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Repeated games
Lesson 1: Finitely repeated games
Universidad Carlos III de Madrid
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Finitely repeated games
§ A finitely repeated games is a dynamic game in which a simultaneous game (the stage game) is played finitely many times, and the result of each stage is observed before the next is played.
§ Example: Play the prisoners’ dilemma several times. The stage game is the simultaneus prisoners’ dilemma game.
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Results
§ The repeated game has only one SPNE if the stage game (the simultaneous game) has only a NE. – In the SPNE players’ play the strategies in the NE in
each stage. § If the stage game has 2 or more NE, one can
find a SPNE where, at some stage, players play a strategy that is not part of a NE of the stage game.
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A game repeated twice Ø Think of the following game repeated twice:
Ø Two players play the same simultaneous game twice, at t=1 and at t=2.
Ø After the first time the game is played (after t=1) the result is observed before playing the second time.
Ø The payoffo of the repeated game is the sum of the payoffs in each stage (t=1, t=2)
Ø Which is the SPNE?
Player 2
L2 R2
Player 1 L1 1 , 1 5 , 0
R1 0 , 5 4 , 4
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Extensive Form
R2
1+1 1+1
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
1+5 1+0
1+0 1+5
1+4 1+4
1 1 1 1
5+1 0+1
5+5 0+0
5+0 0+5
5+4 0+4
0+1 5+1
0+5 5+0
0+0 5+5
0+4 5+4
4+1 4+1
4+5 4+0
4+0 4+5
4+4 4+4
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Information sets and strategies
1.1 L1 R1
2.1
L2 R2 L2 R2
L1 R1
L2 R2
2. 2
L2 R2
L1 R1
L2 R2
2. 3
L2 R2
L1 R1
L2 R2
2.4
L2 R2
L1 R1
L2 R2
2.5
L2
1+5 1+0
1+0 1+5
1+4 1+4
1.2 1.3 1.4 1.5
5+1 0+1
5+5 0+0
5+0 0+5
5+4 0+4
0+1 5+1
0+5 5+0
0+0 5+5
0+4 5+4
4+1 4+1
4+5 4+0
4+0 4+5
Each Player: 5 Info sets
Example of a strategy: L1 R1 R1 L1 L1
4+4 4+4
1+1 1+1
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Subgames: 4 + Whome game
1 L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2 R2
L1 R1
2
L2 R2
2
L2
6 1
1 6
5 5
1 1 1 1
6 1
10 0
5 5
9 9
1 6
5 5
0 10
4 9
5 5
9 4
4 9
2 2
Subgame 1 Subgame 2 Subgame 3
8 8
Subgame 4
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Another representation
R2
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
1 1
5 0
0 5
4 4
1 1 1 1 (1, 1) (5, 0) (0, 5) (4, 4)
1 1
5 0
0 5
4 4
1 1
5 0
0 5
4 4
1 1
5 0
0 5
4 4
Total payoffs are (1, 1)
+ payoffs in the subgame
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Compute the NE in Subgame 1 The result is the same if one takes only the payoffs of the
stage or total payoffs Payoffs t=2 Player 2
L2 R2
Player 1 L1 1 , 1 5 , 0
R1 0 , 5 4 , 4
Payoffs t=1 + t=2 Player 2
L2 R2
Player 1 L1 2 , 2 6 , 1
R1 1 , 6 5 , 5
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NE in subgames
§ In each of the four subgames there is a unique EN:
§ Substitute, by using backward induction, the subgame with payoffs in NE and then solve the whole game.
NE = {L1, L2}
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Substitute the subgame with NE payoffs
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
5 0
0 5
4 4
1 1 1 1 (2, 2) (6, 1) (1, 6) (5, 5)
1 1
5 0
0 5
4 4
1 1
5 0
0 5
4 4
1 1
5 0
0 5
4 4
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Compute NE in the whole game with these “substituted payoffs”
L2 R2
L1 2 , 2 6 , 1
R1 1 , 6 5 , 5
The payoffs in the NE (1, 1) of the second stage have been added to the payoffs in t=1
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SPNE § SPNE:
(L1 L1L1L1L1, L2 L2L2L2L2)
§ Player 1 plays L1 in t= 1, and plays L1 in t=2 after any result in t=1.
§ Player 2 plays L2 in t= 1, and L2 in t=2 after any result in t=1.
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A repeated game with two NE in the stage game
L2 M2 R2 L1 1 , 1 5 , 0 0 , 0
M1 0 , 5 4 , 4 0 , 0
R1 0 , 0 0 , 0 3 , 3
§ Play twice the stage game depicted below. § Note that there are 2 NE, and that (M1, M2) is not a NE,
but that its payoffs Pareto dominate the payoffs in any NE.
§ Could (M1, M2) be played at t=1 in a SPNE?
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Extensive Form (informal) 1
L1 R1
2 2
L2 R2 M2 L2 R2 M2
L2 R2 M2
2
L1 R1 2 2
L2 R2 M2 L2 R2 M2
L2 R2 M2
2 M1
(1, 1) (5, 0) (0, 5) (4, 4) (0, 0)
M1
(0, 0) (0, 0) (0, 0) (3, 3) 1
(1, 1) (5, 0) (0, 5) (0, 0) (0, 0) (0, 0) (0, 0) (3, 3) (4, 4)
1 1
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Can (M1, M2) be played at t=1 in a SPNE?
§ Yes, if we use strategies with credible “prizes” and “punishments. – For this, both the prize and the punishment
must be NE. § Prize: Play (R1, R2)
– Payoffs: (3, 3)
§ Punishment: Play (L1, L2) – Payoffs: (1, 1)
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Strategias in a SPNE § Strategias in a SPNE
Ø t=1, player 1 plays M1, and 2 plays M2. Ø t=2,
Ø 1 plays R1 if ( M1, M2 ) was played at t=1, and plays L1 otherwise.
Ø 2 plays R2 if ( M1, M2 ) was played at t=1, and plays L2 otherwise.
§ Why is this a ENPS? – In every subgame at t=2, in any NE either ( R1, R2 )
is played, or ( L1, L2 ) is played – Are they NE of the whome game?
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Extensive Form 1
L1 R1
2 2
L2 R2 M2 L2 R2 M2
L2 R2 M2
2
L1 R1 2 2
L2 R2 M2 L2 R2 M2
L2 R2 M2
2 M1
(1, 1) (5, 0) (0, 5) (4, 4) (0, 0)
M1
(0, 0) (0, 0) (0, 0) (3, 3)
1
(1, 1) (5, 0) (0, 5) (0, 0) (0, 0) (0, 0) (0, 0) (3, 3) (4, 4)
(1, 1) (1, 1) (1, 1) (3, 3) (1, 1) (1, 1) (1, 1) (1, 1) (1, 1) +
1 1
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NE in the whole game
Player 2 L2 M2 R2
Player 1 L1 2 , 2 6 , 1 1 , 1
M1 1 , 6 7 , 7 1 , 1
R1 1 , 1 1 , 1 4 , 4
-By backward induction, substitute subgames with payoffs in the NE.
-The resultin normal form game has (M1, M2) as a EN
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Intuition
§ Let’s look at the stage game: If 1 plays M1 then player 2 may be tempted to deviate and play L2
(he wins 5 rather than 4). For him not to deviate: 4 + prize > 5 + punishment, i.e.: 4+3 > 5+1.
The same applies to player 2. Further more, for the strategy to be a SPNE, prizes and
punsishments must NE.
L2 M2 R2 L1 1 , 1 5 , 0 0 , 0
M1 0 , 5 4 , 4 0 , 0
R1 0 , 0 0 , 0 3 , 3
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§ If the payoffs after the deviation were higher (to deviate is more atractive) (M1, M2) in t=1 may not be sustained in a SPNE If 1 plays M1 Player 2 may be tempted to deviate and play L2 (he
wins 7 rather than 4). For him not to deviate we must have: 4 + prize >7 + punishment, but this IS NOT satisfied (7<8). Player 2 deviates, and we do not support (M1, M2) in a SPNE.
L2 M2 R2 L1 1 , 1 5 , 0 0 , 0
M1 0 , 7 4 , 4 0 , 0
R1 0 , 0 0 , 0 3 , 3
