Game Theory 2:

Extensive-Form Games and

Subgame Perfection

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Dynamics in Games

How should we think of strategic interactions that occur insequence?

Who moves when?

And what can they do at different points in time?

How do people react to different histories?

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Modeling Games with Dynamics

Players

Player function

I Who moves when

Terminal histories

I Possible paths through the game

Preferences over terminal histories

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Strategies

A strategy is a complete contingent plan

Player i’s strategy specifies her action choice at each pointat which she could be called on to make a choice

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An Example: International Crises

Two countries (A and B) are competing over a piece ofland that B occupies

Country A decides whether to make a demand

If Country A makes a demand, B can either acquiesce orfight a war

If A does not make a demand, B keeps land (game ends)

A’s best outcome is Demand followed by Acquiesce, worstoutcome is Demand and War

B’s best outcome is No Demand and worst outcome isDemand and War

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An Example: International Crises

A can choose: Demand (D) or No Demand (ND)

B can choose: Fight a war (W ) or Acquiesce (A)

Preferences

uA(D,A) = 3 > uA(ND,A) = uA(ND,W ) = 2 > uA(D,W ) = 1

uB(ND,A) = uB(ND,W ) = 3 > uB(D,A) = 2 > uB(D,W ) = 1

How can we represent this scenario as a game (in strategicform)?

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International Crisis Game: NE

Country B

W A

Country AD 1, 1 3, 2

ND 2, 3 2, 3

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International Crisis Game: NE

Country B

W A

Country AD 1, 1 3, 2

ND 2X, 3 2, 3

7 / 26

International Crisis Game: NE

Country B

W A

Country AD 1, 1 3X, 2

ND 2X, 3 2, 3

7 / 26

International Crisis Game: NE

Country B

W A

Country AD 1, 1 3X, 2X

ND 2X, 3 2, 3

7 / 26

International Crisis Game: NE

Country B

W A

Country AD 1, 1 3X, 2X

ND 2X, 3X 2, 3X

7 / 26

International Crisis Game: NE

Country B

W A

Country AD 1, 1 3X, 2X

ND 2X, 3X 2, 3X

7 / 26

International Crisis Game: NE

Country B

W A

Country AD 1, 1 3X, 2X

ND 2X, 3X 2, 3X

I Is there something funny here?

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International Crisis Game: NE

Country B

W A

Country AD 1, 1 3X, 2X

ND 2X, 3X 2, 3X

I Is there something funny here?

I Specifically, (ND,W )?

7 / 26

International Crisis Game: NE

Country B

W A

Country AD 1, 1 3X, 2X

ND 2X, 3X 2, 3X

I Is there something funny here?

I Specifically, (ND,W )?

I The threat of war deters the demand, but would Bfollow through?

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Non-Credible Threats

The equilibrium (ND,W ) depends on a “non-crediblethreat”

Once A makes a demand, B does not want to fight a war

But to rule out such behavior, we need a stronger solutionconcept

One that incorporates the fact that actions are taken insequence

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Why Rule out Non-credible

Threats

Equilibrium as a steady state

War is only a best-response for B because when no demandis made, B is indifferent

If A accidentally made a demand, war is not a sequentialbest-response for B. B should acquiesce instead

I Read the strategy W as “if A makes a demand, I willgo to war”

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Subgame Perfect Nash

Equilibrium

A strategy specifies what a player will do at every decisionpoint

I Complete contingent plan

Strategy in a SPNE must be a best-response at each node,given the strategies of other players

Backward Induction

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But First!

Let’s introduce a way of incorporating the timing of actionsinto the game explicitly

Use a game tree to represent the sequential aspect ofchoices

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An Example: International Crises

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An Example: International Crises

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12 / 26

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12 / 26

An Example: International Crises

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An Example: International Crises

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12 / 26

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Another Example

Player 2

C D

Player 1A 5, 3 0, 0

B 0, 0 3, 5

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Another Example

Player 2

C D

Player 1A 5X, 3X 0, 0

B 0, 0 3X, 5X

13 / 26

Another Example

Player 2

C D

Player 1A 5X, 3X 0, 0

B 0, 0 3X, 5X

Represent this as a game tree under the assumption thatplayer 1 moves first

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Another Example

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SPNE: (A, (C,D))

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What if Player 2 Moves First?

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SPNE: ((A,B), D)

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The Order of Action Matters

The outcome depends on who chooses first

I When 1 moves first: (A,C) with payoffs 5, 3

I When 2 moves first: (D,B) with payoffs 3, 5

In this game, there is an advantage to moving first

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The Centipede Game

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17 / 26

The Centipede Game

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17 / 26

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17 / 26

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17 / 26

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17 / 26

The Centipede Game

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Unique SPNE: ((E,E), (E,E))

Equilibrium payoffs (1, 1)

Pareto dominated by 3 outcomes!17 / 26

Multiple Equilibria

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Multiple Equilibria

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Multiple Equilibria

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18 / 26

Multiple Equilibria

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Multiple Equilibria

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SPNE 1: (D, A, (R,L))18 / 26

Multiple Equilibria

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SPNE 1: (D, A, (R,L))18 / 26

Multiple Equilibria

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18 / 26

Multiple Equilibria

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SPNE 1: (D, A, (R,L))SPNE 2: (U,B,(R,R)) 18 / 26

A Familiar Example: Public Good

in a Team

Two players: 1 & 2

Each can choose a level to contribute to a public good: si

Payoff for individual i are

ui(s1, s2) = s1 + s2 +s1s2

2− s2i

2

19 / 26

Nash Equilibrium

s∗1 = 2 s∗2 = 2

Individual player’s equilibrium payoff:

2 + 2 +2 · 2

2− 22

2= 4

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Consider an extensive form

version

Player 1 must make her choice first

Before Player 2 decides how much to put in, she observeshow much Player 1 puts in

How might this change contributions?

We will use backward induction

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Best Response for Player 2The payoff function for player 2:

u2(s1, s2) = s1 + s2 +s1s2

2− s22

2

How do we determine the best response of player 2?

∂u2(s1, s2)

∂s2= 1 +

s12− s2

Setting equal to zero (∂u2(s1,s2)∂s1

= 0), Player 2’sbest-response to s1 is

BR2(s1) = 1 +s12

≡ s2X(s1)

22 / 26

Best Response for Player 2The payoff function for player 2:

u2(s1, s2) = s1 + s2 +s1s2

2− s22

2

How do we determine the best response of player 2?

∂u2(s1, s2)

∂s2= 1 +

s12− s2

Setting equal to zero (∂u2(s1,s2)∂s1

= 0), Player 2’sbest-response to s1 is

BR2(s1) = 1 +s12

≡ s2X(s1)

22 / 26

Best Response for Player 2The payoff function for player 2:

u2(s1, s2) = s1 + s2 +s1s2

2− s22

2

How do we determine the best response of player 2?

∂u2(s1, s2)

∂s2= 1 +

s12− s2

Setting equal to zero (∂u2(s1,s2)∂s1

= 0), Player 2’sbest-response to s1 is

BR2(s1) = 1 +s12

≡ s2X(s1)

22 / 26

Best Response for Player 1

Player 1’s best response must account for how Player 2 willrespond to whatever she chooses

:

u1(s1, sX2 (s1))

u1(s1, sX2 (s1)) = s1 + sX2 (s1) +

s1sX2 (s1)

2− s21

2

u1(s1, sX2 (s1)) = s1 +

(1 +

s12

)+

s12

(1 +

s12

)− s21

2

u1(s1, sX2 (s1)) = 1 +

3

2s1 +

s12

+s214− s21

2

23 / 26

Best Response for Player 1

Player 1’s best response must account for how Player 2 willrespond to whatever she chooses :

u1(s1, sX2 (s1))

u1(s1, sX2 (s1)) = s1 + sX2 (s1) +

s1sX2 (s1)

2− s21

2

u1(s1, sX2 (s1)) = s1 +

(1 +

s12

)+

s12

(1 +

s12

)− s21

2

u1(s1, sX2 (s1)) = 1 +

3

2s1 +

s12

+s214− s21

2

23 / 26

Best Response for Player 1

Player 1’s best response must account for how Player 2 willrespond to whatever she chooses :

u1(s1, sX2 (s1))

u1(s1, sX2 (s1)) = s1 + sX2 (s1) +

s1sX2 (s1)

2− s21

2

u1(s1, sX2 (s1)) = s1 +

(1 +

s12

)+

s12

(1 +

s12

)− s21

2

u1(s1, sX2 (s1)) = 1 +

3

2s1 +

s12

+s214− s21

2

23 / 26

Best Response for Player 1

Player 1’s best response must account for how Player 2 willrespond to whatever she chooses :

u1(s1, sX2 (s1))

u1(s1, sX2 (s1)) = s1 + sX2 (s1) +

s1sX2 (s1)

2− s21

2

u1(s1, sX2 (s1)) = s1 +

(1 +

s12

)+

s12

(1 +

s12

)− s21

2

u1(s1, sX2 (s1)) = 1 +

3

2s1 +

s12

+s214− s21

2

23 / 26

Best Response for Player 1

Player 1’s best response must account for how Player 2 willrespond to whatever she chooses :

u1(s1, sX2 (s1))

u1(s1, sX2 (s1)) = s1 + sX2 (s1) +

s1sX2 (s1)

2− s21

2

u1(s1, sX2 (s1)) = s1 +

(1 +

s12

)+

s12

(1 +

s12

)− s21

2

u1(s1, sX2 (s1)) = 1 +

3

2s1 +

s12

+s214− s21

2

23 / 26

Best Response for Player 1We can write Player 1’s problem as:

u1(s1, sX2 (s1)) = 1 + 2s1 −

s214

Solve for Player 1’s optimal choice:

2 − s12

= 0

s∗1 = 4

Go back to Player 2:

s∗2 = s2X(4) = 1 +

1

2(4) = 3

24 / 26

Best Response for Player 1We can write Player 1’s problem as:

u1(s1, sX2 (s1)) = 1 + 2s1 −

s214

Solve for Player 1’s optimal choice:

2 − s12

= 0

s∗1 = 4

Go back to Player 2:

s∗2 = s2X(4) = 1 +

1

2(4) = 3

24 / 26

Public Good in a Team

So each player contributes more:

s∗1 = 4 s∗2 = 3

and equilibrium utilities:

u∗1 = 6 u∗

2 = 8.5

They each are better off, but it’s better to move second

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Public Good in a Team

So each player contributes more:

s∗1 = 4 s∗2 = 3

and equilibrium utilities:

u∗1 = 6 u∗

2 = 8.5

They each are better off, but it’s better to move second

25 / 26

Subgame Perfect Nash

Equilibrium

Subgame Perfect Nash Equilibrium is a refinement of NashEquilibrium

It rules out equilibria that rely on incredible threats in adynamic environment

All SPNE are identified by backward induction

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