Post on 23-Feb-2016
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Subgames and Credible Threats(with perfect information)
Econ 171
Alice and Bob
Bob
Go to A Go to B
Go to A
Alice Alice
Go to B Go to A Go to B
23 0
011
32
Strategies• For Bob – Go to A– Go to B
• For Alice– Go to A if Bob goes A and go to A if Bob goes B – Go to A if Bob goes A and go to B if Bob goes B– Go to B if Bob goes A and go to A if Bob goes B– Go to B if Bob goes A and go B if Bob goes B
• A strategy specifies what you will do at EVERYInformation set at which it is your turn.
Strategic Form
Go where Bob went.
Go to A no matter what Bob did.
Go to B no matter what Bob did.
Go where Bob did not go.
Movie A 2,3 2,3 0,0 0,1
Movie B 3,2 1,1 3,2 1,0
Alice
Bob
How many Nash equilibria are there for this game?A) 1B) 2C) 3D) 4
The Entry Game
Challenger
Stay out
01
Challenge
Incumbent
Give in Fight
10
-1 -1
Are both Nash equilibria Plausible?
• What supports the N.E. in the lower left?• Does the incumbent have a credible threat?• What would happen in the game starting from
the information set where Challenger has challenged?
Entry Game (Strategic Form)
-1,-1
0,0
0,1 0,0
Challenge Do not ChallengeChallenger
Incumbent
Give in
Fight
How many Nash equilibria are there?
Subgames
• A game of perfect information induces one or more “subgames. ” These are the games that constitute the rest of play from any of the game’s information sets.
• A subgame perfect Nash equilibrium is a Nash equilibrium in every induced subgame of the original game.
Backwards induction in games of Perfect Information
• Work back from terminal nodes.• Go to final ``decision node’’. Assign action to the
player that maximizes his payoff. (Consider the case of no ties here.)
• Reduce game by trimming tree at this node and making terminal payoffs at this node, the payoffs when the player whose turn it was takes best action.
• Keep working backwards.
Alice and Bob
Bob
Go to A Go to B
Go to A
Alice Alice
Go to B Go to A Go to B
23 0
011
32
Two subgames
Bob went A Bob went B
Alice Alice
Go to A Go to BGo to A Go to B
23
00
11
32
Alice and Bob (backward induction)
Bob
Go to A Go to B
Go to A
Alice Alice
Go to B Go to A Go to B
23 0
011
32
Alice and Bob Subgame perfect N.E.
Bob
Go to A Go to B
Go to A
Alice Alice
Go to B Go to A Go to B
23 0
011
32
Strategic Form
Go where Bob went.
Go to A no matter what Bob did.
Go to B no matter what Bob did.
Go where Bob did not go.
Movie A 2,3 2,3 0,0 0,1
Movie B 3,2 1,1 3,2 1,0
Alice
Bob
A Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
51
22
Kidnapper
43
Kill ReleaseKill Release
14
In the subgame perfect Nash equilibrium
A) The victim is kidnapped, no ransom is paid and the victim is killed.
B) The victim is kidnapped, ransom is paid and the victim is released.
C) The victim is not kidnapped.
Another Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
41
22
Kidnapper
53
Kill ReleaseKill Release
14
In the subgame perfect Nash equilibrium
A) The victim is kidnapped, no ransom is paid and the victim is killed.
B) The victim is kidnapped, ransom is paid and the victim is released.
C) The victim is not kidnapped.
Does this game have any Nash equilibria that are not subgame perfect?
A) Yes, there is at least one such Nash equilibrium in which the victim is not kidnapped.
B) No, every Nash equilibrium of this game is subgame perfect.
In the subgame perfect Nash equilibrium
A) The victim is kidnapped, no ransom is paid and the victim is killed.
B) The victim is kidnapped, ransom is paid and the victim is released.
C) The victim is not kidnapped.
Twice Repeated Prisoners’ Dilemma
Two players play two rounds of Prisoners’ dilemma. Before second round, each knows what other did on the first round. Payoff is the sum of earnings on the two rounds.
Single round payoffs
10, 10 0, 11
11, 0 1, 1
Cooperate Defect
Cooperate
Defect
PLAyER 1
Player 2
Two-Stage Prisoners’ DilemmaPlayer 1
Cooperate Defect
Player 2
CooperateCooperateDefect Defect
Player 1 Player 1 Player 1 Player 1
C
C
C
C
C CD D D D
C C C D
Player 1Pl. 2 Pl 2
Pl 2 Pl 2
2020
D DC D C D C D D1021
2110
1111
1021
022
1111
112
2110
1111
D
220
121
1111
212
121
22
Two-Stage Prisoners’ DilemmaWorking back
Player 1
Cooperate Defect
Player 2
CooperateCooperateDefect Defect
Player 1 Player 1 Player 1 Player 1
C
C
C
C
C CD D D D
C C C D
Player 1Pl. 2 Pl 2
Pl 2 Pl 2
2020
D DC D C D C D D1021
2110
1111
1021
022
1111
112
2110
1111
D
220
121
1111
212
121
22
Two-Stage Prisoners’ DilemmaWorking back further
Player 1
Cooperate Defect
Player 2
CooperateCooperateDefect Defect
Player 1 Player 1 Player 1 Player 1
C
C
C
C
C CD D D D
C C C D
Player 1Pl. 2 Pl 2
Pl 2 Pl 2
2020
D DC D C D C D D1021
2110
1111
1021
022
1111
112
2110
1111
D
220
121
1111
212
121
22
Two-Stage Prisoners’ DilemmaWorking back further
Player 1
Cooperate Defect
Player 2
CooperateCooperateDefect Defect
Player 1 Player 1 Player 1 Player 1
C
C
C
C
C CD D D D
C C C D
Player 1Pl. 2 Pl 2
Pl 2 Pl 2
2020
D DC D C D C D D1021
2110
1111
1021
022
1111
112
2110
1111
D
220
121
1111
212
121
22
Longer Game
• What is the subgame perfect outcome if Prisoners’ dilemma is repeated 100 times?
How would you play in such a game?