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?
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 )?
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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 )?
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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ND D
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An Example: International Crises
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B
(2, 3)
War Acq
ND D
12 / 26
An Example: International Crises
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B
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12 / 26
An Example: International Crises
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An Example: International Crises
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An Example: International Crises
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12 / 26
An Example: International Crises
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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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Another Example
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14 / 26
Another Example
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Another Example
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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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15 / 26
What if Player 2 Moves First?
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15 / 26
What if Player 2 Moves First?
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15 / 26
What if Player 2 Moves First?
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15 / 26
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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-1 2 1 2
E
C
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C 66
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The Centipede Game
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17 / 26
The Centipede Game
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E
C
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17 / 26
The Centipede Game
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C
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17 / 26
The Centipede Game
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E
C
E
C
E
C
E
C 66
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17 / 26
The Centipede Game
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-1 2 1 2
E
C
E
C
E
C
E
C 66
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03
52
47
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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18 / 26
Multiple Equilibria
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18 / 26
Multiple Equilibria
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18 / 26
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
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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)
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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)
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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
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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
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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
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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
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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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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