Strategic Network Formation and Group Formation

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

Terminal Backup: Each terminal wants to connect to k other terminals.

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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t

0 0 0

1+

Demands:1-t, 2-t, 3-t

Example: Self-Interested Behavior

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Minimum Cost Solution (of cost 1+)

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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1+

Player 3 pays (1+ε)/3,

could pay 1/3

Example: Self-Interested Behavior

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1+

so player 3

would deviate

Example: Self-Interested Behavior

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now player 2

pays (1+ε)/2,

could pay 1/2

Example: Self-Interested Behavior

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so player 2

deviates also

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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eT

eT

e

ip

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.