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Simon Fraser University
Department of Economics
Economics ECON - 383-3 Daniel Monte
Selected Topics in Game Theory Summer 2009
Problem Set 1: Solutions
1) Consider a simultaneous game between two players. The game is as follows. Each player has
to choose a number from the set of integers, Z = f0; 1; 2; 3; :::g :
a) If the rule is: the player with the lowest number gets a payo¤ of 1, whereas the player with
the highest number gets a payo¤ of zero. If they choose the same number, they each get a payo¤
of 1 . Find all Nash Equilibrium in pure strategies.
2
Answer:
Let si be the strategy of player i. If sj > 0, the best response for player i is si = sj 1. If
sj = 0, the best response for player i is si = 0.
Thus, the Nash Equilibrium is s1 = s2 = 0.
b) If the rule is: the player with the highest number gets a payo¤ of 1, whereas the player with
the lowest number gets a payo¤ of zero. If they choose the same number, they each get a payo¤ of
1
2 . Find all Nash Equilibrium in pure strategies.
Answer:
Given any sj 0, the response for player i is si = sj + b, for any b > 0. Thus, there is no Nash
Equilibrium in this game.
c) If the rule is: if the players choose the same number, they each get a payo¤ of 1, otherwise
they each get 0. Find all Nash Equilibrium in pure strategies.
Answer:
Given sj 0, the best response for player i is si = sj . Thus, any pair (s1 ; s2 ) in which s1 = s2
is a Nash Equilibrium.
2) Consider the following simultaneous game between 3 players: player 1, player 2 and player
3. Player 1 can choose between T and B, whereas player 2 can choose between L and R. Player 3
chooses between 1 and 2 :
player 2 player 2
L R L R
player 1 T 1; 2; 1 3; 1; 3 player 1 T 1; 2; 2 3; 5; 7
B 3; 4; 1 2; 4; 5 B 3; 1; 0 7; 2; 1
1 2
s s
Where the …rst payo¤ is player 1’ payo¤, the second is player 2’ and the third is player 3’s.
Find all Nash Equilibria in pure strategies.
Answer:
1
L R
In 1 : T 1; 2 3; 1 there is a unique outcome in which player 1 and 2 don’ want to t
B 3; 4 2; 4
deviate: B; L. In this outcome, the payo¤ for player 3 is 1, whereas if players 1 and 2 choose B; L
respectively, and player 3 chooses 2 , his payo¤ is 0. Thus, B; L; 1 is a Nash Equilibrium.
L R
Note that In 2 : T 1; 2 3; 5 there is a unique outcome in which player 1 and 2 don’ want t
B 3; 1 7; 2
to deviate: B; R. In this outcome, the payo¤ for player 3 is 1, whereas if players 1 and 2 choose
s
B; R respectively, and player 3 chooses 1 , his payo¤ is 5. Thus, player 3’ best response to B; R
is 1 and B; R; 2 is not a Nash Equilibrium.
3)
player 2
L R
player 1 T a,b 3,1
B 5,2 c,d
a) For what values of a; b; c and d is T a strictly dominated strategy?
Answer:
a < 5 and c > 3, any values of b; d:
b) For what values of a; b; c and d is R a strictly dominated strategy?
Answer:
b > 1 and d < 2, any values of a; c.
c) For what values of a; b; c and d is T,L the unique Nash Equilibrium?
Answer:
a > 5 and b > 1 together with c < 3 or d < 2.
Or a = 5, b > 1 and d > 2 and c < 3
Or b = 1, a > 5, and c > 3 and d < 2.
4) In the game below:
L M R
A 0; 3 3; 0 a; a
B 3; 0 0; 3 a; a
C a; a a; a b; b
a) For what values of a and b does the game above have the unique Nash Equilibrium in
which player 1 plays a mixed strategy and chooses A and B with positive probability (but C with
probability 0) and player 2 chooses L and M with positive probability (but R with probability 0)?
Compute this Nash Equilibrium.
Answer:
1:5 > a > b
b) Find values of a and b such that (C; R) is a Nash Equilibrium in the game and there is no
equilibrium as the one in (a) above?
2
Answer:
b > a > 1:5
5) Consider a …rst price sealed bid auction between two bidders. In this auction, each of the
two bidders submit a bid in a sealed envelope at the same time. The auctioneer opens the two
envelopes and the highest bid wins the object and pays its bid. Assume that the payo¤ of player
i; where i = 1; 2 are as follows: v bi if bi > bj , and v 2bi if bi = bj and 0 if bi < bj , where v is
a …nite positive number. Is there an equilibrium in pure strategies? If yes, …nd it, if not, show
formally. Assume that bids can be any non-negative number, but that the smallest unit is a penny,
i.e. bi 2 f0; 0:01; 0:02; :::g.
Answer:
If v 0:01 > bj 0, then player i’ best response is bi = bj + 0:01. If bj v 0:01, player i0 s
s
best response is bi = bj .
Thus, there are three Nash Equilibria:
bi = bj = v 0:01
bi = bj = v
Note that bi = bj = v 0:02 is also a Nash Equilibrium: ui = uj = (v (v 2 0:02)) = 0:01; a
deviation to b = v 0:01; wins the object, but get the same payo¤: ui = v (v 0:01) = 0:01.
6) Consider a simpli…ed version of the game above in question 5. The only di¤erence is that
players are only allowed to submit one of two possible bids: a high bid h or a low bid l. Assume:
v > h > l > 0 and v < 2h l: The payo¤ rule is the same as in 5. Find all Nash Equilibrium, in
pure and mixed strategies.
Answer:
player 2
h l
player 1 h v 2 h ; v 2 h v h; 0
v l v l
l 0; v h 2 ; 2
v h < v 2 l since 2v 2h < v l ) v < 2h l
Thus, there are two Nash Equilibria in pure strategies: (h; h) and (l; l). There is also one Nash
Equilibrium in mixed strategies:
U (h) = p v 2 h + (1 p) (v h) = 1 (pv ph + 2v 2h 2pv + 2ph) = 1 ( pv + ph + 2v 2h)
2 2
U (l) = (1 p) v 2 l
(1 p) v 2 l = 2 ( pv + ph + 2v 2h)
1
(1 p) (v l) = pv + ph + 2v 2h
v l pv + pl = pv + ph + 2v 2h
p (h l) = 2h l v
2h l v
p=
h l
2h l v
Each player plays h with probability p = h l .
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