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
• Compare networks created by self-interested agents with the optimal network– optimal = cheapest
– networks created by self-interested agents = Nash equilibria
• Can realize any Nash equilibrium by finding it, and suggesting it to players– Requires central coordination
– Does not require central control
OPT
NE
s
The Price of Stability
Price of Anarchy = cost(worst NE)
cost(OPT)
Price of Stability = cost(best NE)
cost(OPT)
[Koutsoupias, Papadimitriou]s
t1…tk
1 k
Can think of latter as a network designer proposing a solution.
Single-Source Connection Game[A, Dasgupta, Tardos, Wexler 2003]
Given: G = (V,E), k terminal nodes, costs ce for all e E
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)
s
Goal: minimize payments,while fulfilling connectivity requirements
Other Connectivity Requirements
Survivable: connect to s with two disjoint paths
Sets of nodes: agent i wants to connect set Ti
Group formation
[A, Caskurlu 2009]
[A, Dasgupta, Tardos, Wexler 2003]
Group Network Formation Games
“Group Steiner Tree”: Each terminal wants to connect to at least one terminal from each color.
Terminal Backup: Each terminal wants to connect to k other terminals.
Other Connectivity Requirements
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
[A, Caskurlu 2009]
[A, Dasgupta, Tardos, Wexler 2003]
[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
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)
s
Goal: minimize payments,while fulfilling connectivity requirements
NE
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
Each player selects subnetwork where his connectivity requirements are satisfied.
Players using e pay for it evenly: ci(P) = Σ ce/ke
( ke = # players using e )
s
Goal: minimize payments,while fulfilling connectivity requirements
e є P
Fair Sharing
Fair sharing: The cost of each edge e is 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)
Price of Stability with Fairness
Price of Anarchy is large
Price of Stability is at most log(# players)
Proof: This is a Potential Game, so Nash equilibrium exists Best Response converges Can use this to show existence of good equilibrium
s
t1…tk
1 k
Fair Sharing
Fair sharing: The cost of each edge e is 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)
Example: Self-Interested Behavior
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Each player chooses a path P.Cost to player i is:
cost(i) =
(Everyone shares cost equally)
cost(P)# using P
Example: Self-Interested Behavior
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Player 1 deviates as well, giving a solution with cost 1.833.
This solution is stable/ this solution is a Nash Equilibrium.
It differs from the optimal solution by a factor of 1+ + Hk = Θ(log k)!
1 12 3
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)
Unrestricted Sharing: OPT is a stable solution
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Contrast of Sharing Schemes
Unrestricted Sharing Fair Sharing
NE don’t always exist NE always exist
P.o.S. = O(k) 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 NE always exist
P.o.S. = O(k) P.o.S. = O(log(k))
P.o.S. = 1 for P.o.S. = (log(k)) for
many games almost all games
(P.o.S. = Price of Stability)
Contrast of Sharing Schemes
Unrestricted Sharing Fair Sharing
NE don’t always exist NE always exist
P.o.S. = O(k) P.o.S. = O(log(k))
P.o.S. = 1 for P.o.S. = (log(k)) for
many games almost all games
OPT is an approx. NE OPT may be far from NE
(P.o.S. = Price of Stability)
Unrestricted Sharing Model
What is a NE in this 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.
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
Strategy: a vector of payments
Players choose how much to pay, buy edges together
s
Goal: minimize payments,while fulfilling connectivity requirements
Cost(v) = if v does not satisfy connectivity requirementsPayments of v otherwise
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
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
OPT is a Nash Equilibrium (Price of Stability=1)
If k=n
If k=n
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
OPT is a -approximate Nash Equilibrium(no one can gain more than factor by switching)
=2
=2
=3
=1
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
If we pay for 1-1/ fraction of OPT, then the players will pay for the rest
=2
=2
=3
=1
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 NE always exist
P.o.S. = O(k) P.o.S. = O(log(k))
P.o.S. = 1 for P.o.S. = (log(k)) for
many games almost all games
OPT is an approx. NE 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 NE always exist
P.o.S. = O(k) P.o.S. = O(log(k))
P.o.S. = 1 for P.o.S. = (log(k)) for
many games almost all games
OPT is an approx. NE 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 NE always exist
P.o.S. = O(k) P.o.S. = O(log(k))
P.o.S. = 1 for P.o.S. = (log(k)) for
many games almost all games
OPT is an approx. NE 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
Unrestricted Sharing: OPT is a stable solution
Every player gives what they can afford
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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
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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.