Non-Cooperative Games

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1 N-Cperatve Games a ucerta evrmet Rger J-B Wets rbwets@ucavs.eu Uverst Calra, Davs ONR-MURI, Jul 2002 p.1/??

2 I. Determstc Vers ONR-MURI, Jul 2002 p.2/??

3 Fg a Nash-equlbrum prblem rmulat the Nash-uct asscate wth a game max- pts a Nash equlbrum pts remars abut cmputatal schemes ONR-MURI, Jul 2002 p.3/??

4 Aget s Prblem te (tw?), agets:, ecs aget, ecss all ther agets, -perrmace c : Nash Equlbrum: such that r all ONR-MURI, Jul 2002 p.4/??

5 Startg pst B G ONR-MURI, Jul 2002 p./??

6 . %$ -, + )( ONR-MURI, Jul 2002 p.6/?? Mathematcal Mel -a a $ ' %$ %$ # "! %$ %$ # "! %$ / %$ # "!

7 -perrmace uct u a, x a - z -a ONR-MURI, Jul 2002 p.7/??

8 7 7 The Nash-uct 6! 8 r sme a therwse ONR-MURI, Jul 2002 p.8/??

9 ; : - Equlbrum a max- pts s a Nash equlbrum wth a l = < 6! ; 3 $ 3 : < 6! $ = < 6! ; 3 $ 3 : s a argmax- pt the Nash-uct..e., 6! > ONR-MURI, Jul 2002 p./??

10 I I I C C Q R Exstece Nash Equlbrum L I FHGE ANM E I L I K J I argmax-. exstece exstece, cvex s usc s lsc s usc 7 usc ccave. S R cvex cvex ONR-MURI, Jul 2002 p.10/??

11 Nash equlbrum ONR-MURI, Jul 2002 p.11/??

F @? E I I K J I U T T T T Stablt Nash Equlbrum ANM E I C L I FPO @ A L I I C ANM FHG @BADC stablt Nash Equlbrum = stablt argmax- Nash-uct Nash-ucts

12 E I I K J I U T T T T Stablt Nash Equlbrum ANM E I C L I A L I I C ANM stablt Nash Equlbrum = stablt argmax- Nash-uct Nash-ucts lpse cvergece Nash-cs cvergece argmax- pts! lpse cvergece bvarate ucts. lpse cvergece s relate t ep-cvergece. (uvarate) ucts ONR-MURI, Jul 2002 p.12/??

13 > \ [ Z Z Y Y ' Z Z Z R R Augmete Nash-uct PL-hmtp meths, lw mesal ptmzat-base meth va augmetat - = sale pt V W Zba ' Z = X ^`_ (] V W where a Zba are ual rms. ONR-MURI, Jul 2002 p.13/??

14 c V Z Y ' ' W _ Z Y ' ' _ Z Y ' ' _ h Y Iterats 6!. Set S 6! Za ^ (] be Za ^ (] g! 3 Za be be ^ (] be g! max- pt., as ONR-MURI, Jul 2002 p.14/??

15 II. Stchastc Evrmet ONR-MURI, Jul 2002 p.1/??

16 Exstece, Algrthms -cperatve a ucerta evrmet the agets ptmzat prblems rmat lw -atcpatvt stegrat the stchastc prblem Nash-cs asscate wth a stchastc game remars abut exstece, cmputatal prceures ONR-MURI, Jul 2002 p.16/??

17 Frmulat ucerta (stchastc) evrmet ecs tme 1 (w) ml ecs tme 2 l perrmace estmate aget Nash Equlbrum: R such that r all : R -perrmace estmate ONR-MURI, Jul 2002 p.17/??

18 < $ + )( Aget s Prblem t l t q ;sr p 7 t ml ONR-MURI, Jul 2002 p.18/??

19 < $ + )( Aget s Prblem t l t q ;sr p 7 t ml a tw-stage stchastc ptmzat prblem (geeralzes t -stage, amcall) ONR-MURI, Jul 2002 p.18/??

20 $ + )( 7 Aget s Prblem t l t q ;sr < p t ml a tw-stage stchastc ptmzat prblem (geeralzes t -stage, amcall) Nte: strbut es t epe but the strbut the state the sstem es ONR-MURI, Jul 2002 p.18/??

21 l + )( 7 u u v v Stchastc Optmzat < q;sr p $ l Decs prcess: ecs bservat recurse Irmat prcess: rmat abut the uture s avalable t ca ttall epe realzat.e., there s a -atcpatvt restrct. ONR-MURI, Jul 2002 p.1/??

22 Remvg -atcpatvt: p;sr < $ q;sr < l )( + ml 7 w 7 w x Wth a cstrat qualcat, Q multplers R p such that - a p $ q [ \ l )( + l 7 has the same slut wth R = cstat. ONR-MURI, Jul 2002 p.20/??

23 DISINTEGRATION Oe ca slve: p; r < $ q; r < [ \ l )( + l 7 b slvg r each : p $ q [ \ l )( + ml wth p q, ONR-MURI, Jul 2002 p.21/??

24 z z - { - Prgressve hegg algrthm Step 0. pc R wth,. ONR-MURI, Jul 2002 p.22/??

25 z - { - $ \ [ g Prgressve hegg algrthm., z wth, R $ Step 0. pc Step 1. r each l p< q;}~ g q! p $ p! ONR-MURI, Jul 2002 p.22/??

26 z - { - $ \ [ g Z ƒ {, Prgressve hegg algrthm., z wth, R $ Step 0. pc Step 1. r each l p< q;}~ g q! p $ p! $ $ Step 2. set Z $ $ }! Stp $ $ therwse, a retur t Step 1. wth be ONR-MURI, Jul 2002 p.22/??

27 z - { - $ \ [ g Z ƒ {, Z { Prgressve hegg algrthm., z wth, R $ Step 0. pc Step 1. r each l p< q;}~ g q! p $ p! $ $ Step 2. set Z $ $ }! Stp $ $ therwse, a retur t Step 1. wth be Cvergece Z $ 8 a a prxmal term $ ONR-MURI, Jul 2002 p.22/?? lear cvergece

$t l \ t q p t ml ONR-MURI, Jul 2002$

28 - [ $ + )( Dstegrate equlbrum: ;, let r r each, the etermstc Nash equlbrum whe the agets prblems are: t l \ t q p t ml ONR-MURI, Jul 2002 p.23/??

29 \ [ R Exstece a Algrthm(s) Exstece: ee Nash-ucts r stegrate prblems (=> exstece) a use stablt Nash equlbrum w.r.t. perturbats ( ) Slut Prceure: verall strateg the Prgressve Hegg algrthm t bta cvergece the (-atcpatvt) multplers. Step 1 PHa, terate Augmete Nash-uct t bta argmax- pt. ONR-MURI, Jul 2002 p.24/??

Ordinary Differential Equations. Orientation. Lesson Objectives. Ch. 25. ODE s. Runge-Kutta Methods. Motivation Mathematical Background

Ordinary Differential Equations. Orientation. Lesson Objectives. Ch. 25. ODE s. Runge-Kutta Methods. Motivation Mathematical Background Ordar Deretal Equats C. 5 Oretat ODE s Mtvat Matematcal Bacgrud Ruge-Kutta Metds Euler s Metd Hue ad Mdpt metds Less Objectves Be able t class ODE s ad dstgus ODE s rm PDE s. Be able t reduce t rder ODE