Subgames and Credible Threats. Russian Tanks Quell Hungarian Revolution of 1956.
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Transcript of Subgames and Credible Threats. Russian Tanks Quell Hungarian Revolution of 1956.
Subgames and Credible Threats
Russian Tanks QuellHungarian Revolution of 1956
The background
• After WW II, the Soviet army occupied Hungary. • Ultimately, the government came under Soviet
control. • In 1956, with U.S. encouragement, Hungarians
revolted and threw out the Soviet-backed government.
• Russia did not like this outcome. • The Hungarians appealed to the U.S. for support.
What should U.S. do?
• The U.S. did not have a large enough ground force in Europe to deal effectively with the Soviet army in Eastern Europe.
• The U.S. did have the nuclear capacity to impose terrible costs on Russia.
• But nuclear war would be very bad for everyone. (radioactive fallout, possibility of nuclear retaliation)
Nuclear threat
USSR
Don’t Invade Hungary
01
Invade
US
Give in Bomb USSR
50
-10 -5
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
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?
Nuclear threat
USSR
Don’t Invade Hungary
01
Invade
US
Give in Bomb USSR
50
-10 -5
Now for some theory…
John Nash
Reinhard Selten
John Harsanyi
Thomas Schelling
Subgames in Games of Perfect Information
• 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. (decision nodes)
• 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.
What if the U.S. had installed a Doomsday machine, a la Dr. Strangelove?
The Doomsday Game
Similar structure, but less terrifying: The entry game
Challenger
Stay out
01
Challenge
Incumbent
Give in Fight
10
-1 -1
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
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
Alice and Bob(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
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.
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
Backwards induction in games of Perfect Information
• Work back from terminal nodes.• Go to final ``decision node’’. Assign action to
that maximizes decision maker’s payoff. (Consider the case of no ties here.)
• Reduce game by trimming tree at this node and making terminal payoffs the payoffs to best action at this node.
• Keep working backwards.
A Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
51
22
Kidnapper
43
Kill ReleaseKill Release
14
A Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
51
22
Kidnapper
43
Kill ReleaseKill Release
14
A Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
51
22
Kidnapper
43
Kill ReleaseKill Release
14
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.
Another Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
41
22
Kidnapper
53
Kill ReleaseKill Release
14
Another Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
41
22
Kidnapper
53
Kill ReleaseKill Release
14
Another Kidnapping Game
Kidnapper
Don’t Kidnap
35
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
41
22
Kidnapper
53
Kill ReleaseKill Release
14
Does this game have any Nash equilibria that are not subgame perfect?
A) Yes, there is at least one Nash equilibrium in which the victim is not kidnapped.
B) No, every Nash equilibrium of this game is subgame perfect.
The Centipede Game in extensive form
Backwards induction-Player 1’s last move
Backwards induction- What does 2 do?
One step further. What would 1 do?
Taking it all the way back
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?
The seven goblins
Dividing the spoils
Goblins named A, B, E, G, K, R, and U take turns proposing a division of 100 coins. (no fractions)A proposes a division. He gets 4 or more votes for his division, it is applied. If he does not, then A doesn’t get to vote any more and B proposes a division. If B gets half or more of remaining votes, his division is applied. Otherwise proposal goes to E and B doesn’t get to vote any more.
So it goes, moving down the alphabet.
Backwards induction
• If U gets to propose, then nobody else could vote and he would propose 100 for self.
• But U will never get to propose, because if R gets to propose, R only needs 1 vote (his own) to win. He would give self 100, U gets 0.
• If K gets to propose, he would need 2 votes. He could get U’s vote by offering him 1, offering R 0 and keeping 99.
• Keep working back..
Proposers: A,B,E,G,K,R,U
R proposes: needs 1 vote R-100, U-0K proposes: needs 2 votes K-99, R-0, U-1G proposes: needs 2 votes G-99,K-0, R-1, U-0E proposes: needs 3 votes E-98, G-0,K-1,R-0, U-1B proposes: needs 3 votes B-98,E-0,G-1,K-0,R-1,U-0A proposes: needs 4 votes A-97,B-0,E-1,G-0,K-1,R-0,U-1
Reading Backward and Planning Forward…