The Price of Stability for Network Design Elliot Anshelevich Joint work with: Dasgupta, Kleinberg,...

The Price of Stability for Network Design

Elliot Anshelevich

Joint work with: Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden

Selfish Agents in Networks

• 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

A Connection Game

Given: G = (V,E),

costs ce for all e є E,

k vertex pairs (si,ti)

Each player wants to build a network in which his nodes are connected.

Player strategy: select a path connecting si to ti.

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.

Near-Optimal Network Design with Selfish Agents STOC ‘03 Anshelevich, Dasgupta, Tardos, Wexler.

Another approach: try to ensure some sort of fairness.The Price of Stability for Network Design with Fair Cost AllocationFOCS ’04 Anshelevich, Dasgupta, Kleinberg, Tardos, Wexler, Roughgarden.

Arbitrary Sharing Model

Player i picks payments for each edge e.

e is bought if total payments ≥ ce.

Note: any player can use bought edges

t1

t3

t2

s1

s2

s3

Nash Equilibrium

A Nash Equilibrium (NE) is a set of payments for players such that no player wants to deviate.

• Player i does not care whether other players connect.

• When considering deviations, player i assumes that other players’ payments are fixed.

t1

t3

t2

s1

s2

s3

A Simple Example

t

s

1 k

t1, t2, … tk

s1, s2, … sk

A Simple Example

t

s

1 k

t1, t2, … tk

s1, s2, … sk

t

s

1 k

One NE:

each player

pays 1/k

A Simple Example

t

s

1 k

t1, t2, … tk

s1, s2, … sk

t

s

1 k

One NE:

each player

pays 1/k

t

s

1 k

Another NE:

each player

pays 1

The Price of Stability

Price of Anarchy = cost(worst NE)

cost(OPT)

Price of Stability = cost(best NE)

cost(OPT)

(Min cost Steiner forest)

[Koutsoupias, Papadimitriou][Roughgarden, Tardos]

s1…sk

t1…tk

1 k

Can think of latter as a network designer proposing a solution.

Example 2: No Nash

s1

t1

t2

s2

all edges cost 1

a

b

c

d

Example 2: No Nash

s1

t1

t2

s2

all edges cost 1

a

b

c

d

We know that any NE must be a tree: WLOG assume the tree is a,b,c.

Example 2: No Nash

s1

t1

t2

s2

all edges cost 1

a

b

c

d


Only player 1 can contribute to a. Only player 2 can contribute to c.

Example 2: No Nash

s1

t1

t2

s2

all edges cost 1

a

b

c

d


Only player 1 can contribute to a. Only player 2 can contribute to c. Neither player can contribute to b, since d is a tempting deviation.

Related Work

Generalized Steiner forest [e.g. Goemans, Williamson;…]

• Centralized problem: connect pairs

Price of Anarchy [Koutsoupias, Papadimitriou; Roughgarden, Tardos;…]

Cost sharing [e.g. Jain, Vazirani]

• Get players to pay for a tree

• Players don’t specify edge payment

Network creation game [Fabrikant, Luthra, Maneva, Papadimitriou, Shenker; Bala and Goyal; Heller and Sarangi]

• Players always purchase adjacent edge

• Players care about distances

Results for Arbitrary Sharing Model

• NE do not always exist

• Price of Stability = O(k)

• Price of Stability = 1 for single source

– Directed graphs

– max(i), a price beyond which player i would rather not connect at all

• OPT is an approx. NE

• Approx. NE can be found

Single Source Games

(si = s for all i)

Theorem: In any single source game, there is always a NE that buys OPT.

meaning 2 things…

• There is always a NE

• The Price of Stability is 1!

Simple Case: MST

Easy if all nodes are terminals:

Players buy edge above them in OPT.

Claim: This is a Nash Equilibrium.

( i unhappy => can build cheaper tree )

• Typically we will have Steiner nodes. Who buys the edge above these?

Attempts to Buy Edges

Second node won’t pay more than 5 in total.4 444

4 4

5

3

33

55Both players must help buy top edge.

1) Can we get a single player to pay?

2) Can we split edge costs evenly?

Greedy Algorithm

Given OPT, pay for edges in OPT from the bottom up, greedily, as constrained by deviations.

If we buy all edges, we’re done!

e

In both examples, players were limited by possible deviations.

Proof Idea

e

If greedy fails to pay for e, we will show that the tree is not OPT.

• All players have possible deviations.

• Deviations and current payments must be equal.

• If all players deviate, all connect, but pay less.

A Possible Pitfall

Suppose greedy algorithm cannot pay for e.

e

• Further, suppose 1 & 2 share cost(e’)

• Consider 1 & 2 both deviating…

• Player 1 stops contributing to e’

• Danger: 2 still needs this edge!

e’

1 2 3 4

Safely Selecting Pathse

Shouldn’t allow player 1 to deviate: If only 2 deviates, all players reach the source.

Idea: should use the “highest” deviating paths first.

e’

1 2 3 4

Safely Selecting Paths

e

We may have to select multiple alternate paths.Not trying to find NE, just form contradiction.

Single Source in Polytime

Thm: For single source, can find a (1+ε)-approx. NE in polytime on an α-approx. Steiner tree.

α = best Steiner tree approx. (1.55)

ε > 0, running time depends on ε.

Pf Sketch:

• Greedy algorithm from previous proof either finds a NE or a cheaper tree than it was given.

• Only take significant improvements.

The Fair Connection Game

Can view restricting allowable payments as mechanism design:

What sharing rules induce players to form good solutions?

Natural choice is fair sharing, or Shapley cost sharing:

Players using e pay for it evenly: ci(P) = Σ ce/kee є P

The Fair Connection Game

Can view restricting allowable payments as mechanism design:

What sharing rules induce players to form good solutions?

Natural choice is fair sharing, or Shapley cost sharing:

Players using e pay for it evenly: ci(P) = Σ ce/ke

Each player tries to minimize his cost.e є P

Congestion Games

This is a congestion game!

• Usual congestion games have latency/delay/load:

cost per player increases as the number of players sharing an edge increases.

• Fair Connection Game has edge costs:

cost per player decreases as the number of players sharing an edge increases.

Related Work

Shapley value cost sharing [Feigenbaum, Papadimitriou, Shenker; Herzog, Shenker, Estrin]

Price of anarchy in routing and congestion games[Roughgarden, Tardos]

Potential games[Monderer, Shapley]

A Simple Example

t

s

1 k

t1, t2, … tk

s1, s2, … sk

t

s

1 k

One NE:

each player

pays 1/k

t

s

1 k

Another NE:

each player

pays 1

Contrast with Unfair Cost Sharing

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. = O(log(k)) for

single source single source

OPT is an approx. NE OPT may be far from NE

NE are forests NE can be cyclic

Approx. NE can be found ???

Example: High Price of Stability

1 1k

12

13

1 2 3 k

t

0 0 0 0

1+ . . . k-1

0

1k-1


1 1k

12

13

1 2 3 k

t

0 0 0 0

1+ . . . k-1

0

1k-1

cost(OPT) = 1+ε


1 1k

12

13

1 2 3 k

t

0 0 0 0

1+ . . . k-1

0

1k-1

cost(OPT) = 1+ε

…but not a NE:

player k

pays (1+ε)/k,

could pay 1/k


1 1k

12

13

1 2 3 k

t

0 0 0 0

1+ . . . k-1

0

1k-1

so player k

would deviate


1 1k

12

13

1 2 3 k

t

0 0 0 0

1+ . . . k-1

0

1k-1

now player k-1

pays (1+ε)/(k-1),

could pay 1/(k-1)


1 1k

12

13

1 2 3 k

t

0 0 0 0

1+ . . . k-1

0

1k-1

so player k-1

deviates too


1 1k

12

13

1 2 3 k

t

0 0 0 0

1+ . . . k-1

0

1k-1

Continuing this process, all players defect.

This is a NE!

(the only Nash)

cost = 1 + + … +

Price of Stability is Hk = Θ(log k)!

1 12 k

To Show

• The Hk Price of Stability is worst case possible.

• Proof uses the idea of a Potential Game

[Monderer and Shapley].

• Extend results to many natural generalizations of the Fair Connection Game.

Potential Games

A game is a potential game if there exists a function Ф(S) mapping the current game state Sto a real value s.t.

If player i moves, i’s improvement = change in Ф(S).

Such games have pure NE: just do Best Response!

The Fair Connection Game is a potential game!We extend analysis to bound Price of Stability.

A Potential Function

Define Фe(S) = ce[1+ 1/2 + 1/3 + … 1/ke]

where ke is # players using e in S. Hk

Let Ф(S) = Σ Фe(S)

Consider some solution S (a path for each player).

Suppose player i is unhappy and decides to deviate.

What happens to Ф(S)?

e є S

e

Tracking Player Happiness

Фe(S) = ce[1+ 1/2 + 1/3 + … 1/ke]

Suppose player i’s new path includes e.

i pays ce/(ke+1) to use e.

Фe(S) increases by the same amount.

Likewise, if player i leaves an edge e’,

Фe’(S) exactly reflects the change in i’s payment.

e

e’

ce[1+ 1/2 +… +1/ke]

ce’[1+ 1/2 +… +1/ke’]

i

Tracking Player Happiness

e

e’

ce[1+ 1/2 +… +1/ke]+ce/(ke+1)

ce’[1+ 1/2 +… +1/ke’] -ce’/ke’

i

Фe(S) = ce[1+ 1/2 + 1/3 + … 1/ke]

Suppose player i’s new path includes e.

i pays ce/(ke+1) to use e.

Фe(S) increases by the same amount.

Likewise, if player i leaves an edge e’,

Фe’(S) exactly reflects the change in i’s payment.

Bounding Price of Stability

Consider starting from OPT (central optimum).

From OPT, players will settle on some Nash NE.

OPT

3/2

1

11

1

1




We have argued that

Ф(NE) < Ф(OPT)

NE

3/2

1

1

1

_




We have argued that

Ф(NE) < Ф(OPT)

We also know for any S,

cost(S) < Ф(S) < Hk cost(S).

So cost(NE) < Ф(NE) < Ф(OPT) < Hk cost(OPT).

NE

_

_ _

_ _ _

Extensions: Set Systems

• ground set E of elements with costs.

• player i has allowable set Si of subsets from E.

• player i picks subset, evenly shares element costs.

For networking, can model players who want..

• to connect multiple terminals.

• higher connectivity guarantees.

Ece

sets

i j

Extensions: Buy-at-Bulk Costs

Total cost of edge may increase with # of users, but marginal cost decreases.

(Economies of Scale)

If edge cost is ce(j) for j users

define Фe(S) = Σ ce(j)/j.

Like before, Ф tracks improvement,within log factor of cost => Price of Stability < log(k).

# users

ed

ge c

ost

j=1

ke

Extensions

• All results hold if edges have capacities.• Incorporate distance:

cost to player i = ci(Pi) + length(Pi)

• Utility function of player i can depend on both cost and the set Si picked by i– cost to player i = Σ ce(ke)/ke + fi(Si)

– PoS is still within log(k) if ce is concavee є Si

More Questions

• Cost and Latency• Only Latency

– Nash exist (same potential argument)

– Best NE costs at most OPT w/ twice as many players.

– For large class of functions, worst case Price of Stability is realized on 2 parallel links.

• Best Response Dynamics– Can construct games with k players so that a certain

ordering of moves takes 2O(k) time.

• Weighted Game

Thank you.

Three Observations

1) The bought edges in a NE form a forest.

2) Players only contribute to edges on their si-ti path in this forest.

3) The total payment for any edge e is either c(e) or 0.

Price of Anarchy in Multi-Source Games

s1

t1

t2s2

s3…sk

t3…tk

ε

ε

ε

ε

O(k)

O(k)

1

OPT costs ~1, but it’s not a NE.

The only NE costs O(k), so optimistic price of anarchy is almost k.

Result for Multi Source Games

Theorem: For any game, there exists a 3-approx NE that buys OPT.

Note: this is true even for games where players may have more than 2 terminals.

1

32

1

3 2We know a NE may not exist, so settle for approximate NE.

How bad an approximation must we have if we insist on buying OPT?

Proof Idea

1

32

1

32

• Break up OPT into chunks.

• Use optimality of OPT to show that any player buying a single chunk has no incentive to deviate.

• Each chunk is paid for by a single player.

• Each player pays for at most 3 chunks.

Connection Sets

• A connection set C of player i is a set of edges such that:– C only includes edges on the

path Pi from si to ti in OPT.

– If OPT is bought, and i pays only for C, then i has no incentive to deviate.

• Connection set = chunk

1

1

a b

Connection Sets

1

1

a b

• A connection set C of player i is a set of edges such that:– C only includes edges on the

path Pi from si to ti in OPT.

– If OPT is bought, and i pays only for C, then i has no incentive to deviate.

• Connection set = chunk

Main Challenge

• Form a payment scheme where each player pays for at most 3 connection sets.

• i pays for edges that no other players would pay for in OPT.

• Another connection set for each terminal of i.

1

321

3

2

Tree Decomposition• Decompose OPT into hierarchical paths, where

each path begins at a terminal and ends at a path of higher level.

1

2

1

3

54

5

2

3

4

Tree Decomposition

• Decompose OPT into hierarchical paths, where each path begins at a terminal and ends at a path of higher level.

1

2

1

3

54

5

2

3

4

Payment Scheme• Connection sets in each path P are paid for by

terminals associated with paths entering P.

1

22

3 4

Payment Scheme• Connection sets in each path P are paid for by

terminals associated with paths entering P.

1

2

1

3

54

5

2

3

4

2

5

1

2

33

Approximation Algorithm

Theorem: For multi-source 2-terminal games, can find a (3+ε)-approx. NE in polytime on an 2-approx. to OPT.

For >2 terminals, above approximation becomes (4.65+ε), since need to use best known approx for Steiner tree.

Adding Latency

What if we want to model congestion? …marginal cost increases, so not buy-at-bulk.

Every edge has increasing delay function de(ke).Cost of edge e for player i is ce(ke)/ke+de(ke).Total cost of edge is ce(ke) + kede(ke).

Cost + Latency

From earlier proof, we know that if for all S, cost(S) < AФ(S) < ABcost(S),then the price of stability is < AB.

E.g. if ce is concave, de is polynomial with degree m, then Price of Stability is < (m+1)log(k).

With only latency and no edge costs, we have PoS < m+1 for polynomial delays

-

--

Only Latency

Similar to routing games [Roughgarden, Tardos]

Comparison between these two games:atomic vs. non-atomicPrice of Stability vs. Price of Anarchy

t

s

xm xm

t

s

1/1 0/0

t

s

.5/0 .5/0

[NE is unique]

Latency

In this case Nash Equilibria can be computed.

d(x)

d(1)

d(2)

d(3)

…All edges capacity 1

Claim: A min cost flow corresponds to a NE.Idea: Since d is increasing, flow will use d(1),

then d(2), etc, mirroring a potential function. [Fabrikant, Papadimitriou, Talwar]

Convert all edges

Latency

Results (with single source)

• Nash exist (same potential argument)

• Best NE costs at most OPT w/ twice as many players.

• For large class of functions, worst case Price of Stability is realized on 2 parallel links.

Best Response Dynamics

How long before players settle on a NE?

• In games with 2 players, O(n) time, since shared segment grows monotonically.

• Can construct games with k players so that a certain ordering of moves takes 2O(k) time.

• Can 3-player games run for exponential time?• Can k-player games be scheduled to be

polytime?

Weighted Game

If some player has more traffic, should pay more…

In a weighted game, player i has weight w(i).Players pay for edges proportionally to their weight.

No potential function exists. Do NE always exist?• Best Response converges for single commodity.• Games with at most 2 players per edge have NE.• If NE do exist, Price of Stability will be >> log(k)

The Price of Stability for Network Design Elliot Anshelevich Joint work with: Dasgupta, Kleinberg,...

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