ECON 306 Homework: Sequential Move Games and Repeated Games 2021
PROBLEM SOLVING.
1. Alternating Bargaining Game
In the infinitely repeated 2-player Rubinstein alternating bargaining game, we
stated that the unique subgame perfect equili
ium exhibits no delay. We will
double check that result here in more detail.
Recall the equil. strategies are
Player 1 demands s1,t =
1
(1+ δ)
, Player 2 responds by d2,t = A iff s1,t ≤
1
(1+ δ)
;
Player 2 offers s2,t+1 =
δ
(1+ δ)
, Player 1 responds by d1,t+1 = A iff s2,t+1 ≥
δ
(1+ δ)
,
for any odd periods t = 1, 3, 5, . . ., so t + 1 = 2, 4, 6, . . . are the even periods.
Note: “iff” means “if and only if”.
(a) In odd periods, given d2,t = A if s1,t ≤ 1(1+δ) , can player 1 get a deal by demand-
ing s1,t = 1(1+δ)?
(b) Expecting there will be a deal reached in t = 2, is it profitable for player 1 to
demand more than 1(1+δ)?
(c) Consider period 1(b) where player 1 has made some demand s1,t. Should
player 2 accept any demand s1,t > 1(1+δ)? Why or why not?
2. 3-Period Alternating Bargaining Game
This question walks through the 3-period Rubinstein alternating bargaining game
we covered in class. Below is the setup.
• players 1 and 2 (P1 and P2) bargain over one dollar by making alternating
offers
• players are impatient—they discount payoffs received in later periods by the
factor δ per period
• timing
1a P1 makes an demand s1 ∈ [0, 1] for P1’s share; subscript 1 denotes P1’s
demand
1b P2 makes a decision d2 ∈ {A, R}
– if d2 = A, payoffs realize π1 = s1, π2 = 1− s1;
– if d2 = R, move on to stage 2...
2a P2 makes an offer s2 ∈ [0, 1] for 1’s share; subscript 2 denotes P2’s offer fo
P1’s share
2b P1 makes a decision d1 ∈ {A, R}
– if d1 = A, payoffs realize π1 = s2, π2 = 1− s2;
– if d1 = R, move on to stage 3...
3 payoffs determined exogenously π1 = s, π2 = 1− s
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• Solve the game using backward induction.
(a) In period 2b, what is P1’s strategy on decision rule, given P2’s offer s2?
(b) In period 2a, what offer is P2’s optimal strategy, expecting P1’s responses in
period 2b? (Hint: need to show that inducing P1 to accept is worthwhile fo
P2)
(c) In period 1b, what is P2’s strategy on decision rule, given P1’s demand s1?
(d) In period 1a, what offer is P1’s optimal strategy, expecting P2’s responses in
period 1b? (Hint: need to show that inducing P2 to accept is worthwhile fo
P1)
(e) What is the outcome of the game in the subgame perfect equili
ium?
3. Supporting Cooperation in PD through Infinitely Repeated Game
Consider the following prisoner’s dilemma (PD) game
P2
C D
P1 C (4,4) (0,5)D (5,0) (1,1)
P1 and P2 play an infinitely repeated PD game. Specifically,
• P1 and P2 play the above stage game simultaneously, once the payoffs are
ealized, the players observe the outcome and move on to the next period,
when the game is played again.
• Both players have a discount factor of δ per period.
Consider the strategy of “tit-for-tat”. Specifically, P1 and P2 promise to choose
strategies conditioning on the previous outcomes in the following way
• play C in period 1;
• if (C,C) has always been played in the past, then play “C” this period;
• if the opponent ever played “D”, then play “D” forever.
(a) If
• “D” has never been played in previous periods;
• P1 believes P2 plays the “tit-for-tat” strategy as above;
• P1 will play “tit-for-tat” in the future.
What is the net present value (NPV) of P1’s payoff from playing “C” in the
cu
ent period? (Note: P2’s NPV under the same belief is identical)
(b) If
• “D” has never been played in previous periods;
• P1 believes P2 plays the “tit-for-tat” strategy as above;
• P1 will play “tit-for-tat” in the future.
What is the net present value (NPV) of P1’s payoff from playing “D” in the
cu
ent period? (Note: P2’s NPV under the same belief is identical)
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(c) Derive the condition under which playing C in period 1 is a best response if
players expect each other to play “tit-for-tat”.
(d) If (C,C) is played in period 1, derive the condition under which playing C in
period 2 is a best response if players expect each other to play “tit-for-tat”.
(e) If (C,C) is played in all previous periods up to some period t ≥ 2, derive the
condition under which playing C in the next period t + 1 is a best response if
players expect each other to play “tit-for-tat”.
(f) Use your previous results to show: both players playing “tit-for-tat” is indeed
a Nash equili
ium, and the outcome of the game is (C,C) being played in
every period if δ ≥ 14 .
(g) Use your previous results to show: both players playing “tit-for-tat” is not a
Nash equili
ium if δ < 14 .
4. Review: One-shot Cournot Competition versus One-shot Collusion
Consider a Cournot competition model.
• Firm 1 and 2 (F1 and F2) compete by simultaneously choosing output levels,
q1, q2, to maximize profit.
• The two products are perfect substitutes for the consumers who have inverse
demand
p = a − b(q1 + q2).
• Each firm pays a constant marginal cost of c to produce every unit of good.
(a) Solve for the one-shot NE of this game. (Hint: the equili
ium is (q∗1 =
a−c
3b , q
∗
2 =
a−c
3b ) and their profits are (π
∗
1 =
(a−c)2
9b , π
∗
2 =
(a−c)2
9b ))
(b) If the two firms were integrated into one firm, i.e. a monopoly. Solve fo
the profit-maximizing monopolistic output qm and profit level πm. (Hint: the
monopolistic output is qm = and its profit is πm =.)
(c) If the two firms can collude with each other by agreeing act as if they were
one firm, i.e. a monopoly, and share the market equally. What is each firm’s
output level and profit level? Is it more profitable to collude?(Hint: use you
esult in the previous part, each firm shares half of the monopoly market and
monopoly profit.)
(d) This part investigates if the two firms are able to maintain collusion without a
inding contract. If firm 1 believes that firm 2 will honor the collusion agree-
ment and produce half the monopoly output level qm2 =
a−c
4b .
• What is firm 1’s best response qD1 ? How does the best response compare
against the collusive output level qm1 ?
• If firm 2 indeed produces the collusive output level qm2 , while firm 1 ex-
pected this and plays its best response qD1 , what are each firm’s profit lev-
els?
• How do their profits compare to collusive profits?
• Does firm 1 have incentive to deviate from collusion?
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(Hint: in this part, we assume firm 2 is naive and produces qm2 =
a−c
4b regardless
of firm 1’s play.)
5. Tacit Collusion of Cournot Duopoly through Repeated Play
If we focus on the collusive a
angement (“collude”) versus competitive outcomes
(“dumping” for lack of a better word) in the previous question, we have a payoff
matrix
F2
(C)ollude (D)umping
F1 (C)ollude (
1
8
(a−c)2
,
1
8
(a−c)2
) (
3
32
(a−c)2
,
9
64
(a−c)2
)
(D)umping ( 964
(a−c)2
,
3
32
(a−c)2
) (
1
9
(a−c)2
,
1
9
(a−c)2
)
F1 and F2 repeatedly play this 2-by-2 game infinitely. Specifically,
• P1 and P2 play the above stage game simultaneously, once the payoffs are
ealized, the players observe the outcome and move on to the next period,
when the game is played again.
• Both players have a discount factor of δ per period.
Consider the strategy of “tit-for-tat”. Specifically, F1 and F2 promise to choose
strategies conditioning on the previous outcomes in the following way
• play C in period 1;
• if (C,C) has always been played in the past, then play “C” this period;
• if the opponent ever played “D”, then play “D” forever.
(a) What is the net present value (NPV) of F1’s payoff from playing “C” in the
cu
ent period? (Note: F2’s NPV under the same belief is identical)
(b) If
• “D” has never been played in previous periods;
• P1 believes P2 plays the “tit-for-tat” strategy as above;
• P1 will play “tit-for-tat” in the future.
What is the net present value (NPV) of P1’s payoff from playing “D” in the
cu
ent period? (Note: P2’s NPV under the same belief is identical)
(c) Derive the condition under which playing C in period 1 is a best response if
players expect each other to play “tit-for-tat”.
(d) If (C,C) is played in period 1, derive the condition under which playing C in
period 2 is a best response if players expect each other to play “tit-for-tat”.
(e) If (C,C) is played in all previous periods up to some period t ≥ 2, derive the
condition under which playing C in the next period t + 1 is a best response if
players expect each other to play “tit-for-tat”.
(f) Use your previous results to show: both firms playing “tit-for-tat” is indeed a
Nash equili
ium, and the outcome of the game is (C,C) being played in every
period if δ ≥ 917 .
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(g) Use your previous results to show: both players playing “tit-for-tat” is not a
Nash equili
ium if δ < 14 .
(h) Interpret your results in the context of collusion.