C31: Game Theory. A¢ liate Exam, December 2006
Answer any three questions.
1. Players 1 has an action set X = [0; 1]; while player 2 has action set
Y = [0; 1]: The payo¤ to player 1 as a function of x 2 X and y 2 Y is given by
U (x; y) = 1 + x x2 0:5y:
The payo¤ to player 2 as a function of x and y is given by
V (x; y) = (x y)2 :
In (a) and (b) below, 1 must choose an action x 2 X and 2 must choose
y 2 Y:
a) Solve for a Nash equilibrium in the game where both players choose actions
simultaneously. (10 marks)
b) Consider the extensive form game where player 1 chooses his action …rst.
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Player 2 observes player 1’ choice and chooses his own action. Solve for a
subgame perfect equilibrium of this game. (18 marks)
c) Explain why player 1 has an incentive to choose di¤erently in (b) as
compared to (a). (5 marks)
2. Consider the following game in strategic form depicted in Fig. 1.
L C R
T -1,-1 3,0 0,-2
M 0,3 1,1 1,0
B -1,2 0,0 -1,-1
Fig. 1
a) Eliminate the strictly dominated strategies for each player. After this
elimination, given that player 2 does not use a strictly dominated strategy, is
there any strategy which is strictly dominated for player 1? Similarly, after the
…rst elimination, given that player 1 does not use a strictly dominated strategy,
is there any strategy which is strictly dominated for player 2? (8 marks)
b) Find the pure strategy Nash equilibria of this game or show that a pure
strategy equilibrium does not exist. (8 marks)
c) Find a mixed strategy Nash equilibrium, i.e. where both players random-
ize across two or more pure strategies. (10 marks)
d) Provide one example of one economic or social phenomenon where the
notion of a mixed strategy equilibrium is useful for understanding the phenom-
enon, setting out brie‡y why this is the case. (7 marks)
3. Two bidders are bidding for a painting in an auction. Their valuations
are private information: each bidder knows his own valuation, but only knows
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that his opponent’ valuation is drawn from the uniform distribution on [0; 1]:
Each bidder has to submit a sealed bid. The auctioneer collects both their bids
and the object is allocated to the highest bidder. That is, the auction is an
all-pay auction.
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a) Suppose that bidder i believes that his opponent j’ bid is given by
bj (vj ) = kj (vj )2 ; (1)
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where vj is his opponent’ valuation and kj is a constant.
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Write down i’ expected payo¤ from bidding bi when his valuation is vi :(12
marks)
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b) Maximize the expression derived in (a) to …nd player i’ optimal bid as
a function of vi :(14 marks)
c) Use your results to verify that there is a symmetric Bayes Nash equilibrium
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of this auction of this form, where k1 = k2 = 2 : (7 marks)
4. Consider the following two person parlour game, which starts with n
counters on the table in front of the two players. Players 1 & 2 move sequentially,
player 1 moving …rst. When it is his turn to move, a player must remove
either 1 or 2 counters from the table. The game ends when all the counters are
removed, and the player who moves last wins the game.
a) Suppose that n = 15: Solve the game by backwards induction. (Hint: let
k be the number of counters left on the table. Focus on the winning positions
– the value of k where a player wins – and losing positions for a player. Start
with small values of k; i..e k = 1 & 2 ) (22 marks)
b) How would you generalize your answer in (a), so that for any n; you can
determine which player wins the game in the backwards induction solution? (11
marks).
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