Problem Set 1, MIEG (Part II), Winter Term, 2015
1. For the following game tree find all pure strategy Nash equilibrium and SPNE.
2. For the following game tree find all, pure and mixed strategy, SPNE.
3. Alpha(A) and Beta(B) have been facebook friends for a long time. They decide to go on
a date. They have two options: quick lunch at Pizza Hut, or movie at PVR. A first chooses
where to go. B doesn’t automatically know where A went, but she can learn without any
cost. She can see A0 s FB update to know where he went or she can ignore it consciously
(i.e. without knowing where A went, B first chooses between Learn (without any cost) and
Not-Learn); if she chooses Learn, then she knows where A went and then decides where to
go; otherwise she chooses where to go without learning where A went. The payoffs depend
only on where each player goes. A prefers PVR, and B prefers Pizza Hut. A player gets 3
out of his/her preferred date option, 1 out of his/her less preferred date option, and 0 if they
end up at different places. All these are common knowledge. Find a SPNE of this game.
Compare it with the case when B doesn’t have option of not knowing about A0 s action i.e.
she knows for sure where A went. Does it seem (counter) intuitive.
4. Iraq and Kuwait have common oil field. In each year, simultaneously, each of these countries decide whether to extract high(H) or low(L) amount of oil from this field. Extracting
1
high amount of oil from the common field hurts the other country. In addition, Iraq has the
option of attacking Kuwait (W), which is costly for both countries. The stage game is as
follows (Iraq is row player and Kuwait being column player):
H
L
H 2,2
4,1
L 1,4
3,3
W -1,-1 -1,-2
Consider the infinitely repeated game with this stage game and with different discounting
factors.
(a) Find a SPNE in which each country extracts low (L) amount of oil every year on the
equilibrium path.
(b) Find a SPNE in which Iraq extracts high (H) amount of oil and Kuwait extracts low (L)
amount of oil every year on the equilibrium path.
5. Check whether the Tit-for-Tat strategy played by both players forms a SPNE or not in
the following payoff tables. If its not a SPNE suggest a modification in the strategy.
C
D
C
D
C
2,2
3,0
C
3,3
4,0
D
0,3
1,1
D
0,4
1,1
Note: Tit-for-Tat strategy means:
s1i = C
C
t
si (ht ) =
D
if at−1
=C
j
t−1
if aj = D
Where sti (ht ) means what action to play at time t if the history at the beginning of time t is
ht .
6. This problem deals with the game depicted below, where θ is some parameter.
X
Y
A
6, 2θ + 2
0,0
B
C
0,0 0,−θ
2,θ 9,−θ
(i)If θ = −1, find all of the SPNE of the 8-times repeated version of this game.
(ii) Now suppose that θ = 2 and consider a T-times repeated version of this game. For each
of the following, state whether the proposed strategies could be played in the first round
of a SPNE or not. If they can, describe a full set of SPNE strategies that implement it.
2
(a) T=2, and (Y,C), (b) T=2 and (X,B), (b) T=3 and (X,C)
7. Consider following 3 × 3 stage game G
L M R
U 5,6 2,2 2,3
M 6,3 3,4 0,3
D 2,1 1,0 0,1
In the following restrict attention to only pure strategies.
(i) What are the SPNE of G(T ) when T < ∞
(ii) What are the set of feasible payoff vectors and the payoff vectors that can be obtained in
SPNE of G(∞) by applying Folk Theorem with Nash threats. Nash threat means if players
deviate from the equilibrium action then the deviating players are punished by playing a
(bad) Nash equilibrium.
(iii) Give a SPNE strategy profile that yields the average payoff vector (5, 6). What is the
minimum δ for these strategies to be SPNE
(iv) Does the Folk theorem with Nash threats give an SPNE of G(∞) with payoffs (6, 3). If
so indicate the minimum δ for which it to be a SPNE.
(V) Try above part with (0, 3) instead of (6, 3).
8. Two strategies σi and σˆi are equivalent if they lead to the same probability distribution
over outcomes (or terminal nodes) for all σ−i . Give an example by constructing a game for
each case below.
(i) Some mixed strategy is not equivalent to any behavioral strategy.
(ii) Some behavioral strategy is not equivalent to any mixed strategy.
(iii) Every behavioral strategy is equivalent to some mixed strategy and vice-versa.
9. For the following game tree find all mixed strategy and behavioral strategy Nash equilibrium.
10. For the game tree below find the Nash equilibrium in mixed and behavioral strategies.
(Note: Both nodes a and b are in same information set).
3
11. Alpha(A) and Beta(B) are a couple, playing the infinitely repeated game with the following stage game and discount factor δ. Every day, simultaneously, Alpha and Beta spend
xA ∈ [0, 1] and xB ∈ [0, 1] fraction of their time in their relationship, respectively, receiving
the stage payoffs UA = ln(xA + xB ) + 1 − xA and UB = ln(xA + xB ) + 1 − xB respectively.
(a) Find a subgame-perfect equilibrium along with the conditions on the parameters, which
involves both players to spend all of their time in their relationship (i.e. xA = xB = 1).
(b) Find the conditions on the parameters (δ, xˆ, x˜A , x˜B ) such that the following strategy is
subgame perfect nash equilibrium. There are 4 states: E(Engagement), M(Marriage), DA
and DB . The game starts at state E, in which each player spends xˆ ∈ [0, 1] i.e. xA = xB = xˆ.
If both spend xˆ, they switch to state M, they remain in state E otherwise. In state M, each
spends 1. They remain in state M until one of the player i ∈ {A, B} spends less than 1 while
the other player spends 1, in which case they switch to Di state. In Di state, player i spends
x˜i and the other player spends 1 − x˜i forever.
12. (A game with both long- and short-lived players) Consider an infinite horizon extensive
game in which the strategic game G is played between player 1 and an infinite sequence of
players, each of whom lives for only one period and is informed of the actions taken in every
previous period. Player 1 evaluates sequences of payoffs by the limit of means, and each of
the other players is interested only in the payoff that he gets in the single period in which
he lives.
(i) Find the set of subgame perfect equilibria of the game when G is the second payoff table
in question 5.
(ii) Show that when G is the modification of the game used above, in which the payoff to
player 2 for the action profile (C, D) is 0, then for every rational number x ∈ [1, 3] there is
a subgame perfect equilibrium in which player 1’s average payoff is x.
13. Consider the three-player symmetric infinitely repeated game in which each player’s
preferences are represented by the discounting of all future payoffs. The stage/one shot
simultaneous move game is where for i = 1, 2, 3 we have Ai = [0, 1] and ui (a1 , a2 , a3 ) =
a1 a2 a3 + (1 − a1 )(1 − a2 )(1 − a3 ) ∀(a1 , a2 , a3 ) ∈ A1 × A2 × A3 .
(i) Find the set of enforceable payoffs.
(ii) Show that for any discount factor δ ∈ (0, 1) the payoff of any player in any subgame
1
perfect equilibrium of the repeated game is at least .
4
4