Strategic Network Formation
and Group Formation
Elliot Anshelevich
Rensselaer Polytechnic Institute (RPI)
Centralized Control
A majority of network research has made the
centralized control assumption:
Everything acts according to a centrally defined
and specified algorithm
This assumption does not make sense in many cases.
Self-Interested Agents
• Internet is not centrally
controlled
• Many other settings have
self-interested agents
• To understand these, cannot
assume centralized control
• Algorithmic Game Theory
studies such networks
Agents in Network Design
• Traditional network design
problems are centrally controlled
• What if network is instead built
by many self-interested agents?
• Properties of resulting network may
be very different from the globally
optimum one
s
Goal
s
• Compare networks created by
self-interested agents
with the optimal network
– optimal = cheapest
– networks created by self-interested
agents = Nash equilibria
OPT
• Can realize any Nash equilibrium
by finding it, and suggesting it
to players
– Requires central coordination
– Does not require central control
NE
The Price of Stability
Price of Anarchy =
[Koutsoupias, Papadimitriou]
cost(worst NE)
s
cost(OPT)
1
Price of Stability =
k
cost(best NE)
cost(OPT)
Can think of latter as a network designer
proposing a solution.
t1…tk
Single-Source Connection Game
[A, Dasgupta, Tardos, Wexler 2003]
Given: G = (V,E), k terminal nodes,
costs ce for all e  E
s
Each player wants to build a network
in which his node is connected to s.
Each player selects a path, pays for
some portion of edges in path
(depends on cost sharing scheme)
Goal: minimize payments,
while fulfilling connectivity requirements
Other Connectivity Requirements
 Survivable: connect to s with two disjoint paths
[A, Caskurlu 2009]
 Sets of nodes: agent i wants to connect set Ti
[A, Dasgupta, Tardos, Wexler 2003]
 Group formation
Group Network Formation Games
Terminal Backup:
Each terminal wants to connect
to k other terminals.
Group Network Formation Games
Terminal Backup:
Each terminal wants to connect
to k other terminals.
“Group Steiner Tree”:
Each terminal wants to connect
to at least one terminal from
each color.
Other Connectivity Requirements
 Survivable: connect to s with two disjoint paths
[A, Caskurlu 2009]
 Sets of nodes: agent i wants to connect set Ti
[A, Dasgupta, Tardos, Wexler 2003]
 Group formation: every agent wants to connect to a group
that provides enough resources
satisfactory group specified by a monotone set function
[A, Caskurlu 2009]
Centralized Optimum
 Single-source Connection Game:
Steiner Tree.
 Sets of nodes: Steiner Forest.
 Survivable: Generalized Steiner Forest.
 Terminal Backup: Cheapest network
where each terminal connected to at least
k other terminals.
 “Group Steiner Tree”: Cheapest where
every component is a Group Steiner Tree.
Corresponds to constrained forest problems, has 2-approx.
Connection Games
s
Given: G = (V,E), k players,
costs ce for all e  E
Each player wants to build a network
where his connectivity requirements
are satisfied.
Each player selects subgraph, pays for
some portion of edges in it
(depends on cost sharing scheme)
NE
Goal: minimize payments,
while fulfilling connectivity requirements
Sharing Edge Costs
How should multiple players
on a single edge split costs?
 One approach: no restrictions...
...any division of cost agreed upon by players is OK.
[ADTW 2003, HK 2005, EFM 2007, H 2009, AC 2009]
 Another approach: try to ensure some sort of fairness.
[ADKTWR 2004, CCLNO 2006, HR 2006, FKLOS 2006]
Connection Games with Fair Sharing
Given: G = (V,E), k players,
costs ce for all e  E
s
Each player selects subnetwork
where his connectivity requirements
are satisfied.
Players using e pay for it evenly:
ci(P) = Σ ce/ke
eєP
( ke = # players using e )
Goal: minimize payments,
while fulfilling connectivity requirements
Fair Sharing
Fair sharing: The cost of each edge e
shared equally by the users of e
Advantages:
• Fair way of sharing the cost
• Nash equilibrium exists
• Price of Stability is at most log(# players)
is
Price of Stability with Fairness
Price of Anarchy is large
Price of Stability is at most log(# players)
s
Proof: This is a Potential Game, so
 Nash equilibrium exists
 Best Response converges
 Can use this to show existence of good equilibrium
1
k
t1…tk
Fair Sharing
Fair sharing: The cost of each edge e
shared equally by the users of e
Advantages:
• Fair way of sharing the cost
• Nash equilibrium exists
• Price of Stability is at most log(# players)
Disadvantages:
• Player payments are constrained, need to enforce fairness
• Price of stability can be at least log(# players)
is
Example: Self-Interested Behavior
t
1
1
1+
1
2
1
Demands:
1-t, 2-t, 3-t
3
2
0
0
3
0
Example: Self-Interested Behavior
t
1
1
1+
1
2
1
Minimum Cost Solution
(of cost 1+)
3
2
0
0
3
0
Example: Self-Interested Behavior
Each player chooses a path P.
Cost to player i is:
t
1
1
1+
1
2
1
2
0
0
cost(i) = cost(P)
# using P
3
3
0
(Everyone shares cost equally)
Example: Self-Interested Behavior
t
1
1
1+
1
2
1
3
2
0
0
3
0
Player 3 pays (1+ε)/3,
could pay 1/3
Example: Self-Interested Behavior
t
1
1
1+
1
2
1
3
2
0
0
3
0
so player 3
would deviate
Example: Self-Interested Behavior
t
1
1
1+
1
2
1
3
2
0
0
3
0
now player 2
pays (1+ε)/2,
could pay 1/2
Example: Self-Interested Behavior
t
1
1
1+
1
2
1
3
2
0
0
3
0
so player 2
deviates also
Example: Self-Interested Behavior
Player 1 deviates as well,
giving a solution with cost 1.833.
This solution is stable/
this solution is a Nash Equilibrium.
t
1
1
1+
1
2
1
3
2
0
0
3
0
It differs from the optimal solution
by a factor of 1+ 12 + 13
Hk = Θ(log k)!
Sharing Edge Costs
How should multiple players
on a single edge split costs?
 One approach: no restrictions...
...any division of cost agreed upon by players is OK.
[ADTW 2003, HK 2005, EFM 2007, H 2009, AC 2009]
 Another approach: try to ensure some sort of fairness.
[ADKTWR 2004, CCLNO 2006, HR 2006, FKLOS 2006]
Example: Unrestricted Sharing
Fair Sharing: differs from
the optimal solution
by a factor of Hk = Θ(log k)
t
1
1
1+
1
2
1
3
2
0
0
3
0
Unrestricted Sharing:
OPT is a stable solution
Contrast of Sharing Schemes
Unrestricted Sharing
Fair Sharing
NE don’t always exist
P.o.S. = O(k)
NE always exist
P.o.S. = O(log(k))
(P.o.S. = Price of Stability)
Contrast of Sharing Schemes
Unrestricted Sharing
Fair Sharing
NE don’t always exist
P.o.S. = O(k)
P.o.S. = 1 for
many games
NE always exist
P.o.S. = O(log(k))
P.o.S. = (log(k)) for
almost all games
(P.o.S. = Price of Stability)
Contrast of Sharing Schemes
Unrestricted Sharing
Fair Sharing
NE don’t always exist
P.o.S. = O(k)
P.o.S. = 1 for
many games
OPT is an approx. NE
NE always exist
P.o.S. = O(log(k))
P.o.S. = (log(k)) for
almost all games
OPT may be far from NE
(P.o.S. = Price of Stability)
Unrestricted Sharing Model
• Player i picks payments for each
edge e.
(strategy = vector of payments)
• Edge e is bought if total payments
for it ≥ ce.
• Any player can use bought edges.
What is a NE in this model?
Unrestricted Sharing Model
• Player i picks payments for each
edge e.
(strategy = vector of payments)
• Edge e is bought if total payments
for it ≥ ce.
• Any player can use bought edges.
What is a NE in this model?
Payments so that no players want to
change them
Unrestricted Sharing Model
• Player i picks payments for each
edge e.
(strategy = vector of payments)
• Edge e is bought if total payments
for it ≥ ce.
• Any player can use bought edges.
What is a NE in this model?
Payments so that no players want to
change them
Connection Games with Unrestricted Sharing
Given: G = (V,E), k players,
costs ce for all e  E
s
Strategy: a vector of payments
Players choose how much to pay,
buy edges together
Cost(v) =

if v does not satisfy connectivity requirements
Payments of v otherwise
Goal: minimize payments,
while fulfilling connectivity requirements
Connectivity Requirements
 Single-source: connect to s
 Survivable: connect to s with two disjoint paths
 Sets of nodes: agent i wants to connect set Ti
 Group formation: every agent wants to connect to a group
that provides enough resources
satisfactory group specified by a monotone set function
Some Results
 Single-source: connect to s
 Survivable: connect to s with two disjoint paths
If k=n
 Sets of nodes: agent i wants to connect set Ti
 Group formation: every agent wants to connect to a group
that provides enough resources
satisfactory group specified by a monotone set function If k=n
OPT is a Nash Equilibrium (Price of Stability=1)
Some Results
 Single-source: connect to s
=1
 Survivable: connect to s with two disjoint paths
=2
 Sets of nodes: agent i wants to connect set Ti
=3
 Group formation: every agent wants to connect to a group
that provides enough resources
satisfactory group specified by a monotone set function =2
OPT is a -approximate Nash Equilibrium
(no one can gain more than  factor by switching)
Some Results
 Single-source: connect to s
=1
 Survivable: connect to s with two disjoint paths
=2
 Sets of nodes: agent i wants to connect set Ti
=3
 Group formation: every agent wants to connect to a group
that provides enough resources
satisfactory group specified by a monotone set function =2
If we pay for 1-1/ fraction of OPT,
then the players will pay for the rest
Some Results
 Single-source: connect to s
 Survivable: connect to s with two disjoint paths
 Sets of nodes: agent i wants to connect set Ti
 Group formation: every agent wants to connect to a group
that provides enough resources
satisfactory group specified by a monotone set function
Can compute cheap approximate equilibria in poly-time
Contrast of Sharing Schemes
Unrestricted Sharing
Fair Sharing
NE don’t always exist
P.o.S. = O(k)
P.o.S. = 1 for
many games
OPT is an approx. NE
NE always exist
P.o.S. = O(log(k))
P.o.S. = (log(k)) for
almost all games
OPT may be far from NE
(P.o.S. = Price of Stability)
Contrast of Sharing Schemes
Unrestricted Sharing
Fair Sharing
NE don’t always exist
P.o.S. = O(k)
P.o.S. = 1 for
many games
OPT is an approx. NE
NE always exist
P.o.S. = O(log(k))
P.o.S. = (log(k)) for
almost all games
OPT may be far from NE
(P.o.S. = Price of Stability)
Contrast of Sharing Schemes
Unrestricted Sharing
Fair Sharing
NE don’t always exist
P.o.S. = O(k)
P.o.S. = 1 for
many games
OPT is an approx. NE
NE always exist
P.o.S. = O(log(k))
P.o.S. = (log(k)) for
almost all games
OPT may be far from NE
If we really care about efficiency:
Allow the players more freedom!
Example: Unrestricted Sharing
Fair Sharing: differs from
the optimal solution
by a factor of Hk  log k
t
1
1
1+
1
2
1
3
2
0
0
3
0
Unrestricted Sharing:
OPT is a stable solution
Every player gives what
they can afford
General Techniques
To prove that OPT is
an exact/approximate equilibrium:
 Construct a payment scheme
 Pay in order:
laminar system of witness sets
 If cannot pay, form deviations
to create cheaper solution
Te
T'
u
e
pi
 i ( pi )
Te
Network Destruction Games
• Each player wants to protect itself from untrusted nodes
• Have cut requirements:
must be disconnected from set Ti
• Cutting edges costs money
• Can show similar results for:
Multiway Cut, Multicut, etc.
Thank you.