Introduction to Network Mathematics (3) - Simple Games and applications
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Introduction to Network Mathematics (3)
- Simple Games and applications
Yuedong Xu16/05/2012
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Outline• Overview• Prison’s Dilemma• Curnot Duopoly• Selfish Routing• Summary
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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!
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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?
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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
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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
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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
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Outline• Overview• Prison’s Dilemma• Curnot Duopoly• Selfish Routing• Summary
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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)
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Prison’s DilemmaPrisoner 2
Priso
ner
1
Confess Don’t Confess
Confess -5, -5 0, -10
Don’t Confess
-10, 0 -2, -2
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Prison’s DilemmaPrisoner 2
Priso
ner
1
Confess Don’t Confess
Confess -5, -5 0, -10
Don’t Confess
-10, 0 -2, -2
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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.
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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 ……
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Prison’s Dilemma – Two NEsPrisoner 2
Priso
ner
1
Confess Don’t Confess
Confess -5, -5 -3, -10
Don’t Confess
-10, -3 -2, -2
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Prison’s Dilemma – No NERock-Paper-Scissors game:
If there exists a NE, then it is simple to play!
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Outline• Overview• Prison’s Dilemma• Curnot Duopoly• Selfish Routing• Summary
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Curnot DuopolyBasic 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
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Curnot Duopoly• A’s profit:
• Strategy: quantity that they produce• Market price p: p = 100 - (qA + qB)
• Question: optimal quantity for A&B
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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?
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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
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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.
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Outline• Overview• Prison’s Dilemma• Curnot Duopoly• Selfish Routing• Summary
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Selfish Routing• Braess’s Paradox
s tx 1
x10s t
x 1
x1
Traffic of 1 unit/sec needs to be routed from s to tWant to minimize average delayBraess 1968, in study of road traffic
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Selfish Routing• Before and after
s tx 11
x1 001
0 1s tx 1.5
x1 .5
.5
.5
Think of green flow – it has no incentive to deviateAdding a 0 cost link made average delay worse!!!
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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
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Summary• Present the concept of game and
Nash Equilibrium
• Present a discrete and a continuous examples
• Illustrate the selfish routing
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Thanks!