Introduction to Network Mathematics (3) - Simple Games and applications

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Introduction to Network Mathematics (3) - Simple Games and applications. Yuedong Xu 16/05/2012. Outline. Overview Prison’s Dilemma Curnot Duopoly Selfish Routing Summary. Overview. What is “game theory”? A scientific way to depict the rational behaviors in interactive situations

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### Introduction to Network Mathematics (3)- Simple Games and applications

YuedongXu

16/05/2012

Outline
• Overview
• Prison’s Dilemma
• Curnot Duopoly
• Selfish Routing
• Summary
Overview
• What is “game theory”?
• A scientific way to depict the rational behaviors in interactive situations
• Examples: playing poker, chess; setting price; announcing wars; and numerous commercial strategies
• Why is “game theory” important?
• Facilitates strategic thinking!
Overview
• Four pair of players expelled because they “throw” the matches
• Why are players trying to lose the match in the round-robin stage?
Overview
• Chinese VS Korean
• If Chinese team wins, it may encounter another Chinese team earlier in the elimination tournament. (not optimal for China)

Best strategy for Chinese team: LOSE

• If Korean team wins luckily, it may meet with another Chinese team that is usually stronger than itself in the elimination tournament.

Best strategy for Korean team: LOSE

Overview
• Korean VS Indonesian
• Conditioned on the result: China Lose
• If Korean team wins, meet with another Korean team early in the elimination tournament. (not optimal for Korea)

Best strategy for Korean team: LOSE

• If Indonesian wins, meet with a strong Chinese team in the elimination tournament.

Best strategy for Indonesian team: LOSE

Overview
• What is “outcome”?
• Ugly matches that both players and watchers are unhappy
• By studying this case, we know how to design a good “rule” so as to avoid “throwing” matches
Outline
• Overview
• Prison’s Dilemma
• Curnot Duopoly
• Selfish Routing
• Summary
Prison’s Dilemma
• Two suspects are caught and put in different rooms (no communication). They are offered the following deal:
• If both of you confess, you will both get 5 years in prison (-5 payoff)
• If one of you confesses whereas the other does not confess, you will get 0 (0 payoff) and 10 (-10 payoff) years in prison respectively.
• If neither of you confess, you both will get 2 years in prison (-2 payoff)
Prison’s Dilemma

Prisoner 2

Prisoner 1

Prison’s Dilemma

Prisoner 2

Prisoner 1

Prison’s Dilemma
• Game
• Players (e.g. prisoner 1&2)
• Strategy (e.g. confess or defect)
• Payoff (e.g. years spent in the prison)
• Nash Equilibrium (NE)
• In equilibrium, neither player can unilaterally change his/her strategy to improve his/her payoff, given the strategies of other players.
Prison’s Dilemma
• Some common concerns
• Existence/uniqueness of NE
• Convergence to NE
• Playing games sequentially or repeatedly
• Playing game with partial information
• Evolutionary behavior
• Algorithmic aspects
• and more ……
Prison’s Dilemma – Two NEs

Prisoner 2

Prisoner 1

Prison’s Dilemma – No NE

Rock-Paper-Scissors game:

If there exists a NE, then it is simple to play!

Outline
• Overview
• Prison’s Dilemma
• Curnot Duopoly
• Selfish Routing
• Summary
Curnot Duopoly

Basic setting:

• Two firms: A & B are profit seekers
• Strategy: quantity that they produce
• Market price p:

p = 100 - (qA+ qB)

• Question: optimal quantity for A&B
Curnot Duopoly
• A’s profit:
• Strategy: quantity that they produce
• Market price p:

p = 100 - (qA+ qB)

• Question: optimal quantity for A&B
Curnot Duopoly
• A’s profit:

πA(qA,qB) = qAp = qA(100-qA-qB)

• B’s profit:

πB(qA,qB) = qBp= qB(100-qA-qB)

• How to find the NE?
Curnot Duopoly
• A’s best strategy:

dπA(qA,qB)

—————— = 100 - 2qA – qB= 0

dqA

• B’s best strategy:

dπB(qA,qB)

—————— = 100 - 2qB – qA= 0

dqB

• Combined together: qA* = qB* = 100/3
Curnot Duopoly
• Take-home messages:
• If the strategy is continuous, e.g. production quantity or price, you can find the best response for each player, and then find the fixed point(s) for these best response equations.
Outline
• Overview
• Prison’s Dilemma
• Curnot Duopoly
• Selfish Routing
• Summary
Selfish Routing

x

1

x

1

s

t

s

0

t

x

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1

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Traffic of 1 unit/sec needs to be routed from s to t

Want to minimize average delay

Braess 1968, in study of road traffic

Selfish Routing
• Before and after

x

1

x

1

.5

.5

1

0

s

t

s

0

1

t

.5

.5

0

1

x

x

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Think of green flow – it has no incentive to deviate