Post on 04-Jul-2020
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Repeated Games
Lesson 2: Games repeated infinitely many times
Universidad Carlos III de Madrid
We know that… § In a finitely repeated game with just one NE in
the stage game, by backward induction we observe: Ø Period T: there are no incentives to cooperate.
• No future losses to worry about . Ø Period T-1: no incentives to cooperate either.
• No “oportunity cost” to desviate in T-1, because in T there will be no cooperation.
§ From here, we deduce that there will be no cooperation in any period.
Finite interaction
§ When there is a unique NE cooperation is impossible if the relation between players has a fixed, known duration.
§ This opens different possibilities: Ø Duration is uncertain. Ø Duration is unknown. Ø The game lasts infinitely many periods.
A game repeated infinitely many times
§ A sumultaneous stage game is repeated at periods 1, 2, 3, ..., t-1, t, t+1, ..... In each period t players observe the results of the previous stages, from 1 until t-1.
§ Ecah player discounts future payoffs: δ, 0< δ < 1.
§ A player’s payoff is the present value of future payoffs: ∑ δ t-1 πt. If πt is constant, then:
π+ δ π + δ 2π + δ 3π ….= π 1
1-δ
SPNE § Consider a repeated game that consists of playing
the following stage game infinitely many times. § Is it possible to sustain cooperation? Is it possible to
play (R1, R2) as part of a SPNE?
Player 2
L2 R2
Player 1 L1 1 , 1 5 , 0
R1 0 , 5 4 , 4
1 L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
1 1 1 1 (1, 1) (5, 0) (0, 5) (4, 4)
INFINITELY MANY TIMES
Subgames and Strategies
§ There are infinitely many subgames. § Each is identical to the whole game. § We’ll use trigger strategies:
Ø Cooperate until there is a deviation Ø After a deviation, play the non cooperative strategy
(the NE in the one stage game) for ever.
“Trigger” strategies § Trigger strategy for Player i: play Ri in the first
stage, and in t, if (R1, R2) was played in ALL previous stages, from 1 till t-1; if at any stage (R1, R2) was not played, then play Li.
§ See that theses strategies constitute a SPNE. § Two steps:
Ø 1: Check that they constitute NE of the whole repeated game.
Ø 2: Check that they constitute a NE in every subgame.
Discounted payoffs
δδδδ
−=++++11......1 32
1 xz z...xxxx xz...xxxx 1 z
Define
432
432
=−
++++=
+++++=
x11 z −
=
Step 1 t= 1: (R1, R2)
t= 2: (R1, R2)
t-1: (R1, R2)
t: (R1, L2)
t+1: (L1, L2)
t+2: (L1, L2)
• Assume that Player 1 plays the trigger strategy. • Will Player 2 gain if he deviates at t? • If he does not deviate: he will get a sequence of
payoffs 4, 4, 4, ... (from t till +∞). The discounted sum of these payoffs is:
δδδδ
−=++++14......4444 32
• If he deviates: Player 1 will play L1 from t+1 on.
Player 2 will respond with L2. The sequence of payoffs will be 5, 1, 1, 1 .... The discounted sum of
these payoffs is δ
δδδδ
−+=++++1
5......1115 32
Step 1 (cont.) Stage 1: (R1, R2)
Stege 2: (R1, R2)
t-1: (R1, R2)
t: (R1, L2)
t+1: (L1, L2)
t+2: (L1, L2)
41
15
14
≥⇔−
+≥−
δδδ
δ
• If 41
≥δ , Player 2 does not improve with the
deviation. • Thus, using the trigger by Player 2 is a best
reply against the trigger strategy used by
Player1 if 41
≥δ .
• For Player 1 proceed analogously. • There is a NE in which both play trigger
strategies if 41
≥δ .
Stage 2 § Check that the strategies imply a NE in every subgame. § There are two families of subgames:
Ø Subgames after a sequence of (R1, R2). Ø Subgames after a history in which at some point (R1, R2)
was not played.
§ After the first family, strategies imply a NE (Recall that each subgame is identical to the whole game).
§ The second family implies a NE in which (L1, L2) is played for ever (since that is a NE of the stage game its repetition constitutes a NE in this family of subgames).
Infinite games as games of uncertain duration
§ When there is a unique NE cooperation is impossible if the relation between players has a fixed, known duration
§ Uncertain end Ø The game goes on to the next period with probability p:
§ Equivalent to an infinite game: Ø Recall that if the discount rate is δ, 1 euro tomorrow is
worth δ euros today. Ø If we add the probability p that there is a tomorrow, 1 euro
tomorrow in worth δp euros today. Ø A player’s payoff is the present value of futures payoffs:
∑ (δp) t-1 πt. If πt is constant, then: π+ δp π + (δp)2π + (δp)3π ….= π/(1- δp).
Aplication: collusion § Firms interact an infinite number of times (or
finitely many times, but with unknown end): Ø They can learn to coordinate strategies. Ø They can threaten to use punishment periods (with
low profits) in case of deviations. § Implications:
Ø If firms are sufficiently patience, they can sustain prices close to monopoly in every period.
Ø The higher the number of firms, the more difficult is to achieve the collusion.
Cournot Duopoly repeated infinitely many times
§ Stage game, Player 1:
Max (a-q1-q2-c) q1
q1>0 § In Cournot (symmetric costs):
qi = (a-c)/3 Π i = (a-c)2/9
Cooperative quantity § In a monopoly:
Max (a-Q-c)Q
Q>0
c.p.o.: QM= (a-c)/2
P= (a+c)/2 § In a collusive duopoly:
qi = (a-c)/4 , each one produces half of QM
Π i = (a-c)2/8
§ Trigger strategy: Ø Each player initially produces half of the
monopoly outcome. Ø After a deviation, they produce the Cournot
outcome for ever. Ø If there are no deviations, each continues
producing half the monopoly oucome. § Let’s see if this strategy allows the firms to
sustain the monopoly outcome in a SPNE.
Is it a NE? Assume Player 1 plays her trigger strategy.
Does Player 2 gain if he deviates at t?
• If he does not deviate: he will have a sequence of payoffs 8)( 2ca − . The
discounted value is (a− c)2
8(1−δ)
• If he deviates: The best deviation is top lay the best reply against QM/2 which is
8)(3 ca −
= • His discounted profits are
)1(9)(......
9)(
9)(
9)( 2
32
222
δδ
δδδ−
−+Π=+
−+
−+
−+Π
cacacaca DD
- 64)(9
8)(3
8)(3
4)( 2cacacacacaD −
=−
"#
$%&
' −−
−−−=Π
• His discounted profits if he deviates are
)1(9)(
64)(9 22
δδ
−
−+
− caca
• Thus, for Player 2 not to deviate, it is needed that δ≥9/17, as we see from:
)1(9649
)1(81
)1(9)(
64)(9
)1(8)( 222
δδ
δ
δδ
δ
−+>
−
−
−+
−>
−
− cacaca
Monopoly quantity in a SPNE § The proposed strategies constitute a NE of the whole
game if the discount factor is high enough.
§ In addition they imply a NE in every subgame:
Ø Subgames after a sequence of (qM1, qM
2). Ø Subgames after a history in which at some time (qM
1, qM
2) was not played.
• For the first family, strategies imply a NE. • For the second family, they imply a NE in which they play the
Cournot quantities for ever (since to play the Cournot quantity is a NE in the static game, its repetitio is a NE of any subgame).
Summary
§ Cooperation is feasible if the time horizon is uncertain or infinite.
§ Deviations are avoided by playing credible punishments: play the NE of the stage game.