Subgames and Credible Threats

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?

• What if the U.S. had installed a Doomsday machine, a la Dr. Strangelove?

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

Now for some theory…

John Nash

Reinhard Selten

John Harsanyi

Thomas Schelling

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

Go to A

Alice Alice

Go to B Go to A Go to B

23 0

011

32

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

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.

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.

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.

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.

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?