Game Theory Basics Game theory is designed to model How rational - - PowerPoint PPT Presentation

SLIDE 1

Game Theory Basics

• Game theory is designed to model
• How rational (payoff-maximizing) ``agents” will behave
• When individual outcomes are determined by collective behavior.
• Rules of a game specify agent payoffs as a function of actions taken by

different agents.

SLIDE 2

Let’s play the median game

• On the index card, write down
• Your name
• An integer between 0 and 100 (inclusive).
• After we collect all the index cards, the person (or people) whose

selected number is closest to 2/3 of the median of all the numbers (rounded down) wins a prize.

• E.g., if the numbers are 3, 4, 5, 38, 60, 70, 70, 90, 100

SLIDE 3

Prisoner's Dilemma

Prisoner

prison

Confessbetray is best

response no matter what

dominates

stay

silent

Betray Betray is

a dominant strategy

Not

aPareto

• ptimal strategypair

Gamer

• f

players n

for each player

Si

setof actions that playerI

8

represents

can take

A strategy profiles

s

s

n

si es

Vi

setgsaeeigesdlstmkg.es

Ui 5

is payoff toplayeri

when

the players play

strategyprofiles

SLIDE 4

esin

8 i

i

strategyprofile

for all players but i

strictly

SLIDE 5

SLIDE 6

S

Is P l

s

t

t2

ISP

2

So

to

it

SLIDE 7

Another setting

P2P

networks

free reeding try eachhave

ahh that

is desired byother decision

to upload desired

B

file

• r not

benefit ofrecovery file

3

A

cost ofuploading file not uploadingis

a dominantstrategy

pollution Game

n

countries

Yes or

no to legislationto control

Downtoncontrol costs

3

pollutionemissions

each countrythat pollutes odds

g cost to all countries

k

countries

are

polluting

n

k I

aren't

pollute

don'tpollute dominant strategy

Its

Icts

to pollute

SLIDE 8

Startup Game

Q

whether to enter aantain

Market

• r not

Startup

for Microsoft

Microsoft

f

Entering dominates

staying

• ut

Therefore

startup

can safely

assume

that

Microsoft

will do

so

Microsoft

stamp

Enter

Stay out

is

a

Nasheguilitium

i.e each

player is

best responding

to

the other

SLIDE 9

players plays

si

e Si

n

SLIDE 10

b

is weaklydominated

by

a

ipuilb

O

andFs

www.nfafs i yuiCb5 i

bis stronglydominated

by

a if

V s

iaifa.si

uicb.si

caveat

• nly good predicting

deletedonly

strongly

dominated

• therwise predict

retunigus

strategies

depends on

• rder

SLIDE 11

• On the index card, write down
• Your name
• An integer between 0 and 100 (inclusive).
• After we collect all the index cards, the

person (or people) whose selected number is closest to 2/3 of the median

• f all the numbers (rounded down)

wins a prize.

Back to the median game

deepens

mostmedian

Yzmed 66

SLIDE 12

Coordination Game

Bob

c

Hia

f

G

SLIDE 13

Network coordination

games

each node is

person

achonset

useheappornot

kw

l

ky

2

O

OP.sn

nfiIfY

dgsheEitaEog

x

SITE

O's

all

around

all

used network cascade

SLIDE 14

ProgProji

Nys

T

W

As

Qu neo Ssi ueu

maffine gratify

bsmynofrgy us news

highestquality

synergist

highest synergy

holist highest quality

synergy

random

SLIDE 15

Parking

Game

Inspector

s

i

locip z p Ga p

9

I 9

Effect

inspect

p 344 p P

sp

4

Fete

log aoa g

l p

p

legalbetterthan illegal

loci p

p

dtp

log 90kg

p

p

90

loog 45

is

a Nasihnf

9mixedshatg.es

9

Patent o

alwin rob

al wihnprbtg

mspeetargjnspezf.io

SLIDE 16

Xi G

Prob that

playeri plays

strategy

s

play

Xj

mixed

strategy

Expected payef utilityofplayers

JPY9

when he

plays

s

SLIDE 17

Nash

This game has 2pm equilibria

It

also has

a mixed

NE

where each

player parties

with

prob's

b stays home

• fprob 42

sp

I II

410

2

p I

exp payoff

L

lowerthan

both

pneeg

SLIDE 18

SLIDE 19

Summary so far

• A Nash equilibrium is a set of stable (possibly mixed) strategies.
• Stable means that no player has an incentive to deviate given what the
• ther players are doing.
• Pure equilibrium: there may be none, unique or multiple. Can be identified

with “best response diagrams”.

• A joint mixed strategy for n players:
• A probability distribution for each player (possibly different)
• It is an equilibrium if
• For each player, their distribution is a best response to the others.
• Only consider unilateral deviations.
• Everyone knows all the distributions (but not the outcomes of the coin flips).
• Nash’s famous theorem: every game has a mixed strategy equilibrium.

SLIDE 20

Issues

• Does not suggest how players might choose between different

equilibria

• Does not suggest how players might learn to play equilibrium.
• Does not allow for bargains, side payments, threats, collusions, “pre-

play” communication.

• Computing Nash equilibria for large games is computationally

difficult.

SLIDE 21

Other issues

• Relies on assumptions that might be violated in the real world
• Rationality is common knowledge.
• Agents are computationally unbounded.
• Agents have full information about other players, payoffs, etc.

SLIDE 22

Zero-sum games

are

payop gain

Penalty kicks

grow

player

Goalie

suppose

Kicher says

wer

Luh

pnb p it

loss ofGabeygoesR

tpfo.se

P

E

gosh

20,9 Q4p 0.9 0.48

foiled

• .ee

InpoxmmfEIyE7sr

p

SLIDE 23

kicker goes first

0.2ps 0.8

goaliegoes

ugh

Suppose kicker

0.9

ps

• .scp

must

announce p

0.2

Feast

gained.gs Tuat

is kicker's best

aap0.8 0.9 0.410

choice

for

P

I

• 6p

al

Ete p

p

Choosing

p f

kick leftafprob f

Kick right 4 prob Z

maximizes Kickers

expected gain if

she

has to

announce

first

i.e

is

the p that

maximizes

min

p 0.811 p

0.5 pto 9 Kp

If

kicker plays

p

she guarantees herself

an expected

payoff of

Latosft E

Eg

Y

Goalie

goes first

I

• .iq to 8

kickergoesnght

Supposes

kicker

099 0.811

91

gets to bestrespond

08

to Goalies mixed

1 asg

kicker

goesleft

strategy

q

dive

059

1 9

left

C g

dive

µ

a

g

www.qq.my yes

3 g

9g

choice for q

3

0.025

choosing

GE

dire left

with prob

and

dive

right by prob 73

minimizes

goalies expected

loss if

he has to

announce first

i.e

minimizes

moxfoagto.SC g

0.5Gt

9

If

Goalie goes first

he

can

guarantee himself a

loss of

at most

0.9 zt08Zz

V2

SLIDE 24

Y

exp payoff

the kicker can guarantee himself

if

he

has to go first

V2 exppayoffthe

kicker

can

guarantee himself

if

he

can go second

hp loss

he goalie

can guarantee if

he has

gofirst

V

E Vy

Viva

SLIDE 25

Summary – zero-sum games

• Zero-sum games have a “value”.
• Optimal strategies are well-defined.
• Maximizer can guarantee a gain of at least V by playing p*
• Minimizer can guarantee a loss of at most V by playing q*.
• This is a Nash equilibrium.
• In contrast to general-sum games, optimal strategies in zero-sum

games can be computed efficiently (using linear programming).

SLIDE 26

1500 penaltykicks 0.423

0.5577

actual

• bserved

fractions

942

0.58

• ptimal strategies

in game

0.4

0.38 0.6

0.62

SLIDE 27

Extensive

Form Games

nd White

HID

SLIDE 28

MutualAssured

Destination

A aggressive B benign

gampy

a

SLIDE 29

SLIDE 30

Centipede

so far

gamesg perfectinfo

SLIDE 31

LangeCompany

vs startup

startup announces a

technology that

threatens

big company

Big

isprob that BC

company

canpull together

coup product

Startup

SLIDE 32

Repealed

Prisoners Dilemma

Infinitely

repeated game

with discounting

IIcountdBp

ayff

thefts PTI

P prob game continues

SLIDE 33 Grim Trigger

I p

g

Gnmtriggeri

Cooperate until around

in which your opponent

defects

then

defect

from

then

• n

finthgger

Guthger

NE

Tiffortat DtfortatJ

I

8ptt2

P

vs

6

j PJ

Gj Pt

8ptt2 pJ

when

2ptL4 pi

YE

i p

a

213

posts

SLIDE 34

Titfortat

Cooperate

in round1

mercy round

k

I

play what your

• pponent

played

in

round K l