Consider the two-player game with the following matrix form representation ✗ y а 3,2 -1,0 b с -1,1 1,3 1,-1 4,2 where player 1 is the "row player," player 2 is the "column player," and, for every cell, the left-most number is the utility that player 1 obtains from the corresponding (pure) strategy profile and the right-most number is the utility that player 2 obtains from the corresponding (pure) strategy profile. (a) Find the pure-strategy Nash equilibria of this game. (b) Find the other Nash equilibria of this game. Now, consider the three-player game with the following matrix form representation L ✗ y M ✗ y R X y b с a 2,3,3 0,0,1 1,3,7 0,7,4 3,-1,-2 с 3,1,0 a b -1,0,1 0,-2,5 -7,2,-4 -2,4,10 3,6,7 -1,4,0 a 0,-1,1 2,-3,-5 b 8,4,-2 4,0,6 с 0,7,0 6,-2,-4 where player 1 is the "row player," player 2 is the "column player," player 3 is the "table" player, and, for every cell, the left-most number is the utility that player 1 obtains from the corresponding (pure) strategy profile, the middle number is the utility that player 2 obtains from the corresponding (pure) strategy profile, and the right-most number is the utility that player 3 obtains from the corresponding (pure) strategy profile. (c) Find the pure-strategy Nash equilibria of this game. (d) Find the remaining Nash equilibria of this game.
Consider the two-player game with the following matrix form representation ✗ y а 3,2 -1,0 b с -1,1 1,3 1,-1 4,2 where player 1 is the "row player," player 2 is the "column player," and, for every cell, the left-most number is the utility that player 1 obtains from the corresponding (pure) strategy profile and the right-most number is the utility that player 2 obtains from the corresponding (pure) strategy profile. (a) Find the pure-strategy Nash equilibria of this game. (b) Find the other Nash equilibria of this game. Now, consider the three-player game with the following matrix form representation L ✗ y M ✗ y R X y b с a 2,3,3 0,0,1 1,3,7 0,7,4 3,-1,-2 с 3,1,0 a b -1,0,1 0,-2,5 -7,2,-4 -2,4,10 3,6,7 -1,4,0 a 0,-1,1 2,-3,-5 b 8,4,-2 4,0,6 с 0,7,0 6,-2,-4 where player 1 is the "row player," player 2 is the "column player," player 3 is the "table" player, and, for every cell, the left-most number is the utility that player 1 obtains from the corresponding (pure) strategy profile, the middle number is the utility that player 2 obtains from the corresponding (pure) strategy profile, and the right-most number is the utility that player 3 obtains from the corresponding (pure) strategy profile. (c) Find the pure-strategy Nash equilibria of this game. (d) Find the remaining Nash equilibria of this game.
Chapter8: Game Theory
Section: Chapter Questions
Problem 8.7P
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Transcribed Image Text:1. Consider the two-player game with the following matrix form representation
✗
y
a
3,2 -1,0
b
1,3 1,-1
с
-1,1
4,2
where player 1 is the "row player," player 2 is the "column player,” and, for every cell, the left-most
number is the utility that player 1 obtains from the corresponding (pure) strategy profile and the
right-most number is the utility that player 2 obtains from the corresponding (pure) strategy profile.
(a) Find the pure-strategy Nash equilibria of this game.
(b) Find the other Nash equilibria of this game.
Now, consider the three-player game with the following matrix form representation
L
✗
y
M
✗
a
2,3,3
3,1,0
a
b
0,0,1
1,3,7
b
с
0,7,4
3,-1,-2
с
y
-1,0,1 0,-2,5
-7,2,-4 -2,4,10
3,6,7 -1,4,0
R
X
a
y
0,-1,1 2,-3,-5
8,4,-2 4,0,6
с
0,7,0 6,-2,-4
where player 1 is the "row player," player 2 is the "column player," player 3 is the "table” player, and,
for every cell, the left-most number is the utility that player 1 obtains from the corresponding (pure)
strategy profile, the middle number is the utility that player 2 obtains from the corresponding (pure)
strategy profile, and the right-most number is the utility that player 3 obtains from the corresponding
(pure) strategy profile.
(c) Find the pure-strategy Nash equilibria of this game.
(d) Find the remaining Nash equilibria of this game.
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