Subgames and Credible Threats
(with perfect information)
Econ 171
Alice and Bob
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
The Entry Game
Challenger
Challenge
Stay out
Incumbent
Give in
1
0
0
1
Fight
-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)
Challenger
Challenge
Give in
Do not Challenge
0,1
0,0
-1,-1
0,0
Incumbent
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
Alice
Go to A
2
3
Alice
Go to B
0
0
Go to A
1
1
Go to B
3
2
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
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
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.
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
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?