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# TOP IN CONTEMP MATH II MATH 166

Texas A&M

GPA 3.6

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This 4 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 166 at Texas A&M University taught by Heather Ramsey in Fall. Since its upload, it has received 36 views. For similar materials see /class/226001/math-166-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 155 Spring 2007 Heather Ramsey 1 Math 166 Section 95 Lecture Notes 8m ed by Tan Section 95 Games with Mixed Strategies Pure Strategy a gameplaying strategy in which a player plays the same move over and over again in repeated plays of the game In this section we will look at games that are not strictly determined and the strategies associated with such games Example A Roy and Collin play a game in which they simultaneously show either one or two ngers If both of them show 1 nger then Collin pays Roy 3 If they both show 2 ngers Collin must pay Roy 1 In the other two cases Roy pays Collin 2 a Give the payoff matrix associated with this twoperson zerosum game b Is this game strictly determined Question to consider for the game in Example A What strategy might Roy adopt for this game General question to consider for games that are not strictly determined How can a player choose his moves in a game with mixed strategies Mixed Strategy Vectors In a game with mixed strategies assume that the row player has m moves and the colunm player has n moves I o The entries of the row vectorP 71 p2 pm r t0 11 qz o The entr1es of the column vector Q In Math 155 Spring 2007 Heather Ramsey 2 Expected Value of a Gam the average payoff to the row player when both players adopt a particular set of mixed strategies Calculating the Expected Value of a Game Suppose the row player adopts the mixed strategy P pl 172 l and the column player adopts the mixed strategy 0111 0112 Q ql in a game that has a payoff matrixA 12 0121 0122 J What is the expected payoff to the row player Very Important Note Expected Value of a Game Example B Consider the game Roy and Collin played in Example A Find the expected value of the game if Roy decides to show 1 nger 55 of the time and Collin decides to use his minimax strategy 60 of the time Example C The payoff matrix for a certain game is given by A if g 7 J a Find the expected payoff to the row player if the row player uses the maximin pure strategy and the column player uses the minimax pure strategy h Find the expected payoff to the row player if the row player uses the maximin strategy 70 if the time and the column player uses the minimax strategy 60 of the time and the other two moves 20 of the time each Math 155 Spring 2007 Heather Ramsey 3 Question to Consider Of all the in nitely many different mixed strategies available to the row and column players is there a way to nd the optimal strategies for each Optimal Strategies for Nonstrictly Determined Games Let A be the payoff matrix for a notstrictly determined game Then the optimal mixed strategy for the row b d player is given by P pl 172 L where d 7 C d 17 adibicanl72 P1 P1 and the optimal mixed strategy for the column player is given by Q 1 J where 2 dib d 17 adpbpcan 12 11 11 Also the expected value of the game E PAQ can be computed as 7 adibc iad7bic Example D Consider the game played by Roy and Collin in Example A a Find the optimal mixed strategies for both Roy and Collin b Find the expected value of the game Does it favor one player over the other Math 155 Spring 2007 Heather Ramsey 4 Example E 18 onpg 553 ofTan As part of their investment strategy the Carringtons have decided to put 100000 into stock market investments and also into purchasing precious metals commodity investment The performance of the investments depends on the state of the economy in the next year In an expanding economy it is expected that their stock market investment will outperform their investment in precious metals whereas an economic recession will have precisely the opposite effect In an expanding economy the value of the stock market investment will increase by 20 and the value of the commodity investment will increase by 15 In economic recession the value of the stock market investment will decrease by 5 but the value of the commodity investment will increase by 15 a Determine the optimal investment strategy for the Carringtons investment of 100000 b What pro t can the Carringtons expect to make on their investments over the year if they use their optimal investment strategy

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