subgames and credible threats
DESCRIPTION
Subgames and Credible Threats. Nuclear threat. USSR. Invade. Don’t Invade Hungary. US. 0 1. Give in. Bomb USSR. 5 0. -10 -5. Nuclear threat (strategic form). Soviet Union. Invade Don’t Invade. Give in if USSR Invades Bomb if USSR Invades. - PowerPoint PPT PresentationTRANSCRIPT
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Subgames and Credible Threats
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Nuclear threat
USSR
Don’t Invade Hungary
01
Invade
US
Give in Bomb USSR
50
-10 -5
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Nuclear threat (strategic form)
-5,-10
1,0
0, 5 1,0
Invade Don’t InvadeSoviet Union
United States
Give in ifUSSR Invades
Bomb if USSRInvades
How many pure strategy Nash equilibria are there?
A) 1 B) 2 C) 3 D) 4
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Are all Nash Equilibria Plausible?
• What supports the no-invasion equilibrium?• Is the threat to bomb Russia credible?• What would happen in the game starting from
the information set where Russia has invaded Hungary?
• What if the U.S. had installed a Doomsday machine, a la Dr. Strangelove?
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Similar structure, but less terrifying: The entry game
Challenger
Stay out
01
Challenge
Incumbent
Give in Fight
10
-1 -1
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Alice and Bob Revisited: (Bob moves first)
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
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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.
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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
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Now for some theory…
John Nash
Reinhard Selten
John Harsanyi
Thomas Schelling
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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.
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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.
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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
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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
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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
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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
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How many subgame perfect N.E. does this game have?
A) There is only one and in that equilibrium they both go to movie A.
B) There is only one and in that equilbrium they both go to movie B.
C) There are two. In one they go to movie A and in the other tney go to movie B.
D) There is only one and in that equilibrium Bob goes to B and Alice goes to A.
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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.
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A Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
51
22
Kidnapper
43
Kill ReleaseKill Release
14
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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.
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Another Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
41
22
Kidnapper
53
Kill ReleaseKill Release
14
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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.
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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.
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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.
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Single round payoffs
10, 10 0, 11
11, 0 1, 1
Cooperate Defect
Cooperate
Defect
PLAyER 1
Player 2
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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
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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
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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
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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
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Longer Game
• What is the subgame perfect outcome if Prisoners’ dilemma is repeated 100 times?
How would you play in such a game?