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.