OPERATIONS RESEARCH Chapter 2 Game Theory Prof. Bbhas C. Gr Department of Mathematcs Jadavpur Unversty Kolkata, Inda Emal: bcgr.umath@gmal.com
1.0 Introducton Game theory was developed for decson makng under conflctng stuatons where there are one or more opponents (players). The games lke chess, poker, etc. have the characterstcs of competton and are played accordng to some defnte rules. Game theory provdes optmal solutons to such games, assumng that each of the players wants to maxmze hs proft or mnmze hs loss. Game theory has applcatons n a varety of areas ncludng busness and economcs. In 1994, the Nobel Prze for Economc Scences was won by John F. Nash, Jr., John C. Harsany, and Renhard Selton for ther analyss of equlbra n the theory of noncooperatve games. Later, n 2005, Robert J. Aumann and Thomas C. Schellng won the Nobel Prze for Economc Scences for enhancng our understandng of conflct and cooperaton through game theory analyss.
MODULE - 1: Basc Concept and Termnologes, Two-person Zero-sum Game, and Game wth Pure and Mxed Strateges In ths Module, we wll dscuss some basc termnologes used n Game Theory, twoperson zero-sum game and games wth pure and mxed strateges. 1.1 Basc Termnologes The followng termnologes are commonly used n Game theory. Player : Each partcpant (nterested party) of a game s called a player. Strategy : The strategy of a player s the predetermned rule by whch a player decdes hs course of acton from the lst of courses of acton durng the game. A strategy may be of two types: Pure strategy - It s a decson, n advance of all plays, always to choose a partcular course of acton. Mxed strategy - It s a decson, n advance of all plays, to choose a course of acton for each play n accordance wth some partcular probablty dstrbuton. Optmal strategy : The course of acton whch maxmzes the proft of a player or mnmzes hs loss s called an optmal strategy. Payoff : The outcome of playng a game s called payoff. Payoff matrx : When the players select ther partcular strateges, the payoffs (gans or losses) can be represented n the form of a matrx called the payoff matrx. 3
Saddle pont : A saddle pont s an element of the payoff matrx, whch s both the smallest element n ts row and the largest element n ts column. Furthermore, the saddle pont s also regarded as an equlbrum pont n the theory of games. Value of the game : It refers to the expected outcome per play when players follow ther optmal strategy. 1.2 Two-Person Zero-Sum Game A game wth only two players s called a two-person zero-sum game f the losses of one player are equvalent to the gans of the other so that the sum of ther net gans s zero. Ths game also known as rectangular game. In a two-person game, suppose that the player A has m actvtes and the player B has n actvtes. Then a payoff matrx can be formed by adoptng the followng rules: () Row desgnatons for each matrx are actvtes avalable to the player A. () Column desgnatons for each matrx are actvtes avalable to the player B. () Cell entry v s the payment to the player A n A s payoff matrx when A chooses the actvty and B chooses the actvty. (v) For a zero-sum game, the cell entry n the player B s payoff matrx wll be negatve correspondng to the cell entry v n the player A s payoff matrx so that the sum of payoff matrces for the players A and B s ultmately zero, see Tables 1.1 and 1.2. 1 2 n 1 2 n 1 v 11 v 12 v 1n 1 v 11 v 12 v 1n 2 v 21 v 22 v 2n..... 2 v 21 v 22 v 2n..... m v m1 v m2 v mn m v m1 v m2 v mn Table 1.1: s payoff matrx Table 1.2: s payoff matrx Consder a two-person con tossng game. Each player tosses an unbased con smultaneously. Each player selects ether a head H or a tal T. If the outcomes match (.e., (H, H) or (T, T)) then A wns Rs. 4 from B; otherwse, B wns Rs. 3 from A. s
payoff matrx s gven n Table 1.3. Ths game s a two-person zero-sum game, snce the wnnng of one player s taken as losses for the other. Each player has hs choce from amongst two pure strateges H and T. H T H 4-3 T -3 4 Table 1.3 1.3 Pure Strateges (Mnmax and Maxmn Crteron) The smplest type of game s one where the best strateges for both players are pure strateges. Ths s the case f and only f, the payoff matrx contans a saddle pont. Theorem 1.1: Let (v ) be the m n payoff matrx for a two-person zero-sum game. If v denotes the maxmn value and v the mnmax value of the game, then v v. That s, mn [max {v }] max [mn {v }]. Proof: We have max {v } v f or all = 1,2,...,n and mn {v } v f or all = 1,2,...,m Let the above maxmum and mnmum values be attaned at = 1 and = 1, respectvely,.e., Then, we must have max {v } = v 1 and mn {v } = v 1 v 1 v v 1 f or all = 1,2,...,n; f or all = 1,2,...,m. From ths, we get Therefore, mn mn v 1 v max [max {v }] max v 1 f or all = 1,2,...,n; = 1,2,...,m. [mn {v }].
Note: A game s sad to be far, f v = 0 = v and t s sad to be strctly determnable f v = v = v. Example 1.1: Consder a two-person zero-sum game matrx whch represents payoff to the player A, see Table 1.4. Fnd the optmal strategy, f any. I II III IV V I -2 0 0 5 3 II 4 2 1 3 2 III -4-3 0-2 6 IV 5 3-4 2-6 Table 1.4: Payoff matrx for Example 1.1 Soluton: We use the maxmn (mnmax) prncple to determne the optmal strategy. The player A wshes to obtan the largest possble v by choosng one of hs actvtes (I, II, III, IV), whle the player B s determned to make A s gan the mnmum possble by choce of actvtes from hs lst (I, II, III, IV, V). The player A s called the maxmzng player and B, the mnmzng player. If player A chooses the actvty I then t could I II III IV V Row mnmum I -2 0 0 5 3-2 II 4 2 1 3 2 1 Maxmn III -4-3 0-2 6-4 IV 5 3-4 2-6 -6 Column maxmum 5 3 1 5 6 Mnmax Table 1.5: s payoff matrx happen that the player B also chooses hs actvty I. In ths case, the player B can guarantee a gan of at least 2 to player A,.e., mn{ 2,0,0,5,3} = 2. Smlarly, for other choces of player A,.e., actvtes II, III and IV, B can force the player A to gan only 1, 4 and 6, respectvely, by proper choces from (II, III, IV).e., mn{4,2,1,3,2} = 1, mn{ 4, 3,0, 2,6} = 4 and mn{5,3, 4,2, 6} = 6. For player A, mnmum value n each row represents the least gan to hm f he chooses hs partcular strategy. These
are wrtten n Table 1.5 by row mnmum. wll select the strategy that maxmzes hs mnmum gans,.e., max{ 2,1, 4, 6} = 1.e., player A chooses the strategy II. Ths choce of player A s called the maxmn prncple, and the correspondng gan (here 1) s called the maxmn value of the game. In general, the player A should try to maxmze hs least gans or to fnd max mn v = v. For player B, on the other hand, lkes to mnmze hs losses. The maxmum value n each column represents the maxmum loss to hm f he chooses hs partcular strategy. These are wrtten n Table 1.5 by column maxmum. wll then select the strategy that mnmzes hs maxmum losses. Ths choce of player B s called the mnmax prncple, and the correspondng loss s the mnmax value of the game. In ths case, the value s also 1 and player B chooses the strategy III. In general, the player B should try to mnmze hs maxmum loss or to fnd mn max v = v. If the maxmn value equals the mnmax value then the game s sad to have a saddle pont (here (II, III) cell) and the correspondng strateges are called optmum strateges. The amount at the saddle pont s known as the value of the game. Example 1.2: Solve the game whose payoff matrx s gven below: I II III I -2 15-2 II -5-6 -4 III -5 20-8 Table 1.6: A s payoff matrx Soluton: We use the maxmn (mnmax) prncple to determne the optmal strategy. The game has two saddle ponts at postons (1, 1) and (1, 3). I II III Row mnmum I -2 15-2 -2 Maxmn II -5-6 -4-6 III -5 20-8 -8 Column maxmum -2 20 Mnmax -2 Mnmax
() The best strategy for player A s I. () The best strategy for player B s ether I or III. () The value of the game s 2 for player A and +2 for player B. 1.4 Mxed Strategy: Game wthout A Saddle Pont If maxmn value s not equal to mnmax value then the game s sad to have no saddle pont. In such a case, both the players must determne an optmal mxture of strateges to fnd an equlbrum pont. The optmal strategy mxture for each player may be determned by assgnng to each strategy ts probablty of beng chosen. The strateges so determned are called mxed strateges. The value of game obtaned by the use of mxed strateges represents the least payoff whch player A can expect to wn and the least payoff whch player B can expect to lose. The expected payoff to a player n a game wth payoff matrx [v ] m n can be defned as E(p,q) = m =1 =1 n p v q = pvq T where p = (p 1,p 2,...,p m ) and q = (q 1,q 2,...,q n ) denote probabltes or relatve frequency wth whch a strategy s chosen from the lst of strateges assocated wth m strateges of player A and n strateges of player B, respectvely. Obvously, p 0 ( = 1,2,...m), q 0( = 1,2,...,n) and p 1 + p 2 +... + p m = 1; q 1 + q 2 +... + q n = 1. Theorem 1.2: For any 2 2 two-person zero-sum game wthout any saddle pont havng B 1 B 2 A 1 v 11 v 12 the payoff matrx for player A gven by A 2 v 21 v, the optmal mxed strateges 22 A 1 A 2 S A = p 1 p and S B 1 B 2 B = 2 q 1 q are determned by p 1 p 2 = v 22 v 21 v 11 v 12, q 1 q 2 = v 22 v 12 v 11 v 21 where 2 p 1 + p 2 = 1 and q 1 + q 2 = 1. The value v of the game to A s gven by v = v 11v 22 v 21 v 12 v 11 +v 22 (v 12 +v 21 ). A 1 A 2 Proof: Let a mxed strategy for player A be gven by S A = p 1 p, where p 1+p 2 = 1. 2 Thus, f player B moves B 1 then the net expected gan of A wll be E 1 (p) = v 11 p 1 +v 21 p 2 and f B moves B 2, the net expected gan of A wll be E 2 (p) = v 12 p 1 + v 22 p 2. B 1 B 2 Smlarly, f B plays hs mxed strategy S B = q 1 q, where q 1 + q 2 = 1, then B s 2 net expected loss wll be E 1 (q) = v 11 q 1 +v 12 q 2 f A plays A 1, and E 2 (q) = v 21 q 1 +v 22 q 2 f A
plays A 2. The expected gan of player A, when B chooses hs moves wth probabltes q 1 and q 2, s gven by E(p,q) = q 1 [v 11 p 1 + v 21 p 2 ] + q 2 [v 12 p 1 + v 22 p 2 ]. would always try to mx hs moves wth such probabltes so as to maxmze hs expected gan. Now, E(p,q) = q 1 [v 11 p 1 + v 21 (1 p 1 )] + (1 q 1 )[v 12 p 1 + v 22 (1 p 1 )] = [v 11 + v 22 (v 12 + v 21 )]p 1 q 1 + (v 12 v 22 )p 1 + (v 21 v 22 )q 1 + v 22 ( = λ p 1 v )( 22 v 21 q λ 1 v ) 22 v 12 + v 11v 22 v 12 v 21 λ λ where λ = v 11 + v 22 (v 12 + v 21 ). We see that f A chooses p 1 = v 22 v 21 λ, he ensures an expected gan of at least (v 11 v 22 v 12 v 21 )/λ. Smlarly, f B chooses q 1 = v 22 v 12 λ, then he can lmt hs expected loss to at most (v 11 v 22 v 12 v 21 )/λ. These choces of p 1 and q 1 wll thus be optmal to the two players. Thus, we get p 1 = v 22 v 21 v = 22 v 21 λ v 11 + v 22 (v 12 + v 21 ) q 1 = v 22 v 12 v = 22 v 12 λ v 11 + v 22 (v 12 + v 21 ) v and v = 11 v 22 v 12 v 21 v 11 + v 22 (v 12 + v 21 ) Hence, we have v and p 2 = 1 p 1 = 11 v 12 v 11 + v 22 (v 12 + v 21 ) v and q 2 = 1 q 1 = 11 v 21 v 11 + v 22 (v 12 + v 21 ) p 1 p 2 = v 22 v 21 v 11 v 12, q 1 = v 22 v 12 v ; and v = 11 v 22 v 21 v 12 q 2 v 11 v 21 v 11 + v 22 (v 12 + v 21 ). (1.1) Note: The above formula for p 1, p 2, q 1, q 2 and v are vald only for 2 2 games wthout saddle pont. Example 1.3: Suppose that n a game of matchng cons wth two players, one player wns Rs. 2 when there are 2 heads, and gets nothng when there are 2 tals and looses Re 1 when there are one head and one tal. Determne the best strateges for each player and the value of the game. Soluton: The payoff matrx for player A s gven n Table 1.7. The game has no saddle pont. Let the player A plays H wth probablty x and T wth probablty 1 x. Then, f the player B plays H, then A s expected gan s E(A,H) = x(2) + (1 x)( 1) = 3x 1. If the player B plays T, A s expected gan s E(A,T ) = x( 1) + (1 x).0 = x.
H T H 2-1 T -1 0 Table 1.7: s payoff matrx If the player A chooses x such that E(A,H) = E(A,T ) = E(A) say, then ths wll determne best strategy for hm. Thus we have 3x 1 = x or x = 1/4. Therefore, the best strategy for the player A s to play H and T wth probablty 1/4 and 3/4, respectvely. Therefore, the expected gan for player A s E(A) = 1 4 (2) + 3 4 ( 1) = 1 4. The same procedure can be appled for player B. If the probablty of B s choce of H s y and that of T s 1 y then for the best strategy of the player B, E(B,H) = E(B,T ) whch gves y = 1/4. Therefore, 1 y = 3/4. Thus A s optmal strategy s (1/4, 3/4) and B s optmal strategy s (1/4, 3/4). The expected value of the game s 1/4 to the player A. Ths result can also be obtaned drectly by usng the formulae (1.1).