Exercise 1
Consider a game of rock-scissor-paper. There are two players, player 1 and player 2, and the winner receives $1 from the loser (if they draw, nothing happens). They play this game simultaneously.
1. What kind of game is this? Express this game in normal form.
2. What are the set of possible actions and strategies?
3. Find all Nash equilibria of this game.
Exercise 2
Consider a game represented by the following payoff matrix. Find all pure strategy Nash equilibria.
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L
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C
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R
|
|
U
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-10, 0
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-10, -1
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-10, -1
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|
M
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-8, -8
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0, -10
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2, -1
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D
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4, 5
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7, -10
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1, 1
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Exercise 3
Consider a game represented by the following payoff matrix. There are two firms competing in a duopoly market. Each should decide how much to produce simultane- ously and their profits are given in the matrix. Find all pure strategy Nash equilibria.
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25
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35
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50
|
100
|
|
25
|
125, 125
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100, 140
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63, 125
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-63, -250
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|
35
|
140, 100
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105, 105
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53, 75
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-123, -350
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|
50
|
125, 63
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75, 53
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0, 0
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-250, -500
|
|
100
|
-250, -63
|
-350, -130
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-500, -250
|
-900, -900
|
Exercise 4
Consider two players simultaneously deciding whether to contribute to a public good
- the good is said to be public because, if it is made available, a player who free-rides by paying nothing gets just as much pleasure from its enjoyment as a player who paid for it. If at least one player contributes to the construction of the public good, both players will enjoy a payoff of four from the public good. To ensure the public good is constructed, player 1 must pay c1 or player 2 must pay c2. Assume that c1 < 4 and c2 < 4. If neither contribute, the good is not constructed and neither player gets enjoyment from the project. If one or both players contribute, then the good is constructed and each player enjoys a payoff of four minus the contribution cost if that player has contributed.
1.Describe this game in normal form.
2.Find any pure-strategy Nash equilibria to the game.
3.Find the mixed strategy Nash equilibria - the probabilities you find will be functions of the cost parameters.
4.If c1 = c2 = 1, write out the mixed strategy Nash equilibria and find the proba- bility that the public good is provided.
5.If c1 = c2 = 3, write out the mixed strategy Nash equilibria and find the proba- bility that the public good is provided.