Nash Equilibria in
Competitive Societies
Eyal Rozenberg
Roy 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
 A  ai  a1 , , ai 1 , ai , ai 1 , , a k 




A  B  ai  bi i 1
k
A  B  ai  bi i 1
k


A \ B  ai \ bi i 1
A  B  1  i  k ai  bi
k
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 f is called non-decreasing if
X  Y  V f  X   f Y 


A function f is called submodular if
X , Y  V f  X   f Y   f  X  Y   f  X  Y 
The discrete directed derivative of f at
X  V in direction D  V \ X is
f D  X   f  X  D  f  X 
Submodular Functions
Equivalent definitions

X , Y  V f  X   f Y   f  X  Y   f  X  Y 

A  B  V , D  V \ B f D  A  f D B

A  B  V , v  V \ B fv  A  fv B 
Notation


Set of available acts Vi
Action space A i - set of feasible actions
ai  Vi


Strategy space S i - set of distributions
on Aki
k
A   Ai , S   S i
i 1
i 1
Notation (2)




Private utility function  i
Social utility function 
For convenience, require    0
For S  S , define f S   A~ES f  A
Utility Systems

Submodular social utility function

A  A  i  A   ai  A   i 

Validity: A  A

Itk follows that
k
   A    A
i
i 1
k
    A        A    A
i 1
ai
i
i 1
i
Ascent Lemma



i
A
 a1 , , ai ,  i 1 , ,  k 
Denote
Ascent Lemmak (special case):

A  A   A    ai A i 1
Generally:

i 1
k

A, B  A  B \ A  A    bi \ ai A  B i 1
i 1

Basic Utility Systems



Equality in 2nd requirement
A  A  i  A   ai  A   i 
3rd requirement follows
For every submodular  there exists a
utility system - the basic one
Mixed Strategies

Requirements hold

 is sumbodular
S  S  i S    s S   i 

S  S

  S    S 
i 1

i
k
i
Ascent Lemma holds
k
i 1

 S , T  S  T \ S S     t \ s S  T
i 1
i
i
Example: The Oil Game



There are k nations, each having n
barrels of oil
The utility of each nation is the square
root of the number of barrels it exports
 i  A  ai
The social utility is the sum of private
utilities
The Price of Anarchy

For an optimal solution   A and any
Nash equilibrium S  S
     S      S           S 
k
i 1
i 1
   2 S  
i \ si
i 1

i 1
si \

i
k

i 1




max

S


,
0




S

 si \i
si
i
i:i  si

k
If  is non-decreasing
   2 S 
i 1

Improved Bound

For a non-decreasing, submodular f ,
define its discrete curvature

f D V \ D  
 
  f   max 1 
D V , f D  0
f D  


f v V \ v 



max
1
1i  k ,vVi , f v    0
f v   



A  B  V , D  V \ B  D  A   D B  1     D  A
   2 S   1     S \ 
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 i improves from A to B
 B     A   i B    i  A  0


Facility Location Problem

A bipartite graph





Locations - cost cv of building a facility
Markets - value  u of serving customers
Edges - cost vu of serving u from v
k-median problem - restricted action set
For a choice A of locations
vu
 The actual cost of serving u is u  A  min
v A
 The price charged from u is pu  A   u
Utility Functions

The player is maximizing
  A    pu  A  u  A   cv
uU

No consumer surplus
  A    u  pu  A  0
vA
uU

The player inadvertently maximizes the
total surplus
  A    A    A    u  u  A   cv
uU
vA
Competitive Version (CFL)
Cost cvi for firm i of building a facility in v
 Value  u of serving u
i
 Cost  vu for firm i of serving u from v
For a choice A of locations
i
i



A

min

u
vu
 The cost for firm i of serving u is
vA
i


I
A

arg
min

u
u  A
 The winning firms are
1i  k
i



A

min

u
 The actual cost of serving
is u
u  A
1i  k
j


p
A

min

u
u  A,
 The price charged from
is u
1 j  k , j i
for some i  I u  A


i
Utility Functions in CFL


i
Denote U i  A  u : I u  A  
Firm i is maximizing
i
i
 i  A    pu  A  u  A   cv
uU i  A 

vAi
The consumer surplus is
  A    u  pu  A
uU

The total
surplus is
k
k
  A    i  A    A    u  u     cvi
i 1
uU
i 1 vAi
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




   2 S   FCS \   2 S   FCS 
In the absence of fixed costs
   2 S   1     S \   2 S 
There is a pure strategy Nash
equilibrium
These results are tight
Selfish Routing





There are many copies of each path
The amount of flow is the number of
copies chosen by a player
Player i gains value  i from each unit of
flow she routes
 i  A  ai   i   ai  p  l p  A
pPi
The social utility is the sum of private
utilities
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

   2 S  

2 S      S 




k

i 1




max

S


,
0




S

 si \i
si
i
i:i  si
i 1
Choose   S   to get the
Roughgarden-Tardos double-rate result
If  is non-decreasing
2 S    

Questions?