Assignment: Games and Economic Behavior
1. Consider the following game in normal form, where Player 1 chooses rows and Player 2 chooses columns (in each cell, Player 1's payoff is listed first and Player 2's second). Find all Pure-Strategy Nash Equilibria for this game.
(A)
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Player 2
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Left
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Center
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Right
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Player 1
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Up
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(0, -1)
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(2, -3)
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(1, 1)
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Middle
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(2, 4)
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(-1, 1)
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(2, 2)
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Down
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(1, 2)
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(0, 2)
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(1, 4)
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(B)
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Player 2
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a
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b
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c
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d
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e
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Player 1
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A
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(63, -1)
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(28, -1)
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(-2, 0)
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(-2, 45)
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(-3, 19)
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B
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(32, 1)
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(2, 2)
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(2, 5)
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(33, 0)
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(2, 3)
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C
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(54, 1)
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(95, -1)
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(0, 2)
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(4, -1)
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(0, 4)
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D
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(1, -33)
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(-3, 43)
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(-1, 39)
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(1, -12)
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(-1, 17)
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E
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(-22, 0)
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(1, -13)
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(-1, 88)
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(-2, -57)
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(-2, 72)
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2. Find all the Pure-Strategy Nash Equilibria in the game Rock, Paper, Scissors.
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Player 2
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Rock
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Paper
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Scissors
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Player 1
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Rock
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(0, 0)
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(-1, 1)
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(1, -1)
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Paper
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(1, -1)
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(0, 0)
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(-1, 1)
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Scissors
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(-1, 1)
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(1, -1)
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(0, 0)
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3. You and a friend are in an Italian restaurant, and the owner offers both of you an 8-slice pizza for free under the following condition. Each of you must simultaneously announce how many slices you would like; that is, each player i ∈ {1, 2} names his desired amount of pizza, 0 ≤ si ≤ 8. If s1 + s2 ≤ 8, then the players get their demands (and the owner eats any leftover slices). If s1 + s2 > 8, then the players get nothing. Assume that you each care only about how much pizza you individually consume, and the more the better.
(A) Write out or graph each player's best response correspondence.
(B) What outcomes can be supported as pure-strategy Nash equilibria?
4. Two division managers can invest time and effort in creating a better working relation- ship. Each invests ei ≥ 0, and if both invest more then both are better off, but it is costly for each manager to invest. In particular, the payoff function for player i from effort levels (ei , ej ) is
νi (ei , ej ) = (a + ej ) × ei - ei2.
(A) What is the best response correspondence of each player?
(B) Find the Pure-Strategy Nash Equilibrium of this game and argue that it is unique.
5. Cnsumers are uniformly distributed along a boardwalk that is 1 mile long. Ice-cream prices are regulated, so consumers go to the nearest vendor because they dislike walking (assume that at the regulated prices all consumers will purchase an ice cream even if they have to walk a full mile). If more than one vendor is at the same location, they split the business evenly. Consider a game in which two ice-cream vendors pick their locations simultaneously. Show that there exists a unique Pure-Strategy Nash Equilibrium and that it involves both vendors locating at the midpoint of the boardwalk.