Memoryle Strategie in Concurrent Game with Reachability Objective Λ Krihnendu Chatterjee y Luca de Alfaro x Thoma A. Henzinger y;z y EECS, Univerity o

Size: px

Start display at page:

Download "Memoryle Strategie in Concurrent Game with Reachability Objective Λ Krihnendu Chatterjee y Luca de Alfaro x Thoma A. Henzinger y;z y EECS, Univerity o"

Kellie Curtis
6 years ago
Views:

1 Memoryle Strategie in Concurrent Game with Reachability Objective Krihnendu Chatterjee, Luca de Alfaro and Thoma A. Henzinger Report No. UCB/CSD Augut 2005 Computer Science Diviion (EECS) Univerity of California Berkeley, California 94720

2 Memoryle Strategie in Concurrent Game with Reachability Objective Λ Krihnendu Chatterjee y Luca de Alfaro x Thoma A. Henzinger y;z y EECS, Univerity of California, Berkeley, USA x CE, Univerity of California, Santa Cruz z EPFL, Switzerland fc krih,tahg@eec.berkeley.edu, luca@oe.ucc.edu Augut 2005 Abtract We preent a imple proof of the fact that in concurrent game with reachability objective, for all ">0, memoryle "-optimal trategie exit. A memoryle trategy i independent of the hitory of play; and an "-optimal trategy achieve the objective with probability within " of the value of the game. In contrat to previou proof of thi fact, which rely on the limit behavior of dicounted game uing advanced Puiieux erie analyi, our proof i elementary and combinatorial. 1 Introduction We conider concurrent reachability game played by two player over finite tate pace. The configuration of uch a game i called a tate. At each round, the two player chooe their move concurrently and independently; the two move and the current tate determine a ucceor tate, or in general, a probability ditribution over the ucceor tate. A play of the game conit in the infinite equence of tate viited while playing the game. The goal of player 1 conit in forcing the game to a pecified et of target tate; the goal of player 2 conit in preventing the game from reaching a Λ Thi reearch wa upported in part by the ONR grant N , the AFOSR MURI grant F , and the NSF ITR grant CCR

3 target tate. Conequently, we aign value 1 to all play that reach the target et, and value 0 to all other play. The player can adopt trategie that are both randomized and hitory-dependent. Player 1 can guarantee avalue v for the game from a tate if player 1 ha a trategy that enure that the expected value of a play from i at leat v, regardle of the trategy choen by player 2. The value at of the reachability game with target T i the upremum of the et of value that player 1 can guarantee from. An optimal trategy for player 1 i a trategy that guarantee the value of the game from each tate. For " > 0, an "-optimal trategy for player 1 i a trategy that guarantee the objective i atified with a probability within " of the value of the game for each tate. Concurrent reachability game belong to the family of repeated game [11, 7], and they have been tudied more pecifically in [5, 4, 6]. It ha long been known that optimal trategie need not exit for concurrent reachability game [7], o that one mut ettle for "-optimality. It i alo known that, for ">0, there alway exit "-optimal trategie that are memoryle, i.e., uch that the probability ditribution over move depend only on the current tate, and not on the pat hitory of the game [8]. Unfortunately, the only previou proof i rather complex. The proof conidered dicounted verion of reachability game, where a play that reache the goal in k tep i aigned a value of ff k, for ome dicount factor 0 < ff» 1, rather than value 1. It i poible to how that, for 0 <ff<1, memoryle optimal trategie alway exit. The reult for the undicounted (ff = 1) cae follow from an analyi of the limit behavior of uch optimal trategie for ff! 1; the limit behavior i tudied with the help of reult on the field of real Puiieux erie [8]. Thi proof idea work not only for reachability game, but alo for total-reward game with non-negative reward (ee [8] again). We how that the exitence of memoryle "-optimal trategie for concurrent reachability game can be etablihed by more elementary mean, which do not require the conideration of dicounted verion of the game, nor reult on real Puiieux erie. In particular, we preent a proof that relie only on combinatorial technique, and on imple reult on Markov deciion procee [1, 3]. A our proof i eaily acceible, we believe that the proof technique we ue may find future application in game theory. 2

4 2 Concurrent Game with Reachability Objective Notation. For a countable P et A, aprobability ditribution on A i a function ffi : A 7! [0; 1] uch that a2a ffi(a) =1. We denote the et of probability ditribution on A by D(A). Given a ditribution ffi 2D(A), we denote by Supp(ffi) =fx 2 A j ffi(x) > 0g the upport of ffi. Definition 1 (Concurrent Game) A (two-player) concurrent game tructure G = hs; Move; 1 ; 2 ;ffii conit of the following component: ffl A finite tate pace S and a finite et Move of move. ffl Two move aignment 1 ; 2 : S 7! 2 Move n;. For i 2 f1; 2g, aignment i aociate with each tate 2 S, the non-empty et i () Move of move available to player i at tate. ffl A probabilitic tranition function ffi : S Move Move! D(S), that give the probability ffi(; a 1 ;a 2 )(t) ofatranition from to t when player 1 play move a 1 and player 2 play move a 2, for all ; t 2 S and a (), a (). At every tate 2 S, player 1 chooe a move a (), and imultaneouly and independently player 2 chooe a move a (). The game then proceed to the ucceor tate t with probability ffi(; a 1 ;a 2 )(t), for all t 2 S. A tate i called an aborbing tate if for all a () and a () we have ffi(; a 1 ;a 2 )() =1. In other word, at for all choice of move of the player, the next tate i alway. For all tate 2 S and move a () and a (), we indicate by Det(; a 1 ;a 2 )=Supp(ffi(; a 1 ;a 2 )) the et of poible ucceor of when move a 1, a 2 are elected. A path or a play! of G i an infinite equence! = h 0 ; 1 ; 2 ;:::i of tate in S uch that for all k 0, there are move a k 1 2 1( k ) and a k 2 2 2( k ) with ffi( k ;a k 1 ;ak 2 )( k+1) > 0. We denote by Ω the et of all path and by Ω the et of all path! = h 0 ; 1 ; 2 ;:::i uch that 0 =, i.e., the et of play tarting from tate. Strategie. A elector ο for player i 2 f 1; 2 g i a function ο : S 7! D(Move) uch that for all 2 S and a 2 Move, if ο()(a) > 0, then a 2 i (). We denote by Λ i the et of all elector for player i 2f1; 2 g. A trategy for player 1 i a function ß : S +! Λ 1 that aociate with every finite non-empty equence of tate, repreenting the hitory of the play o 3

5 far, a elector; we define trategie for player 2 imilarly. A memoryle trategy i independent of the hitory of the play and depend only on the current tate. Memoryle trategie correpond to elector; we write ο 1 for the memoryle trategy coniting in playing forever the elector ο 1. We denote by Π 1 and Π 2 the et of all trategie for player 1 and player 2, repectively. We denote by Π M 1 and Π M 2 the family of memoryle trategie for player 1 and player 2, repectively. Once the tarting tate and the trategie ß 1 and ß 2 for the two player have been choen, the game i reduced to an ordinary tochatic proce. Hence, the probabilitie of event are uniquely defined, where an event A Ω i a meaurable et of path. For an event A Ω, we denote by Pr ß 1;ß2 (A) the probability that a path belong to A when the game tart from and the player follow the trategie ß 1 and ß 2. Similarly, for a meaurable function f :Ω! IR, we denote by E ß 1;ß2 (f) the expected value of f when the game tart from and the player follow the trategie ß 1 and ß 2. For i 0, we denote by i :Ω! S the random variable denoting the i-th tate along a path. Valuation. A valuation i a mapping v : S! [0; 1] aociating a real number v() 2 [0; 1] with each tate. Given two valuation v; w : S! IR, we write v» w when v()» w() for all 2 S. For an event A, we denote by Pr ß 1;ß2 (A) the valuation S! [0; 1] defined for all 2 S by Pr ß1;ß2 (A) () =Pr ß 1;ß2 (A); imilarly, for a meaurable function f :Ω! [0; 1], we denote by E ß 1;ß2 (f) the valuation S! [0; 1] defined for all 2 S by E ß 1;ß2 (f) () =E ß 1;ß2 (f). Given a valuation v, and two elector ο 1 2 Λ 1 and ο 2 2 Λ 2, we define the valuation Pre (v), Pre ο 1;ο2 1:ο1 (v), and Pre 1(v) a follow, for all 2 S: Pre ο 1;ο2 (v)() = X a;b2move X t2s Pre 1:ο 1 (v)() = inf Pre (v)() ο 1;ο2 ο22λ2 Pre 1 (v)() = up ο12λ1 v(t) ffi(; a; b)(t) ο 1 (a) ο 2 (b) inf Pre (v)(): ο 1;ο2 ο22λ2 Note that all of thee valuation are monotonic: for two valuation v; w, if v» w, then for all elector ο 1 2 Λ 1 and ο 2 2 Λ 2 we have Pre ο 1;ο2 (v)» Pre ο 1;ο2 (w), Pre 1:ο1 (v)» Pre 1:ο1 (w), and Pre 1(v)» Pre 1 (w). Reachability objective. Given a ubet T S of target tate, the goal of a reachability game conit in reaching T. Therefore, we define the 4

6 et winning play a the et Reach(T ) = f! = h 0 ; 1 ; 2 ;:::i 2 Ω j k 2 T for ome k 0 g of play that reach T. For any T S, the et Reach(T ) i meaurable for any choice of trategie for the two-player [12]; we denote the probability that a path i in Reach(T ) tarting from tate 2 S, given trategie ß 1 and ß 2 for player 1 and 2, repectively, bypr ß 1;ß2 (Reach(T )). Given a tate 2 S and a reachability objective, Reach(T ), we define the value of the game at for player 1 a hh1ii val (Reach(T ))() = up inf ß12Π1 ß22Π2 Pr ß 1;ß2 (Reach(T )): The quantitative determinacy reult of [10] enure that up inf ß12Π1 ß22Π2 Pr ß 1;ß2 (Reach(T )) + up inf ß22Π2 ß12Π1 Pr ß 1;ß2 (Ω n Reach(T )) = 1: A trategy ß 1 for player 1 i optimal if for all 2 S we have inf ß22Π2 Pr ß 1;ß2 (Reach(T )) = hh1ii val (Reach(T ))(): For ">0, a trategy ß 1 for player1i"-optimal if for all 2 S we have inf Pr ß 1;ß2 (Reach(T )) hh1ii val (Reach(T ))() ": ß22Π2 3 Exitence of Memoryle "-Optimal Strategie 3.1 Markov deciion procee In our proof, we need ome fact about one-player verion of concurrent tochatic game, known a Markov deciion procee (MDP) [1]. For i 2 f 1; 2 g, an i-mdp i a concurrent game where, for all 2 S, we have j 3 i ()j =1. Given a concurrent game G, if we fix a memoryle trategy correponding to elector ο for player 1, the game i equivalent to a 2-MDP G ο with tranition function ffi ο (; b)(t) = X a2 1() ffi(; a; b)(t) ο()(a); for all 2 S and b 2 2 (). Similarly, if we fix elector ο 1 ;ο 2 for both player in a concurrent game G, we obtain a Markov chain, which we denote by G. ο 1;ο2 In an MDP, the et of tate that play an equivalent role to the cloed recurrent clae of Markov chain [9] are called end-component [2, 3]. 5

7 Definition 2 (End component) An end-component (EC) of a 2-MDP i a ubet C S uch that there i a elector for player 2 under which C form a cloed recurrent cla of the reulting Markov chain. It i not difficult to ee that an equivalent characterization of an endcomponent C i the following. For each 2 C, there i a ubet of move M() 2 () uch that: 1. when a move inm() ichoen at, all the tate that can be reached with non-zero probability are in C; 2. the graph (C; E), where E conit of the tranition that occur with non-zero probability when move in M( ) are taken, i trongly connected. Given a path!, denote by Infi(!) the et of tate that occur infinitely often along!. Given a et F 2 S of ubet of tate we denote by Infi(F) the event f! j Infi(!) 2Fg. The following theorem tate that in a 2-MDP, for any trategy of player 2, the et of tate viited infinitely often i an EC with probability 1. Corollary 1 follow eaily from Theorem 1. Theorem 1 ([3]) Let C be the et of end-component of a 2-MDP G ο 1. For all trategie ß 2 2 Π 2 and all tate 2 S, wehave Pr ο 1 ;ß 2 (Infi(C)) = 1. Corollary S 1 Let C be the et of end-component of a 2-MDP G ο 1 and let Z = C2C C be the et of tate of all end-component. For all trategie ß 2 2 Π 2 and all tate 2 S, wehave Pr ο 1 ;ß 2 (Reach(Z)) = From value iteration to elector Conider a reachability game with target T S. Let W 2 = f 2 S j hh1ii val (Reach(T ))() =0g be the et of tate from which player 1 cannot reach the goal with poitive probability; from [4, 6] we know that player 2 ha a trategy that confine the game in W 2. An arbitrary trategy for player1i"-optimal for a tate 2 W 2 [T ; hence, without lo of generality we aume that every tate 2 W 2 [ T i an aborbing tate. Our firt tep toward the proof of memoryle "-optimal trategie for reachability game conit in conidering a value-iteration cheme for the computation of hh1ii val (Reach(T )). Let [T ] : S! [0; 1] be the indicator function of T, defined by [T ]() = 1 for 2 T, and [T ]() =0for62 T. We then define: u 0 =[T ] 8k 0: u k+1 = Pre 1 (u k ) (1) 6

8 Note that the claical equation aign u k+1 = [T ] _ Pre 1 (u k ), where _ i interpreted a the maximum in pointwie fahion. Since we aume that tate in T are aborbing, the claical equation reduce to the impler equation given by (1). From the monotonicity ofpre 1 it follow that u k» u k+1, that i, Pre 1 (u k ) u k, for all k 0. The reult of [6] etablihe by a combinatorial argument that hh1ii val (Reach(T )) = lim k!1 u k, where the limit i interpreted in pointwie fahion. A witne for an "-optimal trategy i contructed by letting k be a elector uch that Pre 1 (u k )=Pre 1: k (u k ), for all k 0, and by conidering the trategy ff k for player 1 coniting in the equence of elector k ; k 1 ;:::; 1 ; 0 ; 0 ; 0 ;:::, where the lat elector, 0, i repeated forever. It i then poible to prove by induction on k that inf Pr ff k;ß2 (9j 2 [0::k]: j 2 T ) u k : ß22Π2 A the trategie ff k, for k 0, are not necearily memoryle, thi proof doe not uffice for howing the exitence of memoryle "-optimal trategie. On the other hand, the following example how that a memoryle trategy k doe not necearily guarantee the value u k. Example 1 Conider the 1-MDP hown in Fig 1. At all tate except tate 3, the et of available move for player 1 i ingleton, and at 3 the available move for player 1 i a and b. The tranition at variou tate i hown in the Fig 1. The objective of player 1 i to reach the tate 0, i.e., Reach(f 0 g). Given the MDP we conider the value-iteration procedure and denote by u k the valuation after k-iteration. We have u 0 = (1; 0; 0; 0; 0) and hence we have u 1 = Pre 1 (1; 0; 0; 0; 0) = (1; 0; 1 = 2 ; 0; 0). Similarly iterating the Pre 1 operator we get u 2 = (1; 0; 1 = 2 ; 1 = 2 ; 0) and u 3 = (1; 0; 1 = 2 ; 1 = 2 ; 1 = 2 ). Thi i a fix-point and we have u 4 = u 3. Now conider the elector k for player 1 that chooe at tate 3 the action a with probability 1. The elector k i optimal w.r.t. to the valuation u 3. However if player 1 play the memoryle trategy k, then the game viit 3 and 4 alternately and reache 0 with probability 0. Any memoryle trategy 0 k for player 1 that play action b at tate 3 with poitive probability enure that the et f 0 ; 1 g of tate i reached with probability 1, and 0 i reached with probability 1 = 2 ; and hence i an optimal trategy. In the example, the problem i that the trategy k may caue player 1 to tay forever in S n (T [ W 2 ) with poitive probability. The following lemma how that, in the cae where the trategy k guarantee reaching T [ W 2 with probability 1, then k alo guarantee the value u k. 7

9 1 =2 1 4 a 3 b 2 1 =2 0 Figure 1: A MDP with reachability objective. Lemma 1 Let v be a valuation uch that Pre 1 (v) v and v() = 0 for all 2 W 2. Let ο 1 be a elector for player 1 uch that Pre 1:ο 1 (v) = Pre 1 (v). For all player 2 trategie ß 2, if Pr ο 1 ;ß 2 (Reach(T [ W 2 )) = 1, then Pr ο 1 ;ß 2 (Reach(T )) v. Proof. Conider an arbitrary ß 2 2 Π 2, and for k 0 let v k =E ο 1 ;ß 2 v( k ) be the expected value of v after k tep under ο 1 and ß 2. By induction on k, we can prove v k v for all k 0: in fact, v 0 = v, and for k 0wehave v k+1 Pre 1:ο 1 (v k) Pre 1:ο 1 (v) = Pre 1(v) v. For all k 0 and 2 S, we can write v k a: v k () =E ο 1 ;ß 2 +E ο 1 ;ß 2 +E ο 1 ;ß 2 v( k ) j k 2 T Pr ο 1 ;ß 2 k 2 T v( k ) j k 2 S n (T [ W 2 ) Pr ο 1 ;ß 2 v( k ) j k 2 W 2 Pr ο 1 ;ß2 k 2 W 2 : k 2 S n (T [ W 2 ) Since v()» 1 when 2 T, the firt term on the right hand ide i at k 2 T. For the econd term, we have lim k!1 Pr ο 1 ;ß 2 k 2 mot Pr ο 1 ;ß 2 S n(t [W 2 ) =0byhypothei, ince Pr ο 1 ;ß 2 (Reach(T [W 2 )) = 1 and every tate 2 T [ W 2 i aborbing. Finally, the third term on the right hand ide i 0, a v() = 0 for all 2 W 2. Hence, taking the limit with k! 1, we obtain Pr ο 1 ;ß 2 Reach(T ) = lim k!1 Prο 1 ;ß 2 k 2 T lim k!1 v k v; where the lat inequality follow from v k v for all k From value iteration to optimal elector Since hh1ii val (Reach(T )) = lim k!1 u k, for every ">0, there exit k, uch that for all tate, we have u k () u k 1 () hh1ii val (Reach(T ))() 8

10 ". Lemma 1 indicate that, in order to contruct a memoryle "-optimal trategy, we need to contruct from u k 1 a elector ο 1 with the following propertie: 1. Pre 1:ο 1 (u k 1) =Pre 1 (u k 1 )=u k ; 2. For all ß 2 2 Π 2,wehave Pr ο 1 ;ß 2 (Reach(T [ W 2 )) = 1. The firt of the above condition i eaily met: it tate imply that ο 1 i an optimal elector for Pre 1 (u k 1 ). To meet the econd condition, however, not every optimal elector uffice, a hown by Example 1. To contruct a uitable elector, we need ome definition. For r > 0, the value cla Ur k = f 2 S j u k () =rg, conit of the tate with value r under the valuation u k. Similarly we define U ḳ /r = f 2 S j u k ()./ rg, for./2 f<;»; ;>g. For a tate 2 S, let `k() = minf j» k j u j () =u k () g be the entry time of in U k u k, i.e., the leat iteration j in which the tate () ha the ame value a in iteration k. For k 0, we define the elector k a follow: Λ k () = `k() = arg up inf Pre (u`k() 1) ο 1;ο2 : ο12λ1 ο22λ2 In word, the elector k () i an optimal elector for at the iteration `k(). It follow eaily that u k = Pre 1: k (u k 1 ). We denote by! k the memoryle player-1 trategy that alway follow k. Once we fix the elector k, the game i equivalent to a 2-MDP G k, and we can analyze it behavior with the help of Corollary 1; the goal i to prove the econd condition, i.e., that for all ß 2 2 Π 2 we have Pr ;ß k 2 (Reach(T [ W 2 )) = 1. To reaon about the end-component of G k, for a tate 2 S, and a player-2 action b 2 2 (), we denote by Det k (; b) =f Det(; a; b) j a 2 1 () ^ k (a) 0 g the et of poible ucceor of tate when player 1 play according to k, and player 2 play according to b. Lemma 2 For all k 0, conider a tate 2 S n (T [ W 2 ), and let 2 U k r, for 0 <r<1. For all move b 2 2 (), we have: 1. either Det k (; b) U k >r 6= ;, 2. or Det k (; b) U k r, and there i t 2 Det k(; b) with `k(t) <`k(). 9

11 Proof. For convenience, let m = `k(), and conider any b 2 2 (). ffl Conider firt the cae in which Det k (; b) 6 U k r. Then, it cannot be Det k (; b) U k»r : otherwie, for all tate t 2 Det k(; b) we would have u k (t)» r, and there would be at leat one t 2 Det k (; b) uch that u k (t) <r, contradicting u k () =r and Pre 1: k (u k 1 )=u k. So, it mut be Det k (; b) U k >r 6= ;. ffl Conider now the cae in which Det k (; b) U k r. Since u m» u k, due to the monotonicity of the Pre 1 operator and (1), we have that u m 1 (t)» r for all t 2 Det k (; b). From r = u k () = u m () = Pre 1: k (u m 1 ), we have that u m 1 (t) = r for all t 2 Det k (; b), implying that `k(t) <mfor all t 2 Det k (; b). The above lemma tate that under k, from each tate i 2 Ur k we are guaranteed a probability bounded away from 0 of either moving to a highervalue cla U>r, k or of moving to tate within the value cla that have a trictly lower entry time. Thi implie that every tate in S n W 2 ha a probability bounded above zero of reaching T in at mot n = jsj tep, o that the probability of taying forever in S n (T [ W 2 ) i 0. To prove thi fact formally, we analyze the end component of G k in light of Lemma 2. Lemma 3 For k 0, if for all 2 S n W 2 we have u k 1 () > 0, then for all ß 2 2 Π 2, we have Pr k;ß2 Reach(T [ W 2 )) = 1. Proof. Since every tate 2 T [ W 2 i aborbing, to prove thi reult, in view of Corollary 1, it uffice to how that there i no end component of G k entirely contained in S n (T [ W 2 ). Toward the contradiction, aume there i uch an end component C (S nt [W 2 ); then, we have C U k [r1;r2] with C U 6= ;, for ome 0 r <r 2 1» r 2» 1, where U k = U k U k [r1;r2] r1»r2 i the union of the value clae for value in the interval [r 1 ;r 2 ]. Conider a tate 2 U k with minimal `k, i.e., uch that `k()» `k(t) for all other r2 t 2 U k. From Lemma 2, we are guaranteed that for any b 2 r2 2(), there i t 2 Det k (; b) uch that (i) either t 2 U k and `k(t) <`k(), (ii) or t 2 U k. r2 >r2 In both cae, we reach a contradiction. The above lemma how that k atifie both the requirement for optimal elector pelt out at the beginning of Section 3.3: hence, k guarantee value u k. Thi prove the exitence of memoryle "-optimal trategie for concurrent reachability game. Theorem 2 (Exitence of memoryle "-optimal trategie) For every " > 0, memoryle "-optimal trategie exit for all concurrent game with reachability objective. 10

12 Proof. Conider a reachability game with target T S. Since lim k!1 u k = hh1ii val (Reach(T )), for every " > 0 we can find k 2 N uch that max 2S hh1iival (Reach(T ))() u k 1 () < ". By contruction, Pre 1: k (u k 1 ) = Pre 1 (u k 1 ) = u k. Hence, from Lemma 1 and 3, for all ß 2 2 Π 2 wehave Pr k ;ß 2 (Reach(T )) u k 1, leading to the reult. Reference [1] D.P. Berteka. Dynamic Programming and Optimal Control. Athena Scientific, Volume I and II. [2] C. Courcoubeti and M. Yannakaki. The complexity of probabilitic verification. Journal of the ACM, 42(4): , [3] L. de Alfaro. Formal Verification of Probabilitic Sytem. PhD thei, Stanford Univerity, Technical Report STAN-CS-TR [4] L. de Alfaro and T.A. Henzinger. Concurrent omega-regular game. In Proc. 15th IEEE Symp. Logic in Comp. Sci., page , [5] L. de Alfaro, T.A. Henzinger, and O. Kupferman. Concurrent reachability game. In Proc. 39th IEEE Symp. Found. of Comp. Sci., page IEEE Computer Society Pre, [6] L. de Alfaro and R. Majumdar. Quantitative olution of omega-regular game. Journal of Computer and Sytem Science, 68: , A preliminary verion appeared in STOC 01: 33rd Annual ACM Sympoium on Theory of Computing, [7] H. Everett. Recurive game. In Contribution to the Theory of Game III, volume 39 of Annal of Mathematical Studie, page 47 78, [8] J. Filar and K. Vrieze. Competitive Markov Deciion Procee. Springer-Verlag, [9] J.G. Kemeny, J.L. Snell, and A.W. Knapp. Denumerable Markov Chain. D. Van Notrand Company, [10] D.A. Martin. The determinacy of Blackwell game. The Journal of Symbolic Logic, 63(4): , [11] L.S. Shapley. Stochatic game. Proc. Nat. Acad. Sci. USA, 39: ,

13 [12] M.Y. Vardi. Automatic verification of probabilitic concurrent finitetate ytem. In Proceeding of the 26th Annual Sympoium on Foundation of Computer Science, page IEEE Computer Society Pre,

Lecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004

Lecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004 18.997 Topic in Combinatorial Optimization April 29th, 2004 Lecture 21 Lecturer: Michel X. Goeman Scribe: Mohammad Mahdian 1 The Lovaz plitting-off lemma Lovaz plitting-off lemma tate the following. Theorem