Subgames and Credible Threats

Russian Tanks QuellHungarian 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

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?

Nuclear threat

USSR

Don’t Invade Hungary

01

Invade

US

Give in Bomb USSR

50

-10 -5

Now for some theory…

John Nash

Reinhard Selten

John Harsanyi

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

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

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

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.

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

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

35

Kidnap

Relative

Pay ransom

Kidnapper

Don’t pay

51

22

Kidnapper

43

Kill ReleaseKill Release

14

A Kidnapping Game

Kidnapper

Don’t Kidnap

35

Kidnap

Relative

Pay ransom

Kidnapper

Don’t pay

51

22

Kidnapper

43

Kill ReleaseKill Release

14

A Kidnapping Game

Kidnapper

Don’t Kidnap

35

Kidnap

Relative

Pay ransom

Kidnapper

Don’t pay

51

22

Kidnapper

43

Kill ReleaseKill Release

14

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.

Another Kidnapping Game

Kidnapper

Don’t Kidnap

35

Kidnap

Relative

Pay ransom

Kidnapper

Don’t pay

41

22

Kidnapper

53

Kill ReleaseKill Release

14

Another Kidnapping Game

Kidnapper

Don’t Kidnap

35

Kidnap

Relative

Pay ransom

Kidnapper

Don’t pay

41

22

Kidnapper

53

Kill ReleaseKill Release

14

Another Kidnapping Game

Kidnapper

Don’t Kidnap

35

Kidnap

Relative

Pay ransom

Kidnapper

Don’t pay

41

22

Kidnapper

53

Kill ReleaseKill Release

14

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

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?

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-0K proposes: needs 2 votes K-99, R-0, U-1G proposes: needs 2 votes G-99,K-0, R-1, U-0E proposes: needs 3 votes E-98, G-0,K-1,R-0, U-1B proposes: needs 3 votes B-98,E-0,G-1,K-0,R-1,U-0A proposes: needs 4 votes A-97,B-0,E-1,G-0,K-1,R-0,U-1

Reading Backward and Planning Forward…