IE 598: Games, Markets, and Mathematical Programmingjugal.ise.illinois.edu/598/lec1.pdf · •InStuttgart,Germany, after investments into the road network in 1969, the traffic situation

Jugal Garg

IE 598: Games, Markets, and Mathematical Programming

Course webpage Grading Policy Office Hours

Topics Games and Equilibrium Concepts Routing Games and Price of Anarchy Mechanism Design Markets

Game Theory

Game Theory

Game Theory

Analysis/Design of a system whererational agents interact to achieve selfish goals

Multiple self-interestedagents interacting in the

same environment

Deciding what to do

Q: What to expect? How good is it? Can it be controlled?

Prisoner’s Dilemma

Two thieves caught for burglaryTwo options: {confess, remain silent}

Prisoner’s DilemmaTwo thieves caught for burglary.Two options: {confess, remain silent}

C

S

SC

-5 -5

-1 -1

0 -20

-20 0

Prisoner’s DilemmaTwo thieves caught for burglary.Two options: {confess, remain silent}

C

S

SC

-5 -5

-1 -1

0 -20

-20 0

Only stable state

Tragedy of commonsLimited but open resource shared by many

Stable: Over use => Disaster

Bad outcome!

Rock-Paper-Scissors

Rock-Paper-Scissors

Rock-Paper-Scissors

R

P

SR

0 0

-1 1

1 -1

1 -1

No pure stable state!

P

S

0 0

0 0-1 1

-1 1

1 -1

Both playing (1/3,1/3,1/3) is the only stable state. Why?

Chicken

S

C

S

S

C

C

C S

Chicken (Traffic Light)

S

C

S

S

C

C

C S

Routing Games and Price of Anarchy

Braess’ Paradox

s t

/100 hours

/100 hours1 hour

1 hour

100 commuters50

50

Commute time: 1.5 hours

u

v

Braess’ Paradox

s t

/100 hours

/100 hours1 hour

1 hour

100 commuters50

50

Commute time: 1.5 hours

0 hours~1 hour

u

v

Braess’ Paradox

s t

/100 hours

/100 hours1 hour

1 hour

100 commuters

Commute time: 2 hours

0 hours

100

u

v

Braess’ Paradox

s t

/100 hours

/100 hours1 hour

1 hour

100 commuters

0 hours

100

Price of Anarchy: .

Can’t be worse!

u

v

• In Stuttgart, Germany, after investments into the road network in 1969, the traffic situation did not improve until a section of newly built road was closed for traffic again.

• In 1990 the temporary closing of 42nd Street in New York City for Earth Day reduced the amount of congestion in the area.

• In 2009, New York experimented with closures of Broadway at Times Square and Herald Square, which resulted in improved traffic flow and permanent pedestrian plazas.…

Source: Wikipedia.org

True Stories

Mechanism Design

Single Item Auction

Vickery Auction Sealed bid auction Highest bidder wins and pays the second highest bid

Properties: Truthful, Incentive Compatible, Welfare Maximization

Adwords Auction

Markets

At equilibrium: Demand = Supply

A Single Good

Pric

e

Quantity∗

∗Equilibrium

More than one good

Demand of a good now depend on price of other goods

Markets

Equilibrium: Demand = Supply

BuyersGoods

$10

$5 $25

Buyers have preferences over goods

demand > supply? demand < supply?

Q: Does there exist an equilibrium? If yes, how to compute it?Do buyers have incentive to lie about their preferences?

want best affordable bundle

Housing Market

$250K

Buyers Houses

Each buyer wants to buy exactly one housewhich maximizes valuation – payment and affordable

if house is allocated to buyer otherwise

price of house

valuation of buyer for house

Maximize ,

if house is allocated to buyer otherwise

price of house

valuation of buyer for house

Maximize ,

Play A Game• Each of you will put $[0, 10] in a pot in a sealed envelope • M ←Total money in the pot • Each will receive ← 2*M/#people

What is the best strategy that maximize your own payoff?

Next Class • Nash equilibrium concept• Zero-sum games• Minimax theorem and LP duality