Extra Credit Homework Exercises

1. Consider the following two-player game.

   	B1	B2	B3
   A1	1,1	3,4	2,1
   A2	2,4	2,5	8,1
   A3	3,3	0,4	0,9
   A3	2,4	4,1	3,6

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
2. Consider the following zero-sum two-player game.

   	B1	B2	B3
   A1	3	2	4
   A2	-2	2	-1
   A3	0	1	3
   A3	-1	2	0

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
3. Consider the following two-player game.

   	B1	B2	B3
   A1	1,1	3,1	2,3
   A2	2,2	6,1	8,1
   A3	3,3	0,2	0,5
   A3	2,4	4,1	3,6

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
4. Consider the following zero-sum two-player game.

   	B1	B2	B3
   A1	-1	3	2
   A2	2	2	-1
   A3	3	4	1
   A3	-1	2	0

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
5. Consider the following two-player game.

   	B1	B2	B3
   A1	1,1	3,4	2,1
   A2	2,2	2,1	1,1
   A3	3,3	4,4	2,5
   A3	2,4	4,1	3,6

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
6. Consider the following zero-sum two-player game.

   	B1	B2	B3
   A1	3	1	4
   A2	2	-1	1
   A3	-1	1	2
   A3	3	0	1

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
7. Consider the following zero-sum two-player game.

   	B1	B2	B3	B4	B5	B6
   A1	1	0	-1	1	0	0
   A2	1	-1	1	0	1	0
   A3	-1	-1	0	-1	0	-1
   A4	1	-1	1	-1	1	1
   A5	-1	0	0	-1	1	0
   A6	1	1	0	0	0	1
   A7	0	0	-1	0	1	1

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
8. Consider the following zero-sum two-player game.

   	B1	B2	B3	B4	B5	B6
   A1	1	0	-1	1	0	1
   A2	1	-1	1	-1	1	0
   A3	0	-1	0	-1	1	-1
   A4	1	-1	1	-1	1	1
   A5	0	0	0	-1	1	1
   A6	1	-1	0	-1	0	1
   A7	0	0	-1	0	1	1

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
9. Consider the following zero-sum two-player game.

   	B1	B2	B3	B4	B5	B6
   A1	1	0	-1	0	-1	-1
   A2	0	1	2	-1	0	-2
   A3	1	0	0	0	0	0
   A4	2	2	1	1	1	-1
   A5	0	-1	0	0	2	-1
   A6	0	0	-1	-1	0	0
   A7	0	0	0	-2	1	1

   • Find the Maximin moves for both players.
   • Which moves are dominated?
   • Find the bimatrix obtained by IESD.
   • Find the bimatrix obtained by IEWD.
   • Are there any Nash equilibria?
10. Consider the following two-player game.

    	B1	B2	B3	B4	B5	B6
    A1	0	-1	0	-1	0	-1
    A2	0	0	1	-1	1	1
    A3	2	2	1	2	0	0
    A4	0	1	-1	-1	-1	1
    A5	0	0	1	0	1	0
    A6	0	2	1	-1	0	-2
    A7	0	0	0	0	1	-1

    • Find the Maximin moves for both players.
    • Which moves are dominated?
    • Find the bimatrix obtained by IESD.
    • Find the bimatrix obtained by IEWD.
    • Are there any Nash equilibria?
11. Consider the following two-player game.

    	B1	B2	B3	B4	B5	B6
    A1	1, 0	0, 0	1, 0	1, 0	1, 0	3, 1
    A2	0, 0	1, 3	0, 2	1, 1	1, 2	1, 0
    A3	0, 2	0, 2	0, 0	2, 0	0, 0	0, 1
    A4	0, 0	1, 0	1, 0	2, 2	2, 0	0, 0
    A5	1, 0	0, 2	0, 1	0, 3	1, 1	2, 0
    A6	0, 1	1, 0	2, 1	1, 0	1, 0	1, 0
    A7	0, 0	0, 0	0, 2	2, 0	0, 2	0, 2

    • Find the Maximin moves for both players.
    • Which moves are dominated?
    • Find the bimatrix obtained by IESD.
    • Find the bimatrix obtained by IEWD.
    • Are there any Nash equilibria?
12. Consider the following two-player game.

    	B1	B2	B3	B4	B5	B6	B7
    A1	0, 1	1, 0	0, 1	0, 1	0, 2	1, 2	2, 0
    A2	2, 0	0, 0	1, 1	1, 2	3, 0	2, 0	3, 1
    A3	2, 2	0, 1	0, 2	0, 0	1, 2	0, 0	0, 0
    A4	0, 1	0, 0	1, 0	2, 1	0, 0	3, 0	2, 0
    A5	2, 0	1, 0	0, 0	1, 0	1, 0	1, 3	2, 1
    A6	0, 2	0, 3	0, 2	0, 3	0, 2	0, 2	0, 1
    A7	0, 0	1, 0	0, 0	1, 0	1, 0	1, 0	1, 0

    • Find the Maximin moves for both players.
    • Which moves are dominated?
    • Find the bimatrix obtained by IESD.
    • Find the bimatrix obtained by IEWD.
    • Are there any Nash equilibria?
13. Consider the following two-player game.

    	B1	B2	B3	B4	B5	B6	B7
    A1	0, 1	1, 0	0, 1	0, 1	1, 2	1, 2	2, 0
    A2	2, 0	0, 2	1, 1	1, 2	3, 0	2, 0	3, 1
    A3	2, 2	0, 1	1, 2	0, 0	1, 2	0, 1	0, 0
    A4	2, 0	1, 0	0, 1	1, 0	1, 0	1, 3	2, 1
    A5	0, 2	0, 3	0, 2	0, 3	0, 2	0, 2	0, 1
    A6	0, 0	1, 0	2, 0	1, 0	1, 0	1, 0	1, 0

    • Find the Maximin moves for both players.
    • Which moves are dominated?
    • Find the bimatrix obtained by IESD.
    • Find the bimatrix obtained by IEWD.
    • Are there any Nash equilibria?
14. Consider the following two-player game.

    	B1	B2	B3	B4	B5	B6	B7
    A1	0, 1	1, 0	0, 1	0, 1	0, 1	1, 2	2, 1
    A3	2, 2	0, 1	0, 2	0, 0	1, 2	0, 0	0, 1
    A4	0, 1	0, 0	1, 0	2, 1	0, 0	2, 0	2, 1
    A5	1, 0	1, 0	0, 0	1, 0	1, 0	1, 2	2, 1
    A6	0, 2	0, 2	0, 2	0, 2	0, 2	0, 2	0, 1
    A7	0, 0	1, 0	0, 0	1, 0	1, 0	1, 0	1, 0

    • Find the Maximin moves for both players.
    • Which moves are dominated?
    • Find the bimatrix obtained by IESD.
    • Find the bimatrix obtained by IEWD.
    • Are there any Nash equilibria?
15. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
16. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
17. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
18. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
19. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
20. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
21. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
22. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
23. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
24. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
25. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
26. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
27. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
28. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
29. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
30. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
31. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
32. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
33. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
34. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
35. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
36. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
37. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
38. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
39. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
40. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
41. Analyze the game described here using backward induction. What are the expected payoffs for Ann and Beth in this game?
42. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
43. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
44. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
45. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
46. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
47. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
48. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
49. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
50. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
51. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
52. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
53. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
54. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
55. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
56. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
57. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
58. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
59. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
60. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
61. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
62. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
63. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
64. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
65. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
66. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
67. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
68. How many information set do Ann and Beth have in this game? How many pure strategies do Ann and Beth have? Describe one of these pure strategies for each of the players.
69. Create the normal form of the game whose extensive form is given here.
70. Create the normal form of the game whose extensive form is given here.
71. Create the normal form of the game whose extensive form is given here.
72. Create the normal form of the game whose extensive form is given here.
73. Create the normal form of the game whose extensive form is given here.
74. Create the normal form of the game whose extensive form is given here.
75. Create the normal form of the game whose extensive form is given here.
76. Create the normal form of the game whose extensive form is given here.
77. Create the normal form of the game whose extensive form is given here.
78. Create the normal form of the game whose extensive form is given here.
79. Create the normal form of the game whose extensive form is given here.
80. Find some mixed Nash equilibrium for the game with the following normal form:

    	B1	B2	B3
    A1	1,1	3,4	2,1
    A2	2,4	2,5	8,1
    A3	3,3	0,4	0,9
    A3	2,4	4,1	3,6

81. Find some mixed Nash equilibrium for the zero-sum game with the following normal form:

    	B1	B2	B3
    A1	-1	3	2
    A2	2	2	-1
    A3	3	4	1
    A3	-1	2	0

82. Find some mixed Nash equilibrium for the game with the following normal form:

    	B1	B2	B3
    A1	1,1	3,1	2,3
    A2	2,2	6,1	8,1
    A3	3,3	0,2	0,5
    A3	2,4	4,1	3,6

83. Find some mixed Nash equilibrium for the game with the following normal form:

    	B1	B2	B3
    A1	1,1	3,4	2,1
    A2	2,2	2,1	1,1
    A3	3,3	4,4	2,5
    A3	2,4	4,1	3,6

84. Find some mixed Nash equilibrium for the zero-sum game with the following normal form:

    	B1	B2	B3
    A1	3	2	4
    A2	-2	2	-1
    A3	0	1	3
    A3	-1	2	0

85. Find some mixed Nash equilibrium for the zero-sum game with the following normal form:

    	B1	B2	B3
    A1	3	1	4
    A2	2	-1	1
    A3	-1	1	2
    A3	3	0	1