Games
with
Sequential Moves
Games with Sequential Moves
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Games where players move one after another.
Possible to combine with simultaneous moves.
(But not considered in this chapter)
Players, when makes moves, have to consider
what the opponents may do.
Game Trees are commonly used to specify all
possible moves by all players and all possible
outcome and payoffs.
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Games in extensive (tree) form.
Games with perfect and complete information
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Game Tree (slightly different from the text)
ANN
(2, 7, 4, 1)
Up
Down
Branches
(1, -2, 3, 0)
1
DEB
2
High
Low
Stop
BOB
ANN
3
(10, 6, 1, 1)
Nodes
Go
CHRIS
Root (Initial Node)
Risky
Safe
(1.3, 2, -11, 3)
(0, -2.718, 0, 0)
Terminal Nodes
Good 50%
(6, 3, 4, 0)
Bad 50%
(2, 8, -1, 2)
NATURE
(3, 5, 3, 1)
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v.s. Decision Tree
Nodes
Places where players make moves.
-Root
-Terminal nodes
Branches
Possible choices of players
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Strategy vs. Moves
Payoffs
-(A, B, C, D)
-Comparison
Nature
Uncertainty
Solving the Game Tree
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Backward Induction
Rollback
Rollback Equilibrium, Subgame Perfect Nash
Equilibrium
Subgame
the part of a game where the subsequent nodes
after the starting nodes can separate from other
nodes not after the starting node of the
subgame
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Subgame
ANN
Ann’s move
(2, 7, 4, 1)
Up
Down
1
Bob’s Move
DEB
2
BOB
Low
CHRIS
Risky
Safe
(0, -2.718, 0, 0)
Deb’s Move
3
Go
(1.3, 2, -11, 3)
High
Stop
ANN
(1, -2, 3, 0)
(10, 6, 1, 1)
Good 50%
(6, 3, 4, 0)
Bad 50%
(2, 8, -1, 2)
NATURE
(3, 5, 3, 1)
Solving the Game Tree
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Expected Utility Theorem (von Neumann
and Morgenstern)
When taking Risky move, Chris expects to
obtain 50% X 4 + 50% X (-1)= 1.5
It guarantees Chris can compare the payoff
of 1.5 by playing Risky move to that of 3 by
playing Safe.
ANN
(2, 7, 4, 1)
Up
Down
(1, -2, 3, 0)
1
(1.3, 2, -11, 3)
(2, 7, 4, 1)High
DEB
2
Low
Stop
BOB
ANN
3
Go
CHRIS
Risky
Safe
(0, -2.718, 0, 0)
(10, 6, 1, 1)
Good 50%
(6, 3, 4, 0)
Bad 50%
(2, 8, -1, 2)
NATURE
(3, 5, 3, 1)
Chris’ Move
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In equilibrium,
A chooses “Go” in the beginning, and “Up”
if she has the chance to go after B .
B chooses “1”
C chooses “Safe”
D chooses “High”
The payoff is 3 to A, 5 to B, 3 to C and 1 to D.
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The Secret Garden Game
(3, 3, 3)
TALIA
C
C
D
NINA
(3, 3, 4)
D
D
C
C
(3, 4, 3)
TALIA
EMILY
D
(1, 2, 2)
D
NINA
C
TALIA
C
(4, 3, 3)
D
D
TALIA
C
D
(2, 1, 2)
(2, 2, 1)
(2, 2, 2)
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In equilibrium, Emily chooses D, Nina
follows C, and then Talia chooses C.
Equilibrium Path (Subgame Perfect
Equilibrium (SPNE))
-Reinhard Selten, 1994 Nobel Laureate
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Strategies
Emily {C, D} 2 strategies
Nina {CC, CD, DC, DD } 4 strategies
Talia {CCCC, CCCD, CCDC, CCDD, …..} 16
strategies for Talia
Nash Equilibrium (NE) is not necessarily a
SPNE, but SPNE must be a NE.
Remarks
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First-mover Advantage?
-Not necessarily!
Tic-tac-toe
-9x8x7x6x5x4x3x2x1=362,880 terminal nodes
Chess?
Existence of the equilibrium?
Zermelo-Theorem: A finite game of perfect
information has (at least) one pure-strategy Nash
equilibrium
Theory vs. Evidence
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A simple bargaining problem
Traveler’s Dilemma
• The Centipede Game
A Pass B Pass A Pass B
Take
Dime
Take
Dime
Take
Dime
Take
Dime
10, 0
0, 20
30, 0
0, 40
Pass
A Pass B Pass
90, 90
Take
Dime
Take
Dime
90, 0
0, 100
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The Survivor
A constant-sum game.
Players
Rich, Rudy, Kelly
Every 3 days, a person will be voted off if not
the immunity winner.
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Homework
question 2, 3, 5, and 10.