3M03_All_Tests_Before_Spring_2008_Answers

3M03_All_Tests_Before_Spring_2008_Answers - Answers to...

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Answers to Practice Test 1 Econ 3M03 – Introduction to Game Theory Summer 2003 Vaticrat All references to 'Nash equilibria' should be taken to mean 'pure strategy Nash equilibria'. 1) I am using A to represent the strategy of Channel 1 showing Alf at 7 o'clock (and Beverly Hills at 7:30) and B to represent the strategy of showing Beverly Hills at 7 o'clock (and Alf at 7:30). Similarly for channel 2, C is the strategy of showing Cheers at 7 o'clock (and Diffrent Strokes at 7:30) and D is the strategy of showing Diffrent Strokes at 7 o'clock (and Cheers at 7:30). This is the resultant extensive form of the game: 1 2 AB CD C D ( ) ( )( ) ( ) 11.5 11 11 10.5 10.5 11 11 11.5 Channel 1's strategy set is { } SA B 1 = , and Channel 2's strategy set is {} SC D 2 = , . 2) Player 2 has 10 different decision nodes (one for each move by player 1) at which an action needs to be specified by any strategy. At each node, player 2 has 9 different actions available (one for each number from 1 to 10 that was not chosen by player 1). Thus at the first decision node, a strategy can specify 9 actions. For each of these, at the second node, there are nine more, leading to 9x9=81 different ways strategies can specify actions for the first two decision nodes. Then for each of these, there are nine ways to specify the action to be taken at the third decision node, leading to 9 3 =729 different ways strategies can specify actions for the first three decision nodes. Continuing this reasoning to accommodate all ten decision nodes, player 2 has 9 10 = 3 486 784 401 different possible strategies. An example of a strategy by player 2 is:
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s 1 = "If player 1 says 1, say 4. If player 1 says 2, say 1. If player 1 says 3, say 3. If player 1 says 4, say 9. If player 1 says 5, say 10. If player 1 says 6, say 10. If player 1 says 7, say 10. If player 1 says 8, say 1. If player 1 says 9, say 3. If player 1 says 10, say 1." 3a) BR 1 (Y)={A,B}, BR 1 (Z)={A}, BR 2 (A)={Z}, BR 2 (B)={Y,Z}. Set of Nash equilibria is {(A,Z),(B,Y)}. b) BR 1 (Y)={A},BR 1 (Z)={A},BR 2 (A)={Y},BR 2 (B)={Y,Z} Unique Nash equilibrium at (A,Y). c) BR 1 (X)={A}, BR 1 (Y)={B}, BR 1 (Z)={B}, BR 2 (A)={Y}, BR 2 (B)={X,Y,Z}, BR 2 (C)={X,Y}, BR 2 (D)={X,Y}. Set of Nash equilibria is {(B,Y),(B,Z)}. d) BR 1 (Y)={A}, BR 1 (Z)={B}, BR 2 (A)={Z}, BR 2 (B)={Y}, BR 2 (C)={Z}. There are no Nash equilibria in this game. 4) Player one would choose x 1 to maximize u 1 by setting u x xx 1 1 21 1 4 3 2 0 =+ = . This solves for 1 1 4 3 2 2 . Both players must pick actions between zero and one, and this value can be greater than 1 depending on what player 2 chooses. So player 1's best response is () {} BR x x 12 1 4 3 2 2 1 min , . Graphically, BR 1 is: Similarly, player two would choose x 2 to maximize u 2 by setting u x 2 2 1 2 2 3 0 =− + = . This solves for 2 3 2 1 3 4 . Again, this can take on values outside of player 2's range of action, depending on what player 1 chooses; it can be negative. So player 2's best response is ( ) { } BR x x 3 2 1 3 4 0 max , . Graphically:
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The Nash equilibria will be wherever the BR's intersect. Graphing them together we get: This suggests three intersections. One at ( ) ( ) xx 12 1 4 0 ,, = , another at () 3 4 1 = , and a third, interior, one where 1 1 4 3 2 2 =+ and 2 3 2 1 3 4 =− .
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This note was uploaded on 09/05/2008 for the course ECONOMICS ECON 3M03 taught by Professor Brucejames during the Fall '08 term at McMaster University.

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3M03_All_Tests_Before_Spring_2008_Answers - Answers to...

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