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Games with Sequential Moves

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Games with Sequential Moves Games with Sequential Moves 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.

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### Games with Sequential Moves

Games with Sequential Moves
• 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.
Games in extensive (tree) form.
• Games with perfect and complete information

Game Tree (slightly different from the text)

(2, 7, 4, 1)

Up

ANN

Down

Branches

(1, -2, 3, 0)

1

(1.3, 2, -11, 3)

High

DEB

2

Low

(0, -2.718, 0, 0)

Stop

BOB

3

ANN

(10, 6, 1, 1)

Terminal Nodes

Nodes

(6, 3, 4, 0)

Good 50%

Go

NATURE

Risky

CHRIS

(2, 8, -1, 2)

Bad 50%

Root (Initial Node)

Safe

(3, 5, 3, 1)

v.s. Decision Tree
• Nodes

Places where players make moves.

-Root

-Terminal nodes

• Branches

Possible choices of players

Strategy vs. Moves
• Payoffs

-(A, B, C, D)

-Comparison

• Nature

Uncertainty

Solving the Game Tree
• 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

Subgame

(2, 7, 4, 1)

Up

ANN

Ann’s move

Down

(1, -2, 3, 0)

1

Bob’s Move

(1.3, 2, -11, 3)

High

DEB

2

Low

(0, -2.718, 0, 0)

Stop

Deb’s Move

BOB

3

ANN

(10, 6, 1, 1)

(6, 3, 4, 0)

Good 50%

Go

NATURE

Risky

CHRIS

(2, 8, -1, 2)

Bad 50%

Safe

(3, 5, 3, 1)

Solving the Game Tree
• 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.

(2, 7, 4, 1)

(2, 7, 4, 1)

Up

ANN

Down

(1, -2, 3, 0)

1

(1.3, 2, -11, 3)

High

DEB

2

Low

(0, -2.718, 0, 0)

Stop

BOB

3

ANN

(10, 6, 1, 1)

(6, 3, 4, 0)

Good 50%

Go

NATURE

Risky

CHRIS

(2, 8, -1, 2)

Bad 50%

Safe

Chris’ Move

(3, 5, 3, 1)

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.

The Secret Garden Game

(3, 3, 3)

TALIA

C

C

D

NINA

(3, 3, 4)

D

C

(3, 4, 3)

D

C

TALIA

EMILY

D

(1, 2, 2)

D

TALIA

C

(4, 3, 3)

C

NINA

D

(2, 1, 2)

D

C

(2, 2, 1)

TALIA

D

(2, 2, 2)

In equilibrium, Emily chooses D, Nina follows C, and then Talia chooses C.
• Equilibrium Path (Subgame Perfect Equilibrium (SPNE))

-Reinhard Selten, 1994 Nobel Laureate

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
• 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
• A simple bargaining problem
• Traveler’s Dilemma

The Centipede Game

B

Pass

A

B

A

Pass

Pass

Pass

Pass

Pass

B

A

90, 90

Take

Dime

Take

Dime

Take

Dime

Take

Dime

Take

Dime

Take

Dime

10, 0

0, 20

30, 0

0, 40

90, 0

0, 100

The Survivor

A constant-sum game.

• Players

Rich, Rudy, Kelly

• Every 3 days, a person will be voted off if not the immunity winner.
Homework

question 2, 3, 5, and 10.