and Credible Threats with perfect information Econ 171 Alice and Bob 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 ID: 195469
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Slide1
Subgames and Credible Threats(with perfect information)
Econ 171Slide2
Alice and Bob
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
2Slide3
Strategies
For Bob
Go to A
Go to B
For AliceGo 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 BGo 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 BA strategy specifies what you will do at EVERYInformation set at which it is your turn.Slide4
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
2,32,30,0
0,1Movie B3,2
1,13,21,0
Alice
Bob
How many Nash equilibria are there for this game?1
234Slide5
The Entry Game
Challenger
Stay out
0
1
Challenge
Incumbent
Give in
Fight
1
0
-1 -1Slide6
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?Slide7
Entry Game (Strategic Form)
-1,-1
0,0
0
,1 0,0
Challenge Do not Challenge
Challenger
Incumbent
Give in
Fight
How many Nash equilibria are there? Slide8
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.Slide9
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.Slide10
Alice and Bob
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
2Slide11
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
2Slide12
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
2Slide13
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
2Slide14
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
2,32,30,0
0,1Movie B3,2
1,13,21,0
Alice
Bob
Slide15
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
4Slide16
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.Slide17
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
4Slide18
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.Slide19
Does this game have any Nash equilibria that are not subgame
perfect?
Yes, there is at least one such Nash equilibrium in which the victim is not kidnapped.
No, every Nash equilibrium of this game is
subgame
perfect.Slide20
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.Slide21
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.Slide22
Single round payoffs
10, 10
0,
11
11, 0
1, 1
Cooperate
Defect
Cooperate
DefectPLAyER
1 Player 2Slide23
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
2Slide24
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
2Slide25
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
2Slide26
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
2Slide27
Longer GameWhat is the
subgame
perfect outcome if
Prisoners’ dilemma is repeated 100 times?
How would you play in such a game?