1 Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב INTRODUCTION.

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Issues on the border of economics and computation

וחישוב כלכלה בגבול נושאים

INTRODUCTION

Instructors

• Dr. Liad Blumrosen "בלומרוזן ליעד ר ד– Department of economics, huji.

• Dr. Michael Schapira "שפירא מיכאל ר ד– School of computer science and engineering, huji.

• Office hours: by appointment.

Course requirements

• Attend (essentially all) classes.

• Solve 3-4 problem sets.– The final problem set might be slightly bigger.

• Problem sets grade is 100% of the final grade.– No exam, no home exam.

Computer science and economics ?!?

Today:– Introduction and examples– Game theory 1.0.1.

Classic computer science

What a single computer can compute?

Classic Economics

Analyzing the interaction between humans, firms, etc.

New computational environments

• Properties:– Large-scale systems, belong to various economic

entities.– Participants are individuals/firms with different goals.– Participants have private information.– Rapid changes in users behavior.

Electronic markets Information providers

Social networks P2P networks

Internet

Mobile and apps

Algorithmic game theory

• Which tools can we use for analyzing such environments?

• Interactions between computers, owned by different economic entities and different goals.

• New tools should be developed: algorithmic game theory

• The theory borrows a lot from each field.

What tools should we use?

“Classic” CS

Not handling, eg:

Incentives

Asymmetric information

Participation constraints

Economics / Game theory

Not handling, eg:

Tractability

Approximation

Various objectives

Algorithmic Game Theory:+ Design & evaluate systems with selfish agents.+ Real need from the industry.

Few examples

Example 1: Single-Item Auctions

2nd-price auction• Buyers submit bids• Highest bid wins• Winner pays the 2nd-

highest bid

In which auction would you bid higher?How do people behave in such auctions?Which one earns greater revenue for the seller?

1st-price auction• Buyers submit bids• Highest bid wins• Winner pays his own bid

Say that you need to sell a single (indivisible) item to a set of bidders.

How can you do that?

Example 1: Single-Item Auctions

• Auctions are part of the mechanism design literature.

• Mechanism design: economists as engineers.Design markets with selfish agent to achieve some desired goals.– Relation to computer science is straightforward.– Once a niche field in economics, now mainstream.

See this year’s Nobel prize (+ 2007, 1994)

Example 2: Sponsored-search auctions

Bla

Search results Advertisements

Example 2: Sponsored-search auctions

A real system: A simple interface short response time robustness

Selfish parties: Google vs. Yahoo vs. MSN Users Advertisers

Economic challenges, eg:

Which auction to use? Private info – how much advertisers will pay? Click Fraud Attract new advertisers payments per impression/click/action

Example 3: FCC spectrum auctions

• Multi-billion dollar auctions.

• Preferences for bundles of frequencies (Combinatorial auctions): Consecutive geographic areas. Overlaps, already owned spectrum.

• Sophisticated bidders– At&t, Verizon, Google.– Again, asymmetric information.

• Bottleneck: communication.

Example 4: selfish routing

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• Many cars try to minimize driving time.• All know the traffic congestion (גלגלצ, WAZE)

Externalities and equilibria

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• Negative externalities: my driving time increases as more drivers take the same route.

• In “equilibrium”: no driver wants to change his chosen route.

• Or alternatively:– Equilibrium: for each driver, all routes have the same

driving time.• (Otherwise the driver will switch to another route…)

Efficiency, equilibrium.

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• Our question: are equilibria socially efficient?– Would it be better for the society if someone told each

driver how to drive?

• We would like to compare:– The socially-efficient outcome.

• What would happen if a benevolent planner controlled traffic.

– The equilibrium outcome.• What happens in real life.

Network 1

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• Socially efficient outcome: splitting traffic equally– expected driving time: ½*1+½*1/2=3/4 – Exercise: prove this is efficient.

• The only equilibrium: everyone use lower edge.– Otherwise, if someone chooses upper link, the cost in

the lower link is less than 1.– Expected cost: 1*1=1

C(n)=n

C(n)=1 (million)• c(n) – the cost (driving time) to

users when n users are using this road.

• Assume that a flow of 1 (million) users use this network.

S T

Network 1

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• Conclusion:– Letting people choose paths incurs a cost– “price of anarchy”

• The immediate question: if we have a ratio of 75% for this small network, can it be much higher in more complex networks? Which networks?

C(n)=n

C(n)=1 (million)

S T

Network 2

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• In equilibrium: half of the traffic uses upper routehalf uses lower route.

• Expected cost: ½*(1/2+1)+1/2*(1+1/2)=1.5

c(n)=n

c(n)=1

S T

c(n)=n

c(n)=1

Network 3

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• The only equilibrium in this graph:everyone uses the svwt route.– Expected cost: 1+1=2

• Building new highways reduces social welfare!?

c(n)=n

c(n)=1

S T

v

W

c(n)=n

c(n)=1

c(n)=0

Now a new highway

was constructed!

Braess’s Paradox

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• This example is known as the Braess’s Paradox:

sometimes destroying roads can be beneficial for society.

• The immediate question: how can we choose which roads to build or destroy?

c(n)=n

c(n)=1

S T

v

W

c(n)=n

c(n)=1

c(n)=0

Now a new highway

was constructed!

Example 5: Internet Routing

Establish routes between the smaller networks that make up the Internet

Currently handled by the Border Gateway Protocol (BGP).

AT&T

Qwest

Comcast

Level3

Why is Internet Routing Hard?

Not shortest-paths routing!!!

AT&T

Qwest

Comcast

Level3

My link to UUNET is for backup purposes only.

Load-balance myoutgoing traffic.

Always chooseshortest paths.

Avoid routes through AT&T if at all possible.

BGP Dynamics

1 2

d

2, I’m available

1, my routeis 2d

1, I’m available

Prefer routes

through 2

Prefer routes

through 1

Two Important Desiderata

• BGP safety

– Guaranteeing convergence to a stable routing state.

• Compliant behaviour.– Guaranteeing that nodes (ASes)

adhere to the protocol.

• We saw examples for modern systems that raise many interesting questions in algorithmic game theory.

• Next:a quick introduction to game theory

• Outline:– What is a game?– Dominant strategy equilibrium– Nash equilibrium (pure and mixed)

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Game Theory• Game theory involves the study of strategic

situations

• Portrays complex strategic situations in a highly simplified and stylized setting– Strategic situations: my outcome depends not only

on my action, but also on the actions of the others.

• A central concept: rationality– A complex concept. Many definitions.– One possible definition:

Agents act to maximize their own utility subject to the information the have and the actions they can take.

Applications• Economics

– Essentially everywhere• Business

– Pricing strategies, advertising, financial markets…• Computer science

– Analysis and design of large systems, internet, e-commerce.• Biology

– Evolution, signaling, …• Political Science

– Voting, social choice, fair division…• Law

– Resolutions of disputes, regulation, bargaining…• …

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Game Theory: Elements• All games have three elements

– players– strategies– payoffs

• Games may be cooperative or noncooperative– In this course, noncooperative games.

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• Let’s see some examples….

Example 1: “chicken”Chicken!!!

Swerve Straight

Swerve 0, 0 -1, 1Straight 1, -1 -10,-10

Example 2: Prisoner’s Dilemma• Two suspects for a crime can:

– Cooperate (stay silent, deny crime).• If both cooperate, 1 year in jail.

– Defect (confess).• If both defect, 3 years (reduced since they confessed).

– If A defects (blames the other), and B cooperate (silent) then A is free, and B serves a long sentence.

Cooperate Defect

Cooperate -1, -1 -5, 0Defect 0, -5 -3,-3

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Lecture Outline• What is a game?

– Few examples.

Best responses

• Dominant strategies

• Nash Equilibrium– Pure– Mixed

• Existence and computation

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Notation• We will denote a game G between two

players (A and B) by

G[ SA, SB, UA(a,b), UB(a,b)]

whereSA = set of strategies for player A (a SA)

SB = set of strategies for player B (b SB)

UA : SA x SB R (utility function for player A)

UB : SA x SB R utility function for player B

Normal-form game: Example

• Example:– Actions:

SA = {“C”,”D”}SB = {“C”,”D}

– Payoffs:uA(C,C) = -1, uA(C,D) = -5, uA(D,C) = 0, uA(D,D) = -3

Cooperate Defect

Cooperate-1, -1 -5, 0

Defect0, -5 -3,-3

A best response: intuition

• Can we predict how players behave in a game?

First step, what will players do when they know the strategy of the other players?

• Intuitively: players will best-respond to the strategies of their opponents.

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A best response: Definition

• When player B plays b. A strategy a* is a best response to b if

UA(a*,b) UA(a’,b) for all a’ SA

(given that B plays b, no strategy gains A a higher payoff than a*)

A best response: example

Example:When row player plays Up,what is the best response of the column player?

Left Right

Up 1,1 0,0Bottom 0,0 1,1

Left Right

Up 1,1 0,0Bottom 0,0 1,1

Dominant Strategies ( / דומיננטיות שולטות (אסטרטגיות

• Definition: action a* is a dominant strategy for player A if it is a best response to every action b of B.

Namely, for every strategy b of B we have:

UA(a*,b) UA(a’,b) for all a’ SA

Dominant Strategies: in the prisoner’s dilemma

Cooperate Defect

Cooperate -1, -1 -5, 0

Defect 0, -5 -3,-3

• For each player: “Defect” is a best response to both “Cooperate” and “Defect.

• Here, “Defect” is a dominant strategy for both players…

• In the prisoner’s dilemma: (Defect, Defect) is a dominant-strategy equilibrium.

Dominant Strategy equilibriumשולטות באסטרטגיות משקל שווי

• Definition: (a,b) is a dominant-strategy equilibrium if a is dominant for A and b is dominant for B.– (similar definition for more players)

Cooperate Defect

Cooperate -1, -1 -5, 0

Defect 0, -5 -3,-3

Dominant strategies: another example

• Who has a dominant strategy in this game?

• Dominant-strategy equilibrium?

Left middle Right

Up 7,2 2,2 0,0Bottom 3,4 5,2 0,4

We allowed ≥ in the

definition. “Weakly

dominant”

Dominant strategies: pros and cons

• Plus: Strong solution. – Why should I play anything else if I have a

dominant strategy?

• Main problem:Does not exist in many games….

Left Right

Up 1,1 0,0Bottom 0,0 1,1

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Lecture Outline• What is a game?

– Few examples.

• Best responses

• Dominant strategies (golden balls)

Nash Equilibrium– Pure– Mixed

• Existence and computation

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Nash Equilibrium• How will players play when dominant-strategy

equilibrium does not exist?– We will define a weaker equilibrium concept: Nash

equilibrium

• A pair of strategies (a*,b*) is defined to be a Nash equilibrium if:a* is player A’s best response to b*, and b* is player B’s best response to a*.

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Nash Equilibrium: Definition

• A direct definition:A pair of strategies (a*,b*) is defined to be a Nash equilibrium if

UA(a*,b*) UA(a’,b*) for all a’ SA

UB(a*,b*) Ub(a*,b’) for all b’ SB

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Nash Eq.: Interpretation• No regret: Even if one player reveals his strategy,

the other player cannot benefit.– this is not the case with non-equilibrium strategies

• Stability: Once we reach a Nash equilibrium, players have no incentive to alter their strategies.– Even after observing the strategies of the other players

• Necessary condition for an outcome chosen by rational players.– If players think that there is obvious outcome to the

game, it must be a Nash equilibrium

(Pure) Nash Equilibrium

• Examples:

Left Right

Up 1,1 0,0Bottom 0,0 1,1

Swerve Straight

Swerve 0, 0 -1, 1Straight 1, -1 -10,-10

Note: when column player plays “straight”, then “straight” is no longer a best response to the row player.

Here, communication between players help.

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Nash vs. Dominant Strategies• Every dominant strategy equilibrium is a Nash

equilibrium.– If a strategy is a best response to all strategies of

the other players, it is of course a best response to the dominant strategy of the other.

• The opposite is not true.

Nash equilibrium: existence

• Does a Nash equilibrium always exist?

– Note:we already saw that multiple equilibria are possible.

Example 4:

• No (pure) Nash equilibrium.

• But how do people play this game?

-1,1 1,-1

1,-1 -1,1

Tail Heads

Tail

Heads

Matching Pennies ( פרט או (זוגIs this an

equilibrium?

“Pure” Nash: pros and cons

• Good:– Describes “stable” outcomes.– May exist when dominant-strategy equilibria

does not exist.– Simple and intuitive (especially when unique).

• Bad:– Not unique.

• What happens when multiple equilibria exist?

– Does not always exist!

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Mixed strategies

• Consider the following strategy:“I will toss a coin. With probability ½ I will choose bottom.With probability ½ I will choose up.”

• If lottery is allowed, now each player has an infinite number of strategies…

Left Right

Up 1,1 0,0Bottom 0,0 1,1

Mixed strategies: Definition• Definition: a “mixed strategy” is a probability

distribution over actions.– If {a1,a2,…,am} are the pure strategies of A,

then {p1,…,pm} is a mixed strategy for A if

-1,1 1,-1

1,-1 -1,1

Tail Heads

Tail

Heads

m

iip

1

1

1/2

1/2

1/3

2/3

9/10

1/10

0

1

1/4

1/2

0ip

(1)

(2) For all i

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Pure and Mixed strategies• Clearly, every pure strategy is a mixed strategy

as well.– That gives probability 1 to one of the pure strategies.

• We will simply use the term “strategies”.

Expected payoff• When the two players play mixed strategies, the

payoff is the expected payoff. (הממוצע)

L R

T 3, -1 4, 2

D 6, -5 1,9

2/3

1/3

3/41/4

• What is the payoff of the row player? when the players play sA=(2/3, 1/3) and sB=(1/4,3/4)

uA(sA,sB) = 2/3 * ¼ * 3 + 1/3 * ¼ * 6 + 2/3 * ¾ * 4 + 1/3 * ¾ * 1 = 3.25

Best response (w. mixed strategies)• Definition:

Consider a mixed strategy sB of player B.

A strategy s* for player A is a best response to sB if no other pure strategy gains A higher expected payoff.

Namely,

– Note: we will later see that this implies that no mixed strategy is better for A than s*.

UA(s*,sB) UA(a’,sB) for all a’ in SA

Best response (w. mixed strategies)

-1,1 1,-1

1,-1 -1,1

זוג פרט

זוג

פרט

3/41/4What is a best response to (1/4,3/4)?

What would you do if you knew that your opponent plays one strategy more frequently?

Will you play pure or mixed?

1

0

Mixed strategies are realistic?• Do people randomize?

– Computers? Evolution? Stock markets? Teacher choosing questions in exams.

• Model long term behavior…• Model uncertainty about the other players.

• פרט או זוג• Basketball• Soccer

– How would you define strategy in penalty kicks?

– “the player that kicks more often to the left”

Nash eq. with mixed strategies• Main idea: given a fixed behavior of the others, I

will not change my strategy.

• Definition: (SA,SB) are in Nash Equilibrium, if each strategy is a best response to the other.

-1,1 1,-1

1,-1 -1,1

זוג פרט

זוג

פרט

1/21/2

1/2

1/2

Example: Battle of the SexesEquilibria in “battle

of the sexes”:

– Two pure equilibria.

– One mixed (2/3,1/3),(1/3,2/3)

2,1 0,0

0,0 1,2

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Lecture Outline• What is a game?

– Few examples.

• Best responses

• Dominant strategies

• Nash Equilibrium– Pure– Mixed

Existence and computation

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Existence of equilibria• Dominant strategies equilibria do not exist

in every game.– Same goes for Pure Nash equilibria.

• What about Nash equilibria (with mixed strategies)?

Good news: always exist.

Nash’s Theorem• Theorem (Nash, 1950):

every game has at least one Nash equilibrium!– With some technical details about the set of strategies.– Proof uses fix-point theorems.

• Nash was awarded the Nobel prize for this work in 1994.

Nash equilibrium (with mixed strategies):– Good: always exists. Models long term stability.– Bad: Less simple and intuitive. Multiple equilibria

exist.

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Computing Equilibria• Dominant strategy: for each player, check if she

has a dominant strategy.

• Pure Nash: for each combination of actions, check if a player has a beneficial deviation.

• How can we find Nash equilibria in general?– This is a real problem in large games.

• Area of extensive research.

– Easy in “small” games.

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Summary• We learned about simultaneous-action games,

represented by a matrix of payoffs. (Games in their “normal form”)– Next topic: sequential games.

• We wanted to predict the steady/stable state behavior on the games, and defined concepts of equilibria.

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Finding mixed equilibria• We will use the following lemma:

Lemma: let sA be a best response to sB.If sA chooses the pure strategies a, a’ with positive probability, then

uA(a,sB)=uA(a’,sB)

Namely, if we sometime choose a and sometime choose a’, they gain us the same expected payoff (given a fixed behavior of the others).

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Proof of Lemma• Assume that in best response:

a is chosen with probability pa

a’ is chosen with probability pa’

– pa ,pa’ >0

• Now if uA(a,sB) > uA(a’,sB), then this is not a best response:– The same strategy that chooses

a with probability pa+pa’ and a’ with probability 0 gains A higher payoff.

pa’

pa

0

pa + pa’

sB

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Finding mixed equilibria

M S

M 2,1 0,0

S 0,0 1,2

1/43/4• What is the best response to sB=(3/4,1/4)?

• Can it be sA=(½, ½)?

uA(M,sB) = ¾*2 + 1/4*0 = 1.5

uA(S,sB) = ¾*0 + 1/4*1 = ¼Expected payoff: ½*1.5 +

½*1/4(1/2,1/2) cannot be a best response, niether (0.99,0.01)

• adding more mass to M will increase expected payoff of A. • Again, here the best response is a pure strategy (“M”),

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Finding mixed equilibria• So how can we find (strictly) mixed-strategy

equilibria?

• We will use the lemma that we proved: if in equilibrium a player plays two pure strategies with positive probability, then the expected payoff from both strategies should be the same.

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Finding mixed equilibria

M S

M 2,1 0,0

S 0,0 1,2

1-qq

• Consider an equilibrium sA=(p,1-p), sB=(q,1-q) (q,p>0)

then: uA(“M”,sB) = uA(“S”,sB) =

• If sA is a best response, we must have:

uA(“M”,sB)=uA(“S”,sB)

that is : 2q = (1-q) q=1/3

• Similarly, if sB is a best response then p=2/3.

1-p

p

q*2 + 0*(1-q)q*0 + (1-q)*1

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Finding mixed equilibria

M S

M 2,1 0,0

S 0,0 1,2

2/31/3

• Note that since all mixed strategies are best response to sB=(1/3,2/3).

But only sA=(2/3,1/3) ensures that sB=(1/3,2/3) is also a best response to sA.

1/3

2/3

uA(“M”,sB)=uA(“S”,sB)

Equilibria• All we said extends to more players:

• (s1,…,sn) is a Nash equilibrium, if for every i, s i is a best-response to the other strategies.

• (s1,…,sn) is a dominant-strategy equilibrium, if for every i, si is the best response to any other set of strategies.

EquilibriaTake home message:

• Dominant-strategy equilibrium:

my strategy is the best no matter what the others do.

Exists in some games.

• Nash equilibrium:

my strategy is the best given what the others are currently doing.

Always exists.

Example 1: coordination games

Left Right

Left 1,1 0,0

Right 0,0 1,1

Row playerהשורות שחקן

Column Playerהעמודות שחקן

Right number: utility for

Row Player

Left number: utility for Column Player

Without laws, when this game is repeated, what will happen?