Decision Methods for Engineers

Transcription

1 OE : itroductio 1 Itroductio to game Repreetig game Iformatio i game layig with other ratioal aget olvig game Zero-um game Mixed trategie

2 Outlie 1 Itroductio to game Repreetig game Iformatio i game layig with other ratioal aget olvig game Zero-um game Mixed trategie Military trategy Itroductio to game Example (attle of imarck) attle theatre: Two poible route for Covoy from Rabaul to Lae, each takig three day to complete llie earch aircraft ca cocetrate o either route ad weather o orther route make earch there difficult Oce Covoy potted, bomber deployed to attack it Deciio: Covoy: which route? llie: earch where?

3 trategic aalyi Itroductio to game llie optio 1: Cocetrate earch i the orth. trategic aalyi Itroductio to game llie optio 2: Cocetrate earch to the outh.

4 ame elemet Itroductio to game Thi ettig ivolve: more tha oe aget (called player): llie ad Covoy move/trategie for each player: choice of route for covoy; earch area for llie outcome that co-deped o the trategie of both player (play): the four poible ceario above preferece over outcome repreeted by payoff for each player: umber of day covoy i bombed ame Theory Itroductio to game Defiitio game i ay ettig i which the outcome co-deped o the actio/trategie of two or more player. Outcome which are the reult of ratioal play by player are called olutio of the game Fidig olutio ivolve determiig what player hould do im of ame Theory The aim of game theory i to develop techique for idetifyig olutio to game ituatio.

5 Repreetig game Military game: table repreetatio Covoy llie llie geeral () actio are row label; Covoy geeral (C) actio are colum Each poible ituatio i a outcome whoe preferece quatified by umber of bombig day uruit ad evaio game Repreetig game Example (urue ad evade) prioer () i plaig a ecape from prio. There are two poible ecape route: i the prio orth or outh wig. prio guard () i o watch. The guard ca patrol oe wig but ot both.

6 rio ecape: game tree aalyi Repreetig game Coider thi ceario repreeted a a game tree: From viewpoit: where i outcome e the prioer ecape, ad i c he caught. Each player ha differet payoff, or utility, fuctio for the outcome; for each player p (here p {, }): u p : Ω R rio ecape: game tree Repreetig game Combie each player payoff ito outcome of a commo tree: Each outcome ha a payoff vector with oe value for each player: (u (ω), u (ω)) I thi cae payoff are complemetary: i.e., u (ω) + u (ω) = 0. uch game are called zero um game

7 ame i exteive form Repreetig game Defiitio (ame tree) game tree i alo called the exteive form of a game. ame tree allow fie modellig of game: idividual move at differet tage for each player tur tructure i which player make move at differet tage: e.g., alteratig, imultaeou, etc. iformatio tate, or epitemic tate ( tate of kowledge ), of player at each deciio poit cotiget/coditioal actio/trategie for each player which deped o it epitemic tate: e.g., if prioer move orth, I ll move orth too rio ecape: epitemic tate Iformatio i game Cae 1: The guard oberve the prioer movemet: dditioal kowledge/iformatio about the prioer move give guard a advatage uard optimal trategy: follow prioer move ; i.e., if move, the move ; if move, the move

8 Ecape revered Iformatio i game Cae 2: The prioer oberve the guard movemet before ecapig: dditioal kowledge give advatage to the prioer Optimal trategy: move oppoite the guard; i.e., if move, the move ; if move, the move. Modelig iformatio Iformatio i game Cae 3: either oberve the other move (e.g., imultaeou move): Defiitio (Iformatio et) iformatio et i a et of deciio ode that are epitemically iditiguihable by a aget. iformatio et defie a aget epitemic tate at ome deciio poit. I a game of perfect iformatio every iformatio et ha oly a igle ode; i.e., i a igleto et.

9 Epitemic modellig Iformatio i game The game graph o the right i a alterative repreetatio of prioer ecape game i Cae 3 Here actio i ukow to : i.e., both poibilitie lead to ame epitemic tate for move are odetermiitic i ee that ame actio lead to differet outcome ormal form Iformatio i game Defiitio game matrix i called the ormal (trategic) form of a game. What do the ormal form of the game tree above look like? 1, ,

10 Modelig iformatio Iformatio i game 1, , y obervig move, i Cae 1 hould have a wiig trategy ; i.e., oe that alway yield payoff 1 to Let F be guard optimal trategy: follow prioer move F 1, , oible trategie Iformatio i game Defiitio trategy for a aget i the pecificatio of a uique move i each of it reachable iformatio et (epitemic tate). oible trategie for i Cae 1: if, the ; if, the if, the ; if, the if, the ; if, the if, the ; if, the

11 ormal form Iformatio i game / / / / / / / / Meet lice ad ob layig with other ratioal aget ob lice

12 Example: lice ad ob layig with other ratioal aget Example (lice, ob, ad a cocout) lice () ad ob () are at the bae of a cocout tree which ha oly oe cocout worth 10 kilocalorie (kc) of eergy i total. To get the cocout, oe (or both) mut climb the tree to hake it looe. It would take lice ome effort (2kc) to climb the tree, wherea ob effort i egligible. If ob climb (c) the tree ad lice wait (W) below the lice will get to the cocout firt whe it drop, eatig mot of it (9kc worth) ad leavig oly a mall portio for ob. If lice climb (C) ad ob wait (w) below the ob will get to it firt ad eat hi fill (4kc worth) before lice get dow ad take it off him. If both climb the ob will climb dow quicker ad eat ome (3kc worth) before lice get dow take the ret. ame tructure: lice move firt layig with other ratioal aget uppoe lice move firt; i which cae ob will gai iformatio about lice move. c 5, 3 C w 4, 4 c 9, 1 W w 0, 0 What hould lice do? Wait below hopig for 9kc ad rik 0kc? Climb herelf, ettlig for omethig i betwee?

13 ame v igle-aget deciio layig with other ratioal aget C W c w c w c w C 5 4 W 9 0 From lice perpective the deciio table would look like the oe above lice might ue oe of the deciio rule uder igorace a he doe t kow what ob will do; e.g., Maximi (C) ut lice i t igorat about ob! lice kow ob i ratioal (i.e., will try to maximie utility) lice What if... aalyi layig with other ratioal aget if I wait... c 5, 3 if I climb... c 5, 3 C w 4, 4 C w 4, 4 W c w 9, 1 0, 0 W c w 9, 1 0, 0... ob will climb... ob will wait lice cocluio lice bet trategy, coiderig ob ratioal repoe, hould be to Wait i preferece to Climbig (payoff to lice of 9 compared to 4).

14 trategie ad couter-trategie layig with other ratioal aget If lice move firt, ob ha more iformatio, ad hece more trategic optio; i.e., ob poible pure trategie are: Regardle of whether lice climb or wait, I will wait Regardle of whether lice climb or wait, I will climb I will do the ame a lice: i.e., if lice climb, I will climb; if lice wait I will wait I will do the oppoite of lice: i.e., if lice climb, I will wait; if lice wait I will climb If lice wait, the ob bet couter-trategy i to climb If lice climb, the ob bet couter-trategy i to wait Combiig thee, ob optimal trategy i to do the oppoite of what lice doe dditioal iformatio of game layig with other ratioal aget The game i modelled a a game matrix by: W/w C/w X W/w C/c b 1 W/c C/w X W/c C/c b 1 C 4, 444 5, 3 4, 4 5, 3 W 0, 000 0, 0 9, 1 9, 1 ob domiat trategy i: if lice wait, the I climb; if lice climb, I wait ; i.e., W/c C/w

15 layig with other ratioal aget Reaoig about other aget preferece reviou example how that multi-aget deciio are more complex tha igle aget deciio Epitemic tate of aget affect the trategic optio available to them Multi-aget deciio hould icorporate the preferece ad epitemic tate of the other aget; e.g., lice what if... aalyi of ob repoe to her move Cocluio Reaoig about other player preferece might improve the outcome for each player. ame olutio olvig game Defiitio (lay ad olutio) I two-player game, a play i a pair ( 1, 2 ) coitig of a trategy form each player. play uiquely determie a outcome to the game. For -player game thi geeralie to -tuple ( 1, 2,..., ). The outcome of ratioal trategie from each player i called a olutio to the game. i about developig method ad techique to idetify the olutio to game Domiace ca help implify the problem baed o the aget preferece Do all game have olutio? (Exitece) re olutio uique? (Uiquee)

16 Zero-um game Two-player trictly competitive game Defiitio (Two-player trictly competitive game) two-player trictly competitive (adverarial) game i oe i which the preferece of each aget are i oppoitio. zero-um game i a trictly competitive game i which the aget payoff are complemetary; i.e., their um i zero. For example: r r R 1, , 0 R , 000 2, Other example: che, poker, football, etc. ecaue payoff are complemetary, by covetio oly the row player are how Domiace-baed olutio Zero-um game Recall that: Defiitio (Domiace) trategy i domiated by trategy if for each of the other player trategie, the outcome of i at leat a preferred a that of the correpodig outcome of, ad for ome trategy of the other player it i trictly more preferred. a b c If i domiated by, the i a better trategy regardle of what trategy player 2 play; i.e., it i a uiverally better repoe Domiated trategie ca be diregarded/dicarded

17 Domiace olutio Zero-um game Exercie pply domiace to implify the followig game by elimiatig domiated trategie. a b c a b c Domiace help fid olutio by elimiatig trategie that either player will play The play left after domiace i the game above are (,a) ad (,b) are thee atifactory olutio? imple problem/olutio Zero-um game Example (Racig Uai olt) lice () ad Uai olt () have bee offered the optio to race for $1000; i.e., the wier get $1000 from the other. ame matrix (i $K): a r 1, , 0 R 0, 000 0, 0 hould lice agree? hould Mr. olt refue? Quetio: What hould the player do? What hould the outcome of thi game be?

18 imple problem/olutio Zero-um game Ituitio ugget that olt hould agree to race ad lice hould refue. a r 1, , 0 R 0, , 0 o the olutio i: (R, a) lice refue to race; olt agree; therefore the race doe t go ahead ad both keep their iitial pure Thi game i domiace olvable olvable by elimiatio of domiated trategie I thi a ituitive outcome for thi ituatio? Zero-um game Military game: table repreetatio Covoy llie llie geeral () actio are row label; Covoy geeral (C) actio are colum Each poible ituatio i a outcome whoe preferece quatified by umber of bombig day

19 The battle of the imarck ea Zero-um game The battle of the imarck ea i a zero um game with imperfect iformatio (either the covoy Captai or llie eeral kow the other move) ecaue the payoff are complemetary i a zero um game, by covetio oly thoe for the row player are how C ccordigly, the colum player prefer outcome with maller value i the table The attle of the imarck ea i iterated domiace olvable Meet lice ad ob Zero-um game ob lice

20 Zero-um game trictly competitive, o zero-um game The cocout game i a competitive game that i ot zero-um: W/w C/w W/w C/c W/c C/w W/c C/c W 0, 0 0, 0 9, 1 9, 1 C 4, 4 5, 3 4, 4 5, 3 W/c C/w W 9, 1 C 4, 4 W/c C/w W 9, 1 Domiace implie that the player hould chooe trategie: : Wait, ob: oppoite of (i.e., climb if wait, ad wait if climb); compare with Maximi (C) which ha a value of 4 Reverig role Zero-um game What if ob move firt? c w C W W w/w c/w w/w c/c c/w w/c c/c w 0, 0 0, 0 4, 4 4, 4 c 1, 9 3, 5 1, 9 3, 5 C 3, 5 1, 9 4, 4 0, 0 ob, by movig firt, caue to climb! w/c c/w w 4, 4 c 1, 9 w/c c/w w 4, 4

21 Mixed trategie: domiace Mixed trategie Let u a ad u b be utilitie for the row player whe the colum player play a ad b repectively: u b a b C u a oe of trategie i the game above are domiated... by aother pure trategy Coider mixture of trategie ad ll of player mixed trategie o egmet domiate C Mixed trategie: domiace Mixed trategie a b C 2 1 M 4 3µ 3µ u M a 4 M 3 b M( 1 4 ) C a C b µ Let M (µ) = µ + (1 µ); i.e., For example, M(µ) = (M a (µ), M b (µ)) = (4 3µ, 3µ) M( 1 4 ) = (3 1 4, 3 4 ) Domiace require: 4 3µ 2; i.e., µ 2 3 imilarly: 3µ 1; i.e., µ 1 3. C domiated whe both of the above hold: i.e., whe 1 3 µ 2 3

22 ummary Mixed trategie ehaviour of other ratioal aget make multi-aget deciio more complex layer may ue mixed trategie