introduction to network mathematics (3) - simple games and applications yuedong xu 16/05/2012
TRANSCRIPT
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
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
Pri
soner
1
Confess Don’t Confess
Confess -5, -5 0, -10
Don’t Confess
-10, 0 -2, -2
Prison’s Dilemma
Prisoner 2
Pri
soner
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 NEs
Prisoner 2
Pri
soner
1
Confess Don’t Confess
Confess -5, -5 -3, -10
Don’t Confess
-10, -3 -2, -2
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.
Selfish Routing
• Braess’s Paradox
s tx 1
x1
0s tx 1
x1
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
s tx 1
1
x10
0
10 1s t
x 1.5
x1.5
.5
.5
Think of green flow – it has no incentive to deviate
Adding 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