Introduction to Network Mathematics (3)

- Simple Games and applications

Yuedong Xu16/05/2012

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• 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?

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 DilemmaPrisoner 2

Priso

ner

1

Confess Don’t Confess

Confess -5, -5 0, -10

Don’t Confess

-10, 0 -2, -2

Prison’s DilemmaPrisoner 2

Priso

ner

1

Confess Don’t Confess

Confess -5, -5 0, -10

Don’t Confess

-10, 0 -2, -2

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

• More advanced games– Playing game with partial information– Evolutionary behavior– Algorithmic aspects– and more ……

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

Prison’s Dilemma – No NERock-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 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

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• 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

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!!!

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• Present the concept of game and

Nash Equilibrium

• Present a discrete and a continuous examples

• Illustrate the selfish routing

Thanks!