuco by uervor of lue force.

9 g w f ~ ~ 8 where w wll clcule he wegh for y ecfe co-rge coule for h e of blue force fro he he robbly of he corge coule ~ g wll eere he g fro he co-rge coule for h e of blue force ccorg o he oo feure uch lfor vlue efee/offee cbly of blue e lfor. Slrly we ob he lower level yoff fuco for he h e of re whe force g w f ~ ~ 9 g w f ~ ~ 0 er : For oe yerc hre uch uce bober he yoff fuco y oly coer he lo of he blue e. For oe couflge cocele ee her obecve re o he heelve ove cloe o he blue u. Oher eceo u wll o oe rrol ovee o he her rue gol wh he co he e. er : The whe u oly cre bou her oble loe. For exle whe gerou o eece orl whe u wll f COA o ee heelve fr oble fro he o. The o level yoff fuco re ue o evlue he overll erforce of ech lyer. f J ~ f J ~ f J ~ where he e ex. I our roch he lower lever yoff re clcule rbuely e bc o coer/uervor v couco ewor. er : Sce he g fuco ~ g for blue force ~ g for re force ~ g for whe force re ffere fuco yerc force co ule c be rghforwrly rereee our oel. I o fer rregulr verry eece ffere ye of g fuco wll be ge yclly.

10 The rege --- I h er we ry everl well ow ye of rege. M-x rege [5]. Th of rege wll gve coervve oluo o ze he oble xu lo. Acully our roble x oluo he ee h ech lyer xze he oble u h yoff. So h of rege lo clle fe oluo. Pure Nh Srege wh fe horzo. I ge heory he Nh equlbru e fer Joh Nh [6] who rooe of ol collecve regy ge volvg wo or ore lyer where o lyer h yhg o g by chgg oly h or her ow regy. If ech lyer h choe regy o lyer c beef by chgg h or her regy whle he oher lyer ee her uchge he he curre e of regy choce he correog yoff coue Nh equlbru. I our roch we ue ge erch ree how Fg. o f he oluo. I our rooe roch he oluo o he Mrov ge ree obe v K e-e loo-he roch whch we oly oze he oluo he K e-e horzo. The ubol echque ue uccefully for reog ge uch che bcgo oooly. Fg. A ge ree erch roch o f ure Nh rege wh fe lg horzo ovg lg wow Mxe Nh Srege. A xe regy ue ge heory o ecrbe regy core of oble co oce robbly whch correo o how frequely he co choe. Mxe regy Nh

11 equlbr re equlbr where le oe lyer lyg xe regy. I w rove by Nh h h every fe ge h Nh equlbr bu o ll h ure regy Nh equlbru. Correle Equlbr [7]. Ule Nh equlbr whch re he coce of equlbr forule eee rege he correle equlbr were eveloe fro he correle rege ocooerve ge. The correle equlbru of Mrov ge ecrbe oluo for lyg yc ge whch lyer re ble o couce bu re elf-eree. e o he gl geere by he correle evce ouce o he ech eco er lyer chooe her co ccorg o he receve rve gl. There re wo ye of correlo evce: uooou ory evce. A uooou correlo evce r where M fe e of gl for lyer e e : M M M N M M M M M. A ory correlo evce r where M M N M. Acully ory correlo evce ecl ce of uooou correlo evce where M eee of co fuco h eee of. Gve correlo evce we efe exee ge. The ge lye excly he orgl ge bu he begg of ech ge gl cobo rw ccorg o he robbly fuco N ech lyer fore of. The ech eco er u be h choce of co o he receve gl. Ay evor wll be uhe v h -x vlue. The uhe oly occur f lyer obey he recoeo of he evce. I w rove [7] h every Mrov ge wh uooou correle evce correle equlbru. Sequel Nh Srege [8]. A equel ge oe whch lyer chooe her rege followg cer reefe orer whch le oe lyer c oberve he ove of oher lyer who e eco recee he. The equel ge url frewor o re oe rel roble uch he Aco-eco- Couerco rg ue lry ellgece verg cg rege of everl coeg fr ecooc. I our roch we ue ur-by-ur chee how Fg.. A ech e he corol regy fro oly oe lyer y lyer blue force le he correog Fg. Two-lyer Sequel Ge oucoe wll be oberve by lyer

12 eey force before ece ex co. Th helful eg he ooe e becue ech e oly oe co le he e chge fro he co oberve. Leer-Follower Srege [9]. h he coero of he le oerfec couco we ue he Scelberg coceo o oel he cooero r bewee he coer he locl e. The coer clle he leer eclre ceve o he locl e clle follower orer o uce he o cce h ere ye behvor he coo ere behvor. The Leer-Follower regy ueful our clerly efe herrchcl ye whch hve yerc foro rucure our ce he leer or coer ow he co fucol of every eco er or locl e whle he follower ow oly her ow. To effcely reo he eey e or COA we ve our roch o wo he: rg he reog he. I he fr oe we eure or oberve he eey co core he wh he co geere by our oel. The reul re ue o ue or u he ro rule. I he reog he we fx he ro rule ue he geere re co he e of eey. Deceo [0] re ue o e he oher lyer c our ow vge by g beleve h he ge e oher h he cul oe. I oly oble rl foro ge. e rooe equlbru roch o eceo where eceo efe o be he roce by whch co re choe o uce erroeou ferece o o e vge of he. Th frewor ffer fro he erler lerure o ulge ge wh colee foro h wheher he lyer h erfec uerg of he regy eloye by h ooe. e rouce wo ye of eceo: cogve ye regy evroe. Cogve ye re efe follow. Ech lyer for execo bou he behvor of he oher lyer by oolg ogeher everl oe whch he oher lyer u ove. Ech uch ool of oe referre o cl of logy. Plyer re lo fferee ccorg o wheher or o hey guh bewee he behvor of he vrou ye of her ooe. Forlly cogve ye of lyer chrcerze by A where A for lyer logy ro uy vrble h ecfe wheher or o ye guhe bewee he behvor of he vrou ye of lyer. e le whe ye guhe bewee ye behvor 0 oherwe. A regc evroe ecrbe by Yu where eoe he ror o rbuo o he ye ce Q Q Q. To lfy oo we wll ue h he ye of he wo lyer re eeely rbue fro ech oher we wll refer o he ror robbly of lyer ye where eoe he ror robbly of ye.

13 . Sulo Exere I he Sulo r we bul vrul ble-ce ycl urb cero be o he Oology coce whch exlc forl che-reble ec oel h efe he cle or coce her oble er-relo ecfc o oe ecfe o. To ule our fuo roch we leee ee our ble-ce cero lgorh o he MICA Oe Evroe Plfor OEP be o he oeg C4I ulo whch oel he colleco roceg eo of blefel foro. e ue cero how Fg. 4 o eore he erforce of our rooe hre reco uo wree lgorh. I he how urb evroe he blue force o re o cure ecure wo brge whch re gure by he re force. The wo brge loco re well coece v we ro hghlghe by he le. The re force clue re vehcle fgher yerc force hg cg le he whe obec he cvl vehcle. The blue force co of few fgher wh cloe r uor rove by everl ue erl vehcle UAV uch ll eor UAV ll weo UAV whch wll f eee o oe erchg fghg oo. e ue he ol offee force ol efee force re lo he e level. There o o oe. There re everl choce for he re force o gur hee obecve effcely. They c eloy ll re u o roec oe loco. However he blue force c cure oher lce fr. The blue force fce he e le. So he chllege for boh e o uer he uo fro he fue eor rec he e of he ooe uer he beleve wr uo. Fg. 4: A Sule Scero

14 For h cero ecfc ulo ru correle equlbru eho how Fg. 5 blue e blue e were ge o cure rge rge reecvely lo he whole ulo ero of 0 ue. O he oher h e e e Te were gurg rge rge lo ll he e. lue Te w regclly ovg be o he hre reco fro he Mrov ge. A he e e he e Force w yclly eloyg e Te ryg o ee blce bewee he lue force he e force ech brge. A how he ove of he eo we c ee h fer bou 5 ue of ovee rge he lue force cheve oo wh lue e lue Te lue Te o c re e e Te. The lue force cure ecure rge before e Te reche rge o hel e Te. Durg h Phe ble our lgorh eece wo yerc verre wh eceo ero vehcle whch were he v of bcgrou hrle cvl cvy. Afer he cure of rge he lue e eloye r of he reg force of Te Te o ecure he brge oher o hel Te o cure rge. Flly he lue e wo he urb ble co of 5 lue oler 6 ll weo UAV. Aoher he yerc hre wh eceo error w eece lle he Phe II ble of curg rge. Fg. 5: eul of ulo ru

15 I o o he exle ru we erfore y exere. e core he reul ug he vrou oo uch whou ge heorec fuo whou level whou yerc Thre Preco wh level bu he yoff fuco of ge oel level oe chge yclly Ge roch wh xe Nh Sregy ge roch wh Correle Equlbr. The reul ce he ulo ochc reul re Fg. 6: Dge coro of vrou oo he e of 0 ru for ech ce re how Fg. 6 Oly he ge foro of he lue e how. Fro he ge coro reul we c ee h our rooe Mrov ge frewor wh correle equlbru eceo coero for hre eeco uo wree beer h he oher eho. 4. Cocluo Ge heorec ool hve oel for hre reco h e rel ucere e l eceo oble o coero. I h er we hve evlue he febly of he vce verry e ferece lgorh her effecvee hrough exeve ulo. I h verfe h our rooe lgorh re clble ble erfor fcorly. Acowlegee Th reerch w uore by he US Nvy uer corc uber N C-005. The vew cocluo coe here re hoe of he uhor houl o be erree ecerly rereeg he offcl olce or eoree eher exree or le of he Nvy.

16 eferece [] J. vo Neu O. Morgeer Theory of Ge Ecooc ehvor Prceo Uvery Pre 944. [] E. Sol N. Velle Correle Equlbru Sochc Ge Ge Ecooc ehvor vol. 8 o Feb. 00. [] Norh Alc Trey Orgzo The L C Iforo Exchge D Moel ATCCIS ele.0 Mrch 00 [4] Uer Gue for he Oe Exerel Plfor OEP vero. oeg Mr 00 [5] T. r G. J. Oler Dyc Nocooerve Ge Theory SIAM Sere Clc Ale Mhec.Phlelh eco eo Jury 999. [6] Joh Nh Nocooerve ge Al of Mhec vol [7] E. Sol N. Velle Correle Equlbru Sochc Ge Ge Ecooc ehvor vol. 8 o Feb. 00 [8] D. M. Kre. lo Sequel equlbr Ecooerc vol. 50 o July 98. [9] M. S J.. Cruz Jr. "O he Scelberg Sregy Nozero-Su Ge" Jourl of Ozo Theory Alco Vol. No. 5 My [0] J. P. Hehz Y. S. Ae H. H. Kzloc Deceo No-Cooerve Ge wh Prl Iforo I Proc. of he DAPA-JFACC Syou o Avce Eerre Corol July 000. [] Cruz J.. Jr. S M. A. Gcc A. Jg H. Leeller. L M. Lu Y. Ge-Theorec Moelg Corol of Mlry Ar Oero IEEE Trco o Aeroce Elecroc Sye Vol. 7 No. 4 Ocober 00. [] Cruz J.. Jr. S M. A. Gcc A. Lu Y. Movg horzo Nh rege for lry r oero IEEE Trco o Aeroce Elecroc Sye [] Cruz J.. Jr. Che G. Grgc D. T X. L D. She D. e M. g X. Te yc cc for o lg Proceeg IEEE Coferece o Deco Corol Deceber 00. [4] D. She J.. Cruz G. Che C. Kw S. P. le S. Cox C. Mhew A Ge Theorec Aroch o Mo Plg For Mulle Aerl Plfor AIAA foech@eroce Se Arlgo VA. [5] G. Che D. She J.. Cruz C. Kw S. Cox S. P. le C. Mhew A ovel cooerve h lg for ulle eroce vehcle Proceeg of AIAA foech@eroce Se Arlgo VA.