Subgames 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 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
Invade
Don’t Invade Hungary
US
0
1
Give in
5
0
Bomb
USSR
-10
-5
Nuclear threat (strategic form)
Soviet Union
Invade
Give in if
USSR Invades
Don’t Invade
0, 5
1, 0
United States
Bomb if USSR
Invades
-5,-10
1, 0
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
Invade
Don’t Invade Hungary
US
0
1
Give in
5
0
Bomb
USSR
-10
-5
Now for some theory…
Reinhard Selten
John Harsanyi
John Nash
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
Challenge
Stay out
Incumbent
Give in
1
0
0
1
Fight
-1
-1
Alice and Bob Revisited:
(Bob moves first)
Bob
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
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.
Strategic Form
Alice
Bob
Go where
Bob went.
Go to A no
matter what
Bob did.
Go to B no
Go where
matter what Bob did not
Bob did.
go.
Movie A
2,3
2,3
0,0
0,1
Movie B
3,2
1,1
3,2
1,0
How many Nash equilibria are there for this game?
A) 1
B) 2
C) 3
D) 4
Alice and Bob
(Bob moves first)
Bob
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
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
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
Alice and Bob (backward induction)
Bob
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
Alice and Bob Subgame perfect N.E.
Bob
Go to A
Go to B
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
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
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
5
1
3
5
Kidnapper
Release
4
3
Kill
2
2
Release
1
4
A Kidnapping Game
Kidnapper
Don’t
Kidnap
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
5
1
3
5
Kidnapper
Release
4
3
Kill
2
2
Release
1
4
A Kidnapping Game
Kidnapper
Don’t
Kidnap
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
5
1
3
5
Kidnapper
Release
4
3
Kill
2
2
Release
1
4
A Kidnapping Game
Kidnapper
Don’t
Kidnap
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
5
1
3
5
Kidnapper
Release
4
3
Kill
2
2
Release
1
4
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
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
4
1
3
5
Kidnapper
Release
5
3
Kill
2
2
Release
1
4
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
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
4
1
3
5
Kidnapper
Release
5
3
Kill
2
2
Release
1
4
Another Kidnapping Game
Kidnapper
Don’t
Kidnap
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
4
1
3
5
Kidnapper
Release
5
3
Kill
2
2
Release
1
4
Another Kidnapping Game
Kidnapper
Don’t
Kidnap
Kidnap
Relative
Don’t
pay
Pay ransom
Kidnapper
Kill
4
1
3
5
Kidnapper
Release
5
3
Kill
2
2
Release
1
4
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
Player 2
Cooperate
P
L
A Cooperate
y
E
R
1
Defect
Defect
10, 10
0, 11
11, 0
1, 1
Two-Stage Prisoners’ Dilemma
Player 1
Cooperate
Defect
Player 2
Cooperate
Player 1
C
C
20
20
Playe
Pl. 2
r1
D
10
21
D
C
Cooperate
Defect
Player 1
D
C
Player 1
C
Pl 2
D C D
Defect
C D C
Player 1
D
C
Pl 2
D
C
D
Pl 2
D C
21 11 10 0 11 1 21 11 22 12
10 11 21 22 11 12 10 11 0 1
D
C
11 2 12
11 12 1
D
2
2
Two-Stage Prisoners’ Dilemma
Working
back
Player 1
Cooperate
Defect
Player 2
Cooperate
Player 1
C
C
20
20
Playe
Pl. 2
r1
D
10
21
D
C
Cooperate
Defect
Player 1
D
C
Player 1
C
Pl 2
D C D
Defect
C D C
Player 1
D
C
Pl 2
D
C
D
Pl 2
D C
21 11 10 0 11 1 21 11 22 12
10 11 21 22 11 12 10 11 0 1
D
C
11 2 12
11 12 1
D
2
2
Two-Stage Prisoners’ Dilemma
Working
back further
Player 1
Cooperate
Defect
Player 2
Cooperate
Player 1
C
C
20
20
Playe
Pl. 2
r1
D
10
21
D
C
Cooperate
Defect
Player 1
D
C
Player 1
C
Pl 2
D C D
Defect
C D C
Player 1
D
C
Pl 2
D
C
D
Pl 2
D C
21 11 10 0 11 1 21 11 22 12
10 11 21 22 11 12 10 11 0 1
D
C
11 2 12
11 12 1
D
2
2
Two-Stage Prisoners’ Dilemma
Working
back further
Player 1
Cooperate
Defect
Player 2
Cooperate
Player 1
C
C
20
20
Playe
Pl. 2
r1
D
10
21
D
C
Cooperate
Defect
Player 1
D
C
Player 1
C
Pl 2
D C D
Defect
C D C
Player 1
D
C
Pl 2
D
C
D
Pl 2
D C
21 11 10 0 11 1 21 11 22 12
10 11 21 22 11 12 10 11 0 1
D
C
11 2 12
11 12 1
D
2
2
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-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-1
Reading Backward
and Planning Forward…