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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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Presentation Transcript
outline
Outline
  • Overview
  • Prison’s Dilemma
  • Curnot Duopoly
  • Selfish Routing
  • Summary
overview
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!
overview1
Overview
  • Olympic Badminton Match 2012
    • Four pair of players expelled because they “throw” the matches
    • Why are players trying to lose the match in the round-robin stage?
overview2
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

overview3
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

overview4
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
outline1
Outline
  • Overview
  • Prison’s Dilemma
  • Curnot Duopoly
  • Selfish Routing
  • Summary
prison s dilemma
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 dilemma1
Prison’s Dilemma

Prisoner 2

Prisoner 1

prison s dilemma2
Prison’s Dilemma

Prisoner 2

Prisoner 1

prison s dilemma3
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 dilemma4
Prison’s Dilemma
  • Some common concerns
    • Existence/uniqueness of NE
    • Convergence to NE
    • Playing games sequentially or repeatedly
  • More advanced games
    • Playing game with partial information
    • Evolutionary behavior
    • Algorithmic aspects
    • and more ……
prison s dilemma two nes
Prison’s Dilemma – Two NEs

Prisoner 2

Prisoner 1

prison s dilemma no ne
Prison’s Dilemma – No NE

Rock-Paper-Scissors game:

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

outline2
Outline
  • Overview
  • Prison’s Dilemma
  • Curnot Duopoly
  • Selfish Routing
  • Summary
curnot duopoly
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 duopoly1
Curnot Duopoly
  • A’s profit:
  • Strategy: quantity that they produce
  • Market price p:

p = 100 - (qA+ qB)

  • Question: optimal quantity for A&B
curnot duopoly2
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 duopoly3
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 duopoly4
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.
outline3
Outline
  • Overview
  • Prison’s Dilemma
  • Curnot Duopoly
  • Selfish Routing
  • Summary
selfish routing
Selfish Routing
  • Braess’s Paradox

x

1

x

1

s

t

s

0

t

x

x

1

1

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 routing1
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

1

1

Think of green flow – it has no incentive to deviate

Adding a 0 cost link made average delay worse!!!

selfish routing2
Selfish Routing
  • Braess’s paradox illustrates non-optimality of selfish routing
  • Think of the flow consisting of tiny “packets”
  • Each chooses the lowest latency route
  • This would reach an equilibrium (pointed out by Wardrop) – Wardrop equilibrium
  • = Nash equilibrium
summary
Summary
  • Present the concept of game and Nash Equilibrium
  • Present a discrete and a continuous examples
  • Illustrate the selfish routing