Lecture 5: Dynamic games with complete information
II
1 Dynamic games (cont.)
• Last time:
� Dynamic games with multiple stages; only one player moves at each stage.
� There can be many Nash equilibria because players can design di�erent threats
to induce di�erent outcomes
� But many of those equilibria involve non-credible threats, making them unlikely
predictions.
� We should focus on equilibria without non-credible threats. These are called
subgame perfect equilibria. SPE requires that all players play best responses
even o� the equilibrium path, i.e., in scenarios that they didn’t encounter in
equilibrium.
� We �nd SPE by backward induction
• This lecture:
� Apply the same concept to more complicated games: at each stage, there may be
multiple players that move simultaneously
1.1 Bank Runs
1.1.1 Set up
• Two investors have each deposited D with a bank
• Investment takes two periods to mature
� If investment liquidated before maturity: a total of 2r can be recovered. D > r >
D/2
� If investment matures: pays a total of 2R, R > D
• Timeline of the game:
1
� Stage 1
∗
investor 2
withdraw don’t
investor 1
withdraw r, r D, 2r – D
don’t 2r – D, D next stage
� Stage 2
∗
investor 2
withdraw don’t
investor 1
withdraw R, R 2R – D, D
don’t D, 2R – D R, R
1.1.2 Find subgame perfect equilibrium by backward induction
• Stage 2: unique NE = (withdraw, withdraw) with payo� (R, R)
• Update stage 1 payo�:
�
investor 2
withdraw don’t
investor 1
withdraw r, r D, 2r – D
don’t 2r – D, D R, R
� Two pure NE: (withdraw, withdraw) and (don’t, don’t)
• Two pure-strategy subgame perfect outcomes:
1. Both withdraw in stage 1 and get payo�s (r, r) � �Bank run�
2. Both withdraw in stage 2 and get payo�s (R, R)
• Lesson: Bank runs can arise in equilibrium even without any negative shock to the
economy
1.2 Tari�s and International Competition
1.2.1 Set up
• Timeline:
1. Two countries; the governments simultaneously set up tari� rates (t1, t2)
2. One �rm in each country that produces goods for both home consumption (h) and
export (e). After observing (t1, t2), the �rms simultaneously choose (h1,e1) and (h2, e2).
• Payo�s:
� market-clearing price/ inverse demand curve in country i is Pi(hi, ej) = 100−hi−
ej
� total production cost for �rm i is Ci(hi, ei) = 10 (hi + ei)
2
� �rm i’s payo� is its pro�t
πi (ti, tj, hi, hj, ei, ej) = hi (100− hi − ej) + ei (100− hj − ei)− 10 (hi + ei)− tjei
� government i’s payo� is country i’s total welfare = consumer surplus + �rm pro�t
+ tari� revenue
Wi (ti, tj, hi, hj, ei, ej) =
1
2
(hi + ej)
2︸ ︷︷ ︸+πi (ti, tj, hi, hj, ei, ej) + tiej
∗ where 1
2
(hi + ej)
2
is the consumer surplus:
1.2.2 Find SPE by backward induction
• Stage 2: suppose that the governments have chosen (t1, t2)
� We can separate a �rm’s maximization problem into domestic + overseas
max
hi
πhomei (hi) = hi [100− hi − ej]− 10hi
90− 2hi − ej = 0
h∗i =
90− ej
2
�
max
ei
π
export
i (ei) = ei [100− hj − ei]− 10ei − tjei
e∗i =
90− tj − hj
2{
h1 =
90−e2
2
e2 =
90−t1−h1
2
⇒
{
h∗1 =
90+t1
3
e∗2 =
90−2t1
3
similarly,
{
h∗2 =
90+t2
3
e∗1 =
90−2t2
3
3
� Given that hi =
90+ti
3
and ei =
90−2tj
3
,
πhomei = hi [100− hi − ej]− 10hi
=
(
90 + ti
3
)(
100−
90 + ti
3
−
90− 2ti
3
)
− 10
(
90 + ti
3
)
=
(
90 + ti
3
)(
90−
90 + ti
3
−
90− 2ti
3
)
=
(
90 + ti
3
)(
90−
180− ti
3
)
=
(
90 + ti
3
)(
90 + ti
3
)
=
(90 + ti)
2
9
π
export
i = ei [100− hj − ei]− 10ei − tjei
=
(
90− 2tj
3
)(
100−
90 + tj
3
−
90− 2tj
3
)
− 10
(
90− 2tj
3
)
− tj
(
90− 2tj
3
)
=
(
90− 2tj
3
)(
90− tj −
90 + tj
3
−
90− 2tj
3
)
=
(
90− 2tj
3
)(
90− tj −
180− tj
3
)
=
(
90− 2tj
3
)(
90− 2tj
3
)
=
(90− 2tj)
2
9
• Stage 1: since the government can predict the impact of their tari�s, their maximization
problems are
max
ti
Wi(ti) = CSi + π
home
i + π
export
i + Tari� Revenuei, where
CSi =
(180− ti)
2
18
πhomei =
(90 + ti)
2
9
π
export
i =
(90− 2tj)
2
9
Tari� Revenuei = tiej =
ti (90− 2ti)
3
W
′
i (ti) =
− (180− t1)
9
+
2 (90 + t1)
9
+
90− 4t1
3
W
′
i (ti) = 0⇒ t
∗
i = 30
4
• In the unique subgame perfect equilibrium:
t∗1 = t
∗
2 = 30
h∗1 = h
∗
2 = 40, e
∗
1 = e
∗
2 = 10
2 Repeated games
2.1 Finitely repeated games
Example 1. Two-stage Prisoner’s dilemma
Prisoner 2
Confess Not Confess
Prisoner 1
Confess 1, 1 5, 0
Not Confess 0, 5 4, 4
• Play this game twice.
• Both prisoners observe the outcome of the �rst play before the second play begins.
• Payo� for the entire game = sum of the payo�s from the two stage games (no discount-
ing).
• Q: Since the prisoners will meet again in the future, will they cooperate in the �rst
period?
Solve the game using backward induction:
• Stage 2: unique NE = (confess, confess)
� Both prisoners know that, regardless of what they do in the �rst stage, both will
confess in the second stage and the payo� for the second stage will be (1,1)
• Therefore, the total payo� at the beginning of the game is
Prisoner 2
Confess in stage 1 Not Confess in stage 1
Prisoner 1
Confess in stage 1 1+1, 1+1 5+1, 0+1
Not Confess in stage 1 0+1, 5+1 4+1, 4+1
� Unique NE: (confess, confess)
• Unique SPE outcome: both prisoners confess in both stages and get total payo�s (2,
2)
5
• Having a future stage does not make the prisoners cooperate because any promise
such as �I’ll not confess next time if you don’t confess this time� is not credible. The
prisoners’ best responses in the second stage is irrelevant to their moves in the �rst
stage.
• Same conclusion even if the prisoners play this game for 1 million times:
� Using backward induction, both must confess in the last stage game
� Since both must confess in the last game, both confess in the second-last stage
game, and therefore both confess in the third-last game…
� Both confess in every stage game.
• This result generalizes to any �nitely repeated game.
De�nition 1. Given a stage game G, let G(T) denote the �nitely repeated game in
which G is played T times, with the outcomes of all preceding plays observed before the next
play begins. G is called the stage game of the repeated game. The payo�s for G(T) are
simply the sum of the payo�s from the T stage games.
Proposition 1. If the stage game G has a unique Nash equilibrium then, for any �nite T,
the repeated game G(T) has a unique subgame-perfect outcome: the Nash equilibrium of G
is played in every stage.
Remark 1. This result does not hold if the stage game has multiple NE. See the following
example.
Example 2. Finitely Repeated Game with multiple equilibria. Suppose we extended the
previous example:
Prisoner 2
Confess Not Confess Run
Prisoner 1
Confess 1, 1 5, 0 0,0
Not Confess 0, 5 4, 4 0,0
Run 0, 0 0, 0 3, 3
• A strategy pro�le must specify
1. Which pair of actions do the players choose in stage 1?
2. Following each of the 9 possible outcomes in stage 1, which pair of actions do the
players choose in stage 2?
• A subgame-perfect equilibrium strategy pro�le must satisfy the following conditions:
� The pair of actions chosen in stage 2 must be a Nash equilibrium of the stage
game
� For each of the 9 possible outcomes in stage 1, calculate players’ total payo�s in
the two stages based on the follow-up strategy speci�ed in [2]. Write down these
payo�s in a new game matrix. Then, the pair of actions chosen in stage 1 must
be a Nash equilibrium in this new game matrix.
6
• Observe: the stage game has two pure NE: (Confess, Confess) and (Run, Run)
• Let’s show that there exists a SPE in which both prisoners choose �not confess� in
stage 1.
� Suppose the prisoners anticipate that the second-stage outcome is (R, R) if both
chose �not confess� in the �rst stage and (confess, confess) otherwise.
� Then, the total payo� at the beginning of the game is
Prisoner 2
Confess Not Confess R
Prisoner 1
Confess 1+1, 1+1 5+1, 0+1 0+1,0+1
Not Confess 0+1, 5+1 4+3, 4+3 0+1,0+1
R 0+1, 0+1 0+1, 0+1 3+1, 3+1
⇒
Prisoner 2
Confess Not Confess R
Prisoner 1
Confess 2,2 6,1 1,1
Not Confess 1,6 7,7 1,1
R 1,1 1,1 4,4
� There are three pure NE in this new game: (C, C), (NC, NC), and (R, R).
� The strategy �play R in stage 2 unless both chose NC in stage 1� leads to three
SPE outcomes: ((C, C),(C, C)), ((NC, NC), (R, R)), and ((R, R), (C, C))
� Therefore, the following is a subgame-perfect equilibrium (but it is not the only
SPE):
Both players
Play NC in stage 1
Play R in stage 2 if stage 1 outcome is (NC, NC)
Play C in stage 2 if stage 1 outcome is not (NC, NC)
� The outcome of this SPE is ((NC, NC), (R, R)).
� The outcome in stage 1 is not a NE of the stage game!
Remark 2. If the stage game G has multiple Nash equilibria, then there can be subgame-
perfect outcomes of the repeated game G(T) in which, for any t < T , the outcome in stage
t is not a Nash equilibrium of G.
We will extend this intuition to in�nitely repeated games next week.
7
3 Practice problems
1. Three oligopolists operate in a market with inverse demand given by P (Q) = a − Q,
where Q = q1 + q2 + q3 and qi is the quantity produced by �rm i. Each �rm has a
constant marginal cost of production, c, and no �xed cost. The �rms choose their
quantities as follows:
(1) �rm 1 chooses q1 ≥ 0
(2) �rms 2 and 3 observe q1 and then simultaneously choose q2 and q3, respectively.
What is the subgame-perfect outcome?
2. The accompanying simultaneous-move game is played twice, with the outcome of the
�rst stage observed before the second stage begins. The payo� of the repeated game
the sum of the payo�s from the two stage games. The variable x is greater than 4, so
that (4, 4) is not an equilibrium payo� in the one-shot game. For what values of x is
the following strategy (played by both players) a subgame-perfect Nash equilibrium?
Play Qi in the �rst stage.
If the �rst-stage outcome is (Q1, Q2), play Pi in the second stage.
If the �rst-stage outcome is (y,Q2) where y 6= Q1, play Ri in the second
stage.
If the �rst-stage outcome is (Q1, z) where z 6= Q2, play Si in the second
stage.
If the �rst-stage outcome is (y, z) where y 6= Q1 and z 6= Q2, play Pi in the
second stage.
P2 Q2 R2 S2
P1 2, 2 x, 0 -1, 0 0, 0
Q1 0, x 4, 4 -1, 0 0, 0
R1 0, 0 0, 0 0, 2 0, 0
S1 0, -1 0, -1 -1, -1 2, 0
3. The simultaneous-move game below is played twice, with the outcome of the �rst stage
observed before the second stage begins. The payo� of the repeated game the sum of
the payo�s from the two stage games. Can the payo� (4, 4) be achieved in the �rst
stage in a pure-strategy subgame-perfect Nash equilibrium? If so, give strategies that
do so. If not, prove why not.
L C R
T 3, 1 0, 0 5, 0
M 2, 1 1, 2 3, 1
B 1, 2 0, 1 4, 4
8