MATH4321_hw1.pdf - MATH 4321 Game Theory Homework One...

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MATH 4321 – Game Theory Homework One Course Instructor: Prof. Y.K. Kwok 1. In a Nim game that starts with 4 pennies, each player may take 1 or 2 pennies from the pile. Suppose player I movers first. The game ends when there are no pennies left and the player who took the last penny pays 1 to the other player. (a) Draw the game as we did in 2 × 2 Nim. (b) Write down all the strategies for each player and then the game matrix. (c) Find the upper and lower values of the game, v + , v - . Would you rather be player I or player II? 2. Each of two players must choose a number between 1 and 5. If a player’s choice equals op- posing player’s choice + 1, she loses $2; if a player’s choice opposing player’s choice + 2, she wins $1. If both players choose the same number, the game is a draw. (a) What is the game matrix? (b) Find v + and v - and determine whether a saddle point exists in pure strategies. If so, find it. 3. In the Russian roulette, suppose that if player I spins and survives and player II decides to pass, then the net gain to I is $1000. In this case, I gets all of the additional money that II had to put into the pot in order to pass. Draw the game tree and find the game matrix. What are the upper and lower values of the game? Find the saddle point in pure strategies. 4. In a football game, the offense has two strategies: run or pass. The defense also has two strategies: defend against the run, or defend against the pass. A possible game matrix is A = ( 3 6 x 0 ) . This is the game matrix with the offense as the row player I. The numbers represent the number of yards gained on each play. The first row is run, the second is pass. The first column is defend the run and the second column is defend the pass. Assuming that x > 0, find the value of x so that this game has a saddle point in pure strategies. 5. Let f ( x, y ) = ( x - y ) 2 , the two intervals for x and y are C = D = [ - 1 , 1], respectively. Find v + = min y D max x C f ( x, y ) and v - = max x C min y D f ( x, y ). 6. Nash and iterated dominance (a) Show that every iterated dominance equilibrium s * is a Nash equilibrium. 1 This study resource was shared via CourseHero.com
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(b) Show by counterexample that not every Nash equilibrium can be generated by iter- ated dominance.
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