UNIT 11 TWO-PERSON ZERO-SUM GAMES WITH SADDLE POINT

Transcription

1 UNIT TWO-PERSON ZERO-SUM GAMES WITH SADDLE POINT Structure. Introducton Obectves. Key Terms Used n Game Theory.3 The Maxmn-Mnmax Prncple.4 Summary.5 Solutons/Answers. INTRODUCTION In Game Theory, the word game s not used n the way t s commonly used n dfferent types of sports such as hockey, crcket, football, etc. Also, t does not refer to computer games. In the usual sports or games, the man obectve of the opponents s to wn the game. ut n the games dscussed under game theory between two opposng partes wth conflctng nterests, wnnng means selectng an optmal strategy, e.g., selectng an optmal course of acton as we have dscussed n Unts 9 and 0. Game theory deals wth decson makng processes of players n conflctng and compettve stuatons where the strategy (or acton or move) of a player depends upon the move of the opponent. Recall that n Sec. 9.5 of Unt 9, we have dscussed four types of envronments under whch a decson maker may have to make decsons, namely: Decson makng under certanty; Decson makng under uncertanty; Decson makng under rsk; and Decson makng under conflct. We have dscussed the frst three types of envronments n detal n Unts 9 and 0. We now dscuss the fourth type of envronment whch s present n games between two or more players. In Sec.., we ntroduce the key terms used n game theory. Saddle pont (explaned n Sec..3) s one of the key concepts n game theory. On the bass of whether a saddle pont exsts n the game or not, games can be further classfed as: Games wth saddle pont, and Games wthout saddle pont. The games wth saddle pont are dscussed n the present unt and the games wthout saddle pont are dscussed n Unt. In Sec..3, we dscuss Maxmn-mnmax prncple for solvng two-person zero-sum games wth saddle pont. In the next unt, we shall dscuss the games wthout saddle pont. Obectves After studyng ths unt, you should be able to: defne the key terms nvolved n game theory; solve the prsoner s dlemma game; and solve two-person zero-sum games usng the maxmn-mnmax prncple. The noton of game theory exsted even before John Von Neumann ( ). ut t s John Von Neumann who s known as the father of game theory. He frst publshed a paper on a mathematcal treatment of game theory n 98. In 944 he publshed a book n collaboraton wth Oskar Morgenstern on game theory enttled Theory of Games and Economc ehavour.

2 . KEY TERMS USED IN GAME THEORY In ths secton, we ntroduce the key terms and termnology used commonly n game theory. Then we explan what s meant by a game n game theory. ut before dong so, let us consder the followng stuaton: Suppose two chldren X and Y agree that: Each one of them wll smultaneously place a con on the table. Each one of them wll show the outcome (head or tal). If the faces of both cons match (.e., ether both cons show head or both show tal), chld X wns and gets Rs from chld Y. If the cons do not match, chld Y wns and gets Rs from chld X. We can present ths nformaton n the form of a matrx as shown below. The frst numeral n the four cells havng entres (, ), (, ), (, ), (, ), respectvely, represents the amount won by chld X and the second numeral represents the amount won by chld Y. When chld X wns Rs from chld Y, the wnnng amount for X s and the wnnng amount for Y s (.e., loss of ). Chld X (Player I) Chld Y (Player II) Head Tal Head,, Tal,, In game theory, the stuaton dscussed above s known as Con Matchng Game and ts soluton s provded n E n Sec..8 of Unt. Let us consder another stuaton. Two persons, say X and Y, are arrested by the polce wth enough evdence for a mnor crme. The polce suspect that they are responsble for a murder, but do not have enough evdence. oth persons are put n separate cells so that they have no way of communcatng wth each other. The polce starts nterrogatng them n separate rooms (as shown n Fg..). Each person ether confesses or does not confess. Also, each one knows the consequences of confesson, whch are gven below: If both of them confess, both go to al for 5 years. If one of them confesses and the other does not, then the one who confessed turns government s wtness whle the other who dd not confess goes to al for 0 years. If both do not confess, both go to al for one year. Assume that each one of them has to protect hs self-nterest, whch means that each person tres to act n such a way that he would have to go to al for a shorter perod of tme, regardless of the way the other acts. Also assume that they have no way of communcatng wth each other. What should X and Y do? We can present ths nformaton n matrx form as gven below. The frst numeral n the four cells havng entres ( 5, 5),(0, 0),( 0, 0),(, ) represents the tme to be spent n prson by person X and the second numeral represents the tme to be spent n prson by person Y. A negatve sgn s attached to the numerals because the tme spent n prson s smlar to a loss. Person X (Player I) Person Y (Player II) Confesses Does not Confess Confesses 5, 5 0, 0 Does not Confess 0, 0,

3 In game theory, the stuaton dscussed above s known as the Prsoner s Dlemma. Its soluton s provded n Example 4 n Sec..3. Fg..: Persons X and Y facng questons of polcemen n separate rooms. We now defne some key terms used n game theory. Player: A partcpant or compettor who makes decsons s known as a player. A player may be an ndvdual or a group of ndvduals. For example, n con matchng game, chld X and chld Y are players, and n the Prsoner s Dlemma game, person X and person Y are players. Acton: The optons avalable to the players are known as actons or courses of acton or moves. For example, n con matchng game, the actons are head and tal ; and n the prsoner s dlemma game, confesses and does not confess are actons. Play: A play s sad to occur when each player selects a course of acton. Strategy: A predetermned rule by whch a player decdes hs/her course(s) of acton among the actons avalable to hm/her s known as strategy for the player. A strategy may be of two types: () () Pure Strategy: A strategy s sad to be a pure strategy f the player selects a partcular course of acton each tme,.e., f a player selects, say, the th course of acton ( A ) each tme from among n courses of actons, n A, A,..., A, avalable to hm or her. Ths means that he/she assgns probablty to the th course of acton and zero probablty to each of the other courses of acton A, A,...,A, A,..., An. So the pure strategy s denoted by (0, 0,..., 0,, 0,...,0) havng values 0 at (n ) places except the th poston havng value. For example, n the prsoner s dlemma game, person X wll use pure strategy (, 0) as you wll see n the soluton of ths game, whch s provded n Example 4 n Sec..3. Mxed Strategy: A strategy for a player s sad to be a mxed strategy f the player selects a combnaton of more than one courses of acton by assgnng a fxed probablty to each course of acton. That s, f there are n courses of acton A,( n) avalable to the player, he/she assgns probabltes p,p,..., p n to the courses of acton A,A,...,A n, respectvely, such that p p...p n and p 0 for all, n. For example, n the con matchng game, chld X wll use the mxed strategy (/, /) as you wll see n the soluton of E, whch s provded n Sec..8 of Unt. Note : Pure strategy s a partcular case of mxed strategy because n the case of pure strategy p for some, and p 0 for all Payoff Values and Payoff Matrx: You have learnt about payoff values and payoff matrces n Unt 9. Here the payoff value means money or anythng else

4 that motvates players and when these payoff values are represented n matrx form, the resultng matrx s known as payoff matrx. For example, n the con matchng game, payoff values are the money that chld X gets from chld Y or vce versa, whle n prsoner s dlemma game, the tme (n years) to be spent n prson represents payoff values. Optmal Strategy: The strategy for a player, whch optmses hs/her payoff rrespectve of the strategy of hs /her compettor s known as an optmal strategy. For example, n the con matchng game, the optmal strategy for chld X s (/, /), whch s explaned n the soluton of E n Sec..8 of Unt. Now, we are n a poston to defne what we mean by a game n game theory. Defnton of Game: A compettve stuaton s called a game f () () The number of compettors, called players, s fnte. The number of possble courses of acton for each player s fnte. However, the courses of acton need not be the same for each player. () Each player selects a course of acton smultaneously from among the courses of acton avalable to hm/her wthout drectly communcatng wth the other player. (v) Every combnaton of courses of acton results n an outcome known as payoff value, whch motvates the players. Payoff value may represent loss or gan or any other thng of nterest. The payoff values may be postve, negatve or zero. n-person Game: If the number of players nvolved n the game s n( ), t s known as n-person game. Two-Person Game: If the number of players nvolved n the game s, t s known as two-person game. Zero-Sum Game: If the algebrac sum of the payoff values of all players after each play s zero, the game s known as a zero-sum game. Mathematcally, for zero-sum game, f a, n represents the payoff value of the th player, then n a 0 Non-Zero-Sum Game: A game s sad to be non-zero-sum game f there exsts at least one play such that the algebrac sum of all the payoff values s not equal to zero. For example, the prsoner s dlemma game s a non-zero-sum game. Two-Person Zero-Sum Game: If n a game, each payoff value of one player s negatve of the payoff value of the other player, t s known as two-person zero-sum game. For example, con matchng game s a two-person zero-sum game. Note : In two-person zero-sum game, f we call the two players as player I and player II, we see that: each payoff value of player II s equal n magntude to the payoff value of player I but opposte n sgn. Hence, f the payoff value of player I s known, the payoff value of player II s automatcally known. So, for a two-person zero-sum game, generally, we wrte the payoff values of only player I nstead of wrtng payoff values of both players. For example, payoff matrx n the case of con matchng game can smply be wrtten as follows:

5 Chld X (Player I) Chld Y (Player II) Head Tal Head Tal Note 3: In ths unt and n Unt we shall lmt our dscusson to two-person zero-sum games. Only n Example 4, we provde the soluton of prsoner s dlemma game, whch s a non-zero sum game. So henceforth, n the payoff matrx we shall wrte only payoff values of player I as explaned n Note..3 THE MAXIMIN-MINIMAX PRINCIPLE In Sec. 9.5 of Unt 9, you have studed the maxmn-mnmax crteron for a decson makng stuaton. The basc dea of the crteron remans the same except that here we are workng under the envronment of conflct whereas n Unt 9 we solved problems of decson makng under uncertanty. The second dstngushng feature s that here two players are makng decsons smultaneously, whle n Unt 9, there was only one decson maker. Wth ths clarfcaton, let us defne the prncple: Under ths prncple, frst of all, a player lsts the worst possble payoff values of all strateges avalable to hm/her. Then he/she selects the strategy correspondng to the optmum payoff value from among the worst possble payoff values. Let us consder a two-person zero-sum game to explan ths prncple. The payoff matrx for the game s gven below: Player A Player I II I 4 II 3 III 3 5 If a represents the payoff value when player A chooses hs/her th,( I,II,III) strategy and player chooses hs/her th, ( I,II) strategy, then a Player A wll get Rs from player a 4 Player A wll get Rs 4 from player a Player A wll get Rs from player a 3 Player A wll get Rs 3 from player a3 3 Player A wll pay Rs 3 to player a3 5 Player A wll get Rs 5 from player Let us frst analyse ths game from the pont of vew of player A: Here payoff values n the payoff matrx represent gans of the player A. So when player A chooses a partcular strategy, player would move n such a way that payoff to player A s mnmum for that partcular strategy because the nterests of player A and player are conflctng. Thus, f player A employs strategy I, he/she may gan Rs or Rs 4 dependng upon the strategy adopted by player. Now, whatever strategy s adopted by player, player A wll gan at least mn{, 4} Rs. Smlarly, f player A employs strategy II, he/she wll gan at least mn{, 3} Rs, and f player A employs strategy III, then he/she wll gan at least mn{ 3, 5} Rs3. Obvously,player A would lke to opt for the strategy, whch maxmses hs/her mnmum gans. Snce max{,, 3}, player A should adopt strategy II. You may thnk that f player A adopts strategy III

6 and player adopts strategy II, the gan of player A wll be Rs 5, whch s greater than Rs. You are rght but there s no guarantee that wll adopt strategy II. He/she may employ strategy I, whch wll result n a loss of Rs 3 to player A. So, player A should go for strategy II because, n game theory, we assume that both players are equally ntellgent. Now, we analyse ths game from the pont of vew of player : If player employs strategy I, then he/she may face a loss of Rs, a loss of Rs or a gan of Rs 3 dependng upon the strategy adopted by player A. Now, whatever strategy s adopted by player A, player cannot ncur a loss of more than max{,, 3} Rs. Smlarly, f player employs strategy II, he/she cannot ncur a loss of more than max{4, 3, 5} Rs5. Obvously, player would lke to opt for the strategy, whch mnmses hs/her maxmum losses. Snce mn{, 5} Rs, player should adopt strategy I. From the above dscusson, we note that player A should opt for the strategy whch corresponds to the maxmum payoff value among the row mnmum values,.e., maxmum among the mnma. Hence, t s known as maxmn value and s denoted by max mn{a }. Player should opt for the strategy, whch corresponds to the mnmum payoff value from among the column maxmum values,.e., mnmum among the maxma. Hence, t s known as mnmax value and s denoted by mn max{a }. If maxmn value mnmax value v(say), then v s known as the value of the game and the correspondng strateges of players A and are known as ther optmal strateges. Also, the poston of the element correspondng to the optmal strateges of the two players s known as saddle pont. In the above example, the value of the game s Rs and strategy II of player A s the optmal strategy for player A whle strategy I of player s the optmal strategy for player. Also the poston of the saddle pont s (,),.e., correspondng to second row and frst column. You should follow the steps gven below for applyng maxmn-mnmax prncple to numercal problems: Step : Identfy the mnmum element n each row of the payoff matrx and select the largest element among these row mnma. Ths s the maxmn value. Step : Identfy the maxmum element n each column of the payoff matrx and select the mnmum among these column maxma. Ths s the mnmax value. Step 3: If maxmn value = mnmax value vsay.e., max mn{a } mn max{a } v(say) and les at the ntersecton of the row of maxmn value and the column of the mnmax value, we say that the game s solved by maxmn-mnmax prncple. The maxmn (mnmax) value v s called the value of the game. The strateges correspondng to the row of maxmn value and the column of mnmax value are termed the pure optmal strateges for player A and player, respectvely. Also, the poston of the element where the row of maxmn value and the column of the mnmax value ntersect s known as the saddle pont. Note 4: If maxmn value mnmax value, we say that the game does not have a saddle pont. So we cannot obtan the soluton of the game n

7 Player A terms of pure strateges by applyng the maxmn-mnmax prncple. Games wthout a saddle pont have mxed strateges as ther solutons, whch are dscussed n Unt. Note 5: y soluton of a game, we mean an optmal strategy for each player and the value of the game. We now defne a few more terms before takng up examples. Lower and Upper Values of the Game: If a denotes the payoff value when player I chooses hs/her th, (,,..., m) strategy and player II chooses hs/her th, (,,..., n) strategy, then the maxmn value max mn{a }s known as the lower value of the game and s generally denoted by v. The mnmax value = mn max{a }s known as the upper value of the game and s generally denoted by v. Strctly Determnable Game: A game s sad to be strctly determnable f v vv, where v max mn{a }, v mn max{a }, v value the game Far Game: A game s sad to be far f v v v 0. Let us now apply the maxmn-mnmax prncple to a few games. Example : You are gven a game havng the followng payoff matrx: Player A Obtan the Player 3 A A 4 0 A () Optmal strategy for player A, () Optmal strategy for player, () Value of the game, and (v) Saddle pont. Also answer the questons: Is the game strctly determnable? Is the game far? Soluton: We apply the maxmn-mnmax prncple to the game. The followng three steps are nvolved n applyng ths prncple: Step : We dentfy the mnmum element of each row and then select the maxmum among these mnmum elements. Ths s called the maxmn value as ndcated n Table.. Step : Then we dentfy the maxmum element of each column and select the mnmum among these maxmum elements. It s called the mnmax value as ndcated n Table.. Step Table.: Calculaton of Maxmn-Mnmax Prncple Player Step 3 Row Mnma Maxmn Value A max 4, 0, 4 A A Column Maxma mn 6,7,4 4 Mnmax Value

8 Step 3: We check whether the maxmn and mnmax values are equal or not. In ths case, maxmn value = mnmax value = 4. So, the game s solved usng the maxmn-mnmax prncple and we obtan the followng results: ) Optmal strategy for player A s A as t corresponds to the maxmn value, ) Optmal strategy for player s 3 as t corresponds to the mnmax value, ) The value of the game s 4[ maxmn value mnmax value 4] Now, v thelower valueof thegame maxmn value 4 v the upper valueof thegame mnmax value 4 v the valueof thegame 4 v v( 4) v, thegame s strctly determnable. ut v 4 0. So thegames not far. Note 6: In the above example, player A always has to employ strategy A to maxmse hs/her mnmum gans. So he should assocate probablty to A and a zero probablty to each of A and A 3. Hence, the optmal strategy of player A can also be wrtten as a pure strategy (,0,0). Smlarly, optmal strategy of player can also be wrtten as pure strategy (0,0,). Example : Fnd the maxmn and mnmax values for the game havng the payoff matrx gven below. Does the game have a saddle pont? If the game has a saddle pont, solve t. Player A Soluton: Calculatons for the maxmn and the mnmax values are shown n Table. gven below: Table.: Calculaton of Maxmn-Mnmax Values Player A Step Player 3 A 4 5 A A 3 4 Player Step 3 Row Mnma Maxmn Value A 4 5 A max, 3, 3 A 4 3 Column Maxma Mnmax Value In the above table, we see that mn 6,4,7 4 Maxmn value = 3 and mnmax value = 4. Snce maxmn value mnmax value, the game has no saddle pont. Hence, the soluton of the game cannot be obtaned by usng the maxmnmnmax prncple. Note 7: Solutons of the games wthout saddle ponts are dscussed n Unt.

9 Example 3: Determne the range of values of λ and µ that wll make the poston (, ) a saddle pont for the game havng the payoff matrx gven below: Player Player A 3 A 3 5 A 8 4 λ A 3 µ 9 Soluton: Snce t s gven that the poston (, ) s the saddle pont, the maxmn value = mnmax value 4. Now, maxmn value 4 4 s the mnmum element of the second row. 4 Also mnmax value 4 4 s the maxmun element of the second column. µ 4 Hence, the range of s 4 and the range ofµ s µ 4. Example 4: What are the optmal strateges for person X and person Y n the Prsoner s Dlemma game? Soluton: We have descrbed the Prsoner s Dlemma game n Sec.. and obtaned the payoff matrx for t. Let us rewrte the payoff matrx gven there. Person X (Player I) Person Y (player II) Confesses Does not Confess Confesses 5, 5 0, 0 Does not Confess 0, 0, It s not a zero sum game. Obvously, f players have the faclty to communcate wth each other, then not to confess s the optmal strategy for both players because n ths case both wll get prson for only one year. ut t s gven that they cannot communcate wth each other and both persons have to protect ther self-nterest. What s the optmal strategy for both players? Let us frst fnd the optmal strategy for player X. If Y confesses, then X wll go to al for 5 years f he confesses, and 0 years f he does not confess. So, f Y confesses, t s better for X to confess rather than to not confess. If Y does not confess, then X wll be free (0 year n al) f he confesses, and get year al term f he does not confess. Also f Y does not confess, t s better for X to confess rather than to not confess. Thus, for X t s better to confess rather than to not confess, rrespectve of whether Y confesses or not. Therefore, the optmal strategy for X s to confess. Let us now now fnd the optmal strategy for Y. If X confesses, then Y wll go to al for 5 years f he confesses, and 0 years f he does not confess. So, f X confesses, then t s better for Y to confess rather than to not confess. If X does not confess, then Y wll be free (0 year al) f he confesses, and get year al term f he does not confess. Also f X does not confess, t s better for Y to confess rather than to not confess. Hence, under the gven stuaton the optmal strategy for both X and Y s to confess. So, the optmal pure strategy for both persons s (, 0). You may lke to try the followng exercses to check your understandng of the key terms and concepts explaned n Secs.. and.3. In the exercses E) to E5), choose the correct opton. E) Game theory s the study of: (A) Computer games (C) Identfyng optmal strateges () Usual sports (D) Two-person zero-sum games

10 E) If v and v denote the lower and upper values of the game, then the game s sad to be far f (A) v v () v v 0 (C) v v 0 (D) v 0 v E3) A saddle pont exsts n a game when (A) Maxmax Value = Mnmn Value () Maxmn value = Mnmax Value (C) Maxmax value = Maxmn value (D) Maxmn value = Mnmn value E4) Three strateges are avalable to a player n a game between two players. There s a pure optmal strategy n the game for hm/her. Whch of the followng cannot be hs/her pure strategy? (A) (/3, /3, /3) () (, 0, 0) (C) (0,, 0) (D) (0, 0, ) E5) If a player A assocates probabltes p, p,..., pm wth m strateges avalable to hm, whch one of the followng s the case of pure strategy? (A) p p... pm m () p, p, p3 p 4... pm 0 (C) p p... p 0, p, p m m m (D) p for some and p 0for all E6) Solve the game for whch the payoff matrx s gven by Player A Player 3 A 4 3 A 3 A Let us now summarse the man ponts that we have covered n ths unt..4 SUMMARY ) Player: A partcpant or compettor who makes decsons s known as a player. A player may be an ndvdual or a group of ndvduals. For example, n con matchng game, chld X and chld Y are players, and n the Prsoner s Dlemma game, person X and person Y are players. ) Acton: The optons avalable to the players are known as actons or courses of acton or moves. 3) Play: A play s sad to occur when each player selects a course of acton. 4) Strategy: A predetermned rule by whch a player decdes hs/her course(s) of acton among the actons avalable to hm/her s known as strategy for the player. A strategy may be of two types: Pure Strategy: A strategy s sad to be a pure strategy f the player selects a partcular course of acton each tme. Mxed Strategy: A strategy for a player s sad to be a mxed strategy f the player selects a combnaton of more than one courses of acton by assgnng a fxed probablty to each course of acton.

11 5) Optmal Strategy: The strategy for a player whch optmses hs/her payoff rrespectve of the strategy of hs /her compettor s known as an optmal strategy. 6) Defnton of Game: A compettve stuaton s called a game f () () The number of compettors, called players, s fnte. The number of possble courses of acton for each player s fnte. However, the courses of acton need not be the same for each player. () Each player selects a course of acton smultaneously from among the courses of acton avalable to hm/her wthout drectly communcatng wth the other player. (v) Every combnaton of courses of acton results n an outcome known as payoff value, whch motvates the players. Payoff value may represent loss, gan or any other thng of nterest. The payoff values may be postve, negatve or zero. 7) n-person Game: If the number of players nvolved n the game s n( ), t s known as n-person game. 8) Two-Person Game: If the number of players nvolved n the game s, t s known as two-person game. 9) Zero-Sum Game: If the algebrac sum of the payoff values of all players after each play s zero, the game s known as a zero-sum game. 0) Non-Zero-Sum Game: A game s sad to be non-zero-sum game f there exsts at least one play such that the algebrac sum of all the payoff values s not equal to zero. The prsoner s dlemma game s a non-zerosum game. ) Two-Person Zero-Sum Game: If n a game, each payoff value of one player s negatve of the payoff value of the other player, t s known as two-person zero-sum game. The con matchng game s a two-person zero-sum game. ) Lower and Upper Values of the Game: If a denotes the payoff value when player I chooses hs/her th (,,..., m) strategy and player II th chooses hs/her (,,..., n) strategy, then the maxmn value max mn{a }s known as the lower value of the game and s generally denoted by v. The mnmax value = value of the game and s generally denoted by v. mn max{a }s known as the upper 3) If maxmn value mnmax value v(say), then v s known as the value of the game and the correspondng strateges of players A and are known as ther optmal strateges. Also, the poston of the element correspondng to the optmal strateges of the two players s known as saddle pont. 4) Strctly Determnable Game: A game s sad to be strctly determnable f v v v. 5) Far Game: A game s sad to be far f v v v 0.

12 Step Player A.5 SOLUTIONS/ANSWERS E) In game theory, we try to dentfy the optmal strategy for each player. So, part (C) s the correct opton. E) We know that a game s sad to be far f both lower value and upper value of the game are equal to zero. So, part (C) s the correct opton. E3) We know that saddle pont s the pont of ntersecton of the row of maxmn value and column of mnmax value. Hence the saddle pont exsts n a game f maxmn value = mnmax value. So, part () s the correct opton. E4) We know that among m strateges A, A,..., Am avalable to a player A, a strategy say A, ( m) s sad to be a pure optmal strategy f player A has to use A f he/she wants to be n a comfortable poston rrespectve of the strategy adopted by hs/her opponent. That s, for A to be a pure optmal strategy, we should have p and p 0 for all where p, p,..., p m are the probabltes assocated wth the strateges A, A,...,A m, respectvely. Hence, except for the part (A), all others are cases of pure strateges. So, the correct opton s (A). E5) As explaned n the soluton of E4, the correct opton s (D). E6) We solve the game by usng the maxmn-mnmax prncple. The followng three steps are nvolved n applyng ths prncple: Step : We dentfy the mnmum element of each row and then select the maxmum of these mnmum elements. It s the maxmn value as ndcated n Table.3. Step : We dentfy the maxmum element of each column and then select the mnmum of these maxmum elements. It s the mnmax value as ndcated n Table.3. Table.3: Calculaton of Maxmn-Mnmax Values Player Step 3 Row Mnma Maxmn Value A 4 3 A 3 3 A Column Maxma Mnmax Value 6 3 mn{, 6, 3} Step 3: We check whether the maxmn and mnmax values are equal or not. Here maxmn value = mnmax value =. So, the game s solved usng the maxmn-mnmax prncple, and the soluton s: Optmal strategy for player A s A or (,0,0). Optmal strategy for player s or (0,0,). The value of the game s. max{, 3, 0}