and Credible Threats Russian Tanks Quell Hungarian Revolution of 1956 The background After WW II the Soviet army occupied Hungary Ultimately the government came under Soviet control In 1956 with US encouragement Hungarians revolted and threw out the Sovietbacked government ID: 487261
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Slide1
Subgames and Credible ThreatsSlide2
Russian Tanks QuellHungarian Revolution of 1956Slide3
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. Slide4
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)Slide5
Nuclear threat
USSR
Don’t Invade Hungary
0
1
Invade
US
Give in
Bomb USSR
5
0
-10
-5Slide6
Nuclear threat (strategic form)
-5,-10
1, 0
0, 5 1, 0
Invade Don’t Invade
Soviet Union
United States
Give in if
USSR Invades
Bomb if USSR
Invades
How many pure strategy Nash
equilibria
are there?
A) 1 B) 2 C) 3 D) 4Slide7
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?Slide8
Nuclear threat
USSR
Don’t Invade Hungary
0
1
Invade
US
Give in
Bomb USSR
5
0
-10
-5Slide9
Now for some theory…
John Nash
Reinhard
Selten
John
Harsanyi
Thomas SchellingSlide10
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.Slide11
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.Slide12
What if the U.S. had installed a Doomsday machine, a la
Dr. Strangelove?Slide13
The Doomsday GameSlide14
Similar structure, but less terrifying: The entry game
Challenger
Stay out
0
1
Challenge
Incumbent
Give in
Fight
1
0
-1 -1Slide15
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
2
3
0
0
1
1
3
2Slide16
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 EVERY
Information set at which it is your turn.Slide17
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
3,21,0
Alice
Bob
How many Nash equilibria
are there for this game?12
34Slide18
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
2
3
0
0
1
1
3
2Slide19
How many subgame perfect N.E. does this game have?
There is only one and in that equilibrium they both go to movie A.
There is only one and in that
equilbrium
they both go to movie B.
There are two. In one they go to movie A and in the other
tney
go to movie B.
There is only one and in that equilibrium Bob goes to B and Alice goes to A.Slide20
Two subgames
Bob went A
Bob went B
Alice
Alice
Go to A
Go to B
Go to A
Go to B
2
3
0
0
1
1
3
2Slide21
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
2
3
0
0
1
1
3
2Slide22
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
2
3
0
0
1
1
3
2Slide23
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.Slide24
A Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
5
1
22
Kidnapper
4
3
Kill
Release
Kill
Release
1
4Slide25
A Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
5
1
22
Kidnapper
4
3
Kill
Release
Kill
Release
1
4Slide26
A Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
5
1
22
Kidnapper
4
3
Kill
Release
Kill
Release
1
4Slide27
A Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
5
1
22
Kidnapper
4
3
Kill
Release
Kill
Release
1
4Slide28
In the subgame perfect Nash equilibrium
The victim is kidnapped, no ransom is paid and the victim is killed.
The victim is kidnapped, ransom is paid and the victim is released.
The victim is not kidnapped.Slide29
Another Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
4
1
22
Kidnapper
5
3
Kill
Release
Kill
Release
1
4Slide30
In the subgame perfect Nash equilibrium
The victim is kidnapped, no ransom is paid and the victim is killed.
The victim is kidnapped, ransom is paid and the victim is released.
The victim is not kidnapped.Slide31
Another Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
4
1
22
Kidnapper
5
3
Kill
Release
Kill
Release
1
4Slide32
Another Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
4
1
22
Kidnapper
5
3
Kill
Release
Kill
Release
1
4Slide33
Another Kidnapping Game
Kidnapper
Don’t Kidnap
3
5
Kidnap
Relative
Pay ransom
Kidnapper
Don’t pay
4
1
22
Kidnapper
5
3
Kill
Release
Kill
Release
1
4Slide34
Does this game have any Nash equilibria that are not
subgame
perfect?
Yes, there is at least one Nash equilibrium in which the victim is not kidnapped.
No, every Nash equilibrium of this game is
subgame
perfect.Slide35
The Centipede Game in extensive formSlide36
Backwards induction-Player 1’s last moveSlide37
Backwards induction- What does 2 do?Slide38
One step further. What would 1 do? Slide39
Taking it all the way backSlide40
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.Slide41
Single round payoffs
10, 10
0,
11
11, 0
1, 1
Cooperate
Defect
Cooperate
Defect
P
LAyER
1
Player 2Slide42
Two-Stage Prisoners’ Dilemma
Player 1
Cooperate
Defect
Player 2
Cooperate
Cooperate
Defect
Defect
Player 1
Player 1
Player 1
Player 1
C
C
C
C
C
C
D
D
D
D
C
C
C
D
Player 1
Pl. 2
Pl 2
Pl 2
Pl 2
20
20
D
D
C
D
C
D
C
D
D
10
21
21
10
11
11
10
21
0
22
11
11
1
12
21
10
11
11
D
22
0
12
1
11
11
2
12
12
1
2
2Slide43
Two-Stage Prisoners’ DilemmaWorking back
Player 1
Cooperate
Defect
Player 2
Cooperate
Cooperate
Defect
Defect
Player 1
Player 1
Player 1
Player 1
C
C
C
C
C
C
D
D
D
D
C
C
C
D
Player 1
Pl. 2
Pl 2
Pl 2
Pl 2
20
20
D
D
C
D
C
D
C
D
D
10
21
21
10
11
11
10
21
0
22
11
11
1
12
21
10
11
11
D
22
0
12
1
11
11
2
12
12
1
2
2Slide44
Two-Stage Prisoners’ DilemmaWorking back further
Player 1
Cooperate
Defect
Player 2
Cooperate
Cooperate
Defect
Defect
Player 1
Player 1
Player 1
Player 1
C
C
C
C
C
C
D
D
D
D
C
C
C
D
Player 1
Pl. 2
Pl 2
Pl 2
Pl 2
20
20
D
D
C
D
C
D
C
D
D
10
21
21
10
11
11
10
21
0
22
11
11
1
12
21
10
11
11
D
22
0
12
1
11
11
2
12
12
1
2
2Slide45
Two-Stage Prisoners’ DilemmaWorking back further
Player 1
Cooperate
Defect
Player 2
Cooperate
Cooperate
Defect
Defect
Player 1
Player 1
Player 1
Player 1
C
C
C
C
C
C
D
D
D
D
C
C
C
D
Player 1
Pl. 2
Pl 2
Pl 2
Pl 2
20
20
D
D
C
D
C
D
C
D
D
10
21
21
10
11
11
10
21
0
22
11
11
1
12
21
10
11
11
D
22
0
12
1
11
11
2
12
12
1
2
2Slide46
Longer Game
What is the
subgame
perfect outcome if
Prisoners’ dilemma is repeated 100 times?
How would you play in such a game?Slide47
The seven goblinsSlide48
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.Slide49
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..Slide50
Proposers: A,B,E,G,K,R,U
R proposes: needs 1 vote R-100, U-0
K proposes: needs 2 votes K-99, R-0, U-1
G proposes: needs 2 votes G-99,K-0, R-1, U-0
E proposes: needs 3 votes E-98, G-0,K-1,R-0, U-1
B proposes: needs 3 votes B-98,E-0,G-1,K-0,R-1,U-0
A proposes: needs 4 votes A-97,B-0,E-1,G-0,K-1,R-0,U-1Slide51
Reading Backward and Planning Forward…