Nash Equilibria inCompetitive Societies

Eyal RozenbergRoy Fox

Overview Introducing an interesting model Presenting and proving some

results: Characterizing Nash equilibria Bounding the price of anarchy

Applying the results: Competitive facility location Competitive k-median problem Selfish routing

The Model

Focus on games in which Each player has a set of available

acts The action of a player is a subset

of her available acts Some of the actions are feasible,

though some may not be

Action Profiles A profile is a vector of actions, one

for each player Profile operators

kiii baBA 1 kiii baBA 1 kiii baBA 1\\

kiiii aaaaaaA ,,,,,, 111

ii bakiBA 1

Utility Functions Each player has a private utility

function There is a social utility function Assumptions

All functions are measured in the same units

The social utility function is submodular A player loses more from dropping out

than the society does

Set Functions

A function is called non-decreasing if

A function is called submodular if

The discrete directed derivative of at in direction is

f

f

f

XfDXfXfD

XVD \

YfXfVYX

VX

YXfYXfYfXfVYX ,

Submodular Functions

Equivalent definitions

YXfYXfYfXfVYX ,

BfAfBVDVBA DD \,

BfAfBVvVBA vv \,

Notation

Set of available acts Action space - set of feasible

actions

Strategy space - set of distributions on

,

iV

iA

iAiS

k

ii

1

AA

k

ii

1

SS

ii Va

Notation (2)

Private utility function Social utility function For convenience, require For , define

0SS AfESf

SA~

i

Utility Systems

Submodular social utility function Validity:

It follows that AAA

k

ii

k

iiai

11

iai AAAi

A

AAAk

ii

1

A

Ascent Lemma

Denote Ascent Lemma (special case):

Generally:

kiii aaA ,,,,, 11

k

i

iabAB BAABAii

1

1\\, A

k

i

ia AAAi

1

1A

Basic Utility Systems

Equality in 2nd requirement

3rd requirement follows For every submodular there

exists a utility system - the basic one

iai AAAi

A

Mixed Strategies

Requirements hold is sumbodular

Ascent Lemma holds

SSSk

ii

1

S

isi SSSi

S

k

i

istST TSSTSii

1

1\\, S

Example: The Oil Game

There are nations, each having barrels of oil

The utility of each nation is the square root of the number of barrels it exports

The social utility is the sum of private utilities

k n

ii aA

For an optimal solution and any Nash equilibrium

If is non-decreasing

The Price of Anarchy

ASS

k

i

is

k

i

is SSS

iiii1

1\

1

1\

k

i

is

siis SSS

ii

ii

i1

1\

:

0,max2

S 2

Improved Bound

For a non-decreasing, submodular , define its discrete curvature

f

D

D

fVD f

DVff

D

\1max

0,

v

v

fVvki f

vVf

vi

\1max

0,,1

ABABVDVBA DDD 1\,

\12 SS

Pure Strategy Equilibrium

In a basic utility system, there is a pure strategy Nash equilibrium

The game graph is acyclic Nodes - pure strategies Edges - improving changes for some

player If player improves from to

i A B

0 ABAB ii

Facility Location Problem A bipartite graph

Locations - cost of building a facility Markets - value of serving customers Edges - cost of serving from

k-median problem - restricted action set

For a choice of locations The actual cost of serving is The price charged from is

vc

u

vu u v

u

u

vuAv

u A

min

uu Ap

A

Utility Functions

The player is maximizing

No consumer surplus

The player inadvertently maximizes the total surplus

Av

vUu

uu cAApA

0Uu

uu ApA

Av

vUu

uu cAAAA

Competitive Version (CFL) Cost for firm of building a facility in Value of serving Cost for firm of serving from

For a choice of locations The cost for firm of serving is The winning firms are The actual cost of serving is The price charged from is

, for some

ivc i

i

i

v

u u

u

u

u

ivu v

A i

vuAv

iu

i

A

min AAI i

uki

u

1minarg

AAp ju

ijkju

,1min

AIi u

u AA iu

kiu

1min

Utility Functions in CFL

Denote Firm is maximizing

The consumer surplus is

The total surplus is

ii Av

iv

AUu

iuui cAApA

Uu

uu ApA

k

i Av

iv

Uuuu

k

ii

i

cAAA11

iAIuAU ui :

i

CFL Fits the Model

The total surplus is submodular Marginal costs are supermodular

In the absence of fixed costs, the total surplus is non-decreasing

The system is basic When a player joins, the increase in

consumer surplus matches the decrease in the other players’ profits

Results for CFL

In the absence of fixed costs

There is a pure strategy Nash equilibrium

These results are tight

SFCSSFCS 2\2

SSS 2\12

Selfish Routing There are many copies of each path The amount of flow is the number of

copies chosen by a player Player gains value from each unit

of flow she routes The social utility is the sum of

private utilities

i i

iPp

piiii AlpaaA

Selfish Routing Fits the Model

The social surplus is submodular Latencies is supermodular

The 2nd requirement holds The system is valid We can restrict the action sets to

allow only the correct amount of flow

Results for Selfish Routing

Choose to get the

Roughgarden-Tardos double-rate result

If is non-decreasing

k

i

is

siis SSS

ii

ii

i1

1\

:

0,max2

SS 2 S

S2

Questions?