The Price of Stability for Network Design

1
The Price of Stability
for Network Design

• Elliot Anshelevich
• Joint work with Dasgupta, Kleinberg, Tardos,
  Wexler, Roughgarden

2
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

3
A Connection Game

• Given G (V,E),
• costs ce for all e ? 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.

4
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 Allocation
• FOCS 04 Anshelevich, Dasgupta, Kleinberg,
  Tardos, Wexler, Roughgarden.

5
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

t2
t1
s3
t3
s1
s2
6
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.

t2
t1
s3
t3
s1
s2
7
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.

t2
t1
s3
t3
s1
s2
8
A Simple Example
t1, t2, tk
t
1
k
s
s1, s2, sk
9
A Simple Example
t1, t2, tk
t
t
1
k
1
k
s
s
s1, s2, sk

• One NE
• each player
• pays 1/k

10
A Simple Example
t1, t2, tk
t
t
t
1
k
1
k
1
k
s
s
s
s1, s2, sk

• One NE
• each player
• pays 1/k

Another NE each player pays 1
11
The Price of Stability
cost(worst NE) cost(OPT)
Price of Anarchy
s1sk
Koutsoupias, Papadimitriou Roughgarden, Tardos
(Min cost Steiner forest)
1
k
cost(best NE) cost(OPT)
Price of Stability
t1tk
Can think of latter as a network designer
proposing a
solution.
12
Example 2 No Nash
s1
t2
a
all edges cost 1
b
d
c
t1
s2
13
Example 2 No Nash
s1
t2
a
all edges cost 1
b
d
c
t1
s2
We know that any NE must be a tree WLOG assume
the tree is a,b,c.
14
Example 2 No Nash
s1
t2
a
all edges cost 1
b
d
c
t1
s2

• We know that any NE must be a tree WLOG assume
  the tree is a,b,c.
• Only player 1 can contribute to a.
  Only player 2 can
  contribute to c.

15
Example 2 No Nash
s1
t2
a
all edges cost 1
b
d
c
t1
s2
We know that any NE must be a tree WLOG assume
the tree is a,b,c. 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.
16
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 dont 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

17
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

18
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!

19
Simple Case MST

• Easy if all nodes are terminals
• Players buy edge above them in OPT.

• Claim This is a Nash Equilibrium.
• ( i unhappy gt can build cheaper tree )
• Typically we will have Steiner nodes.
  Who buys the edge above these?

20
Attempts to Buy Edges
1) Can we get a single player to pay?
Both players must help buy top edge.
3
5
5
3
3
2) Can we split edge costs evenly?
Second node wont pay more than 5 in total.
4
4
4
4
4
4
5
21
Greedy Algorithm
In both examples, players were limited by
possible deviations.
e
Given OPT, pay for edges in OPT from the
bottom up, greedily, as constrained by
deviations. If we buy all edges, were done!
22
Proof Idea

• 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.

e
23
A Possible Pitfall
Suppose greedy algorithm cannot pay for e.
e
e
1 2 3 4

• Further, suppose 1 2 share cost(e)
• Consider 1 2 both deviating
• Player 1 stops contributing to e
• Danger 2 still needs this edge!

24
Safely Selecting Paths
e
e
1 2 3 4
Shouldnt allow player 1 to deviate If
only 2 deviates, all players reach the
source. Idea should use the highest deviating
paths first.
25
Safely Selecting Paths
e
We may have to select multiple alternate
paths. Not trying to find NE, just form
contradiction.
26
Single Source in Polytime

• Thm For single source, can find a (1e)-approx.
  NE in polytime on an a-approx. Steiner tree.
• a best Steiner tree approx. (1.55)
• e gt 0, running time depends on e.
• Pf Sketch
• Greedy algorithm from previous proof either
  finds a NE or a cheaper tree than it was given.
• Only take significant improvements.

27
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) S
  ce/ke

e ? P
28
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) S
  ce/ke
• Each player tries to minimize his cost.

e ? P
29
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.

30
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

31
A Simple Example
t1, t2, tk
t
t
t
1
k
1
k
1
k
s
s
s
s1, s2, sk

• One NE
• each player
• pays 1/k

Another NE each player pays 1
32
Contrast with Unfair Cost Sharing

• Unrestricted Sharing Fair Sharing
• NE dont 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 ???

33
Example High Price of Stability
t
1
1
1
1
1
k
2
3
k-1
. . .
1?
1
2
3
k
k-1
0
0
0
0
0
34
Example High Price of Stability
cost(OPT) 1e
t
1
1
1
1
1
k
2
3
k-1
. . .
1?
1
2
3
k
k-1
0
0
0
0
0
35
Example High Price of Stability
cost(OPT) 1e but not a NE player k
pays (1e)/k, could pay 1/k
t
1
1
1
1
1
k
2
3
k-1
. . .
1?
1
2
3
k
k-1
0
0
0
0
0
36
Example High Price of Stability
so player k would deviate
t
1
1
1
1
1
k
2
3
k-1
. . .
1?
1
2
3
k
k-1
0
0
0
0
0
37
Example High Price of Stability
now player k-1 pays (1e)/(k-1),
could pay 1/(k-1)
t
1
1
1
1
1
k
2
3
k-1
. . .
1?
1
2
3
k
k-1
0
0
0
0
0
38
Example High Price of Stability
so player k-1 deviates too
t
1
1
1
1
1
k
2
3
k-1
. . .
1?
1
2
3
k
k-1
0
0
0
0
0
39
Example High Price of Stability
Continuing this process, all players defect.
This is a NE! (the only Nash) cost 1

t
1
1
1
1
1
k
2
3
k-1
. . .
1?
1
2
3
k
k-1
0
0
0
0
0
1 1
2 k
Price of Stability is Hk T(log k)!
40
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.

41
Potential Games

• A game is a potential game if there exists a
• function ?(S) mapping the current game state S
• to a real value s.t.
• If player i moves, is improvement change in
  ?(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.

42
A Potential Function

• Define ?e(S) ce1 1/2 1/3 1/ke
• where ke is players using e in S. Hk
• Let ?(S) S ?e(S)
• Consider some solution S (a path for each
  player).
• Suppose player i is unhappy and decides to
  deviate.
• What happens to ?(S)?

e
e ? S
43
Tracking Player Happiness

• ?e(S) ce1 1/2 1/3 1/ke
• Suppose player is new path includes e.
• i pays ce/(ke1) to use e.
• ?e(S) increases by the same amount.
• Likewise, if player i leaves an edge e,
• ?e(S) exactly reflects the change in
  is payment.

ce1 1/2 1/ke
e
i
e
ce1 1/2 1/ke
44
Tracking Player Happiness

• ?e(S) ce1 1/2 1/3 1/ke
• Suppose player is new path includes e.
• i pays ce/(ke1) to use e.
• ?e(S) increases by the same amount.
• Likewise, if player i leaves an edge e,
• ?e(S) exactly reflects the change in
  is payment.

ce1 1/2 1/kece/(ke1)
e
i
e
ce1 1/2 1/ke -ce/ke
45
Bounding Price of Stability

• Consider starting from OPT (central optimum).
• From OPT, players will settle on some Nash NE.

1
1
3/2
1
1
1
OPT
46
Bounding Price of Stability

• Consider starting from OPT (central optimum).
• From OPT, players will settle on some Nash NE.
• We have argued that
• ?(NE) lt ?(OPT)

1
_
3/2
1
1
NE
47
Bounding Price of Stability

• Consider starting from OPT (central optimum).
• From OPT, players will settle on some Nash NE.
• We have argued that
• ?(NE) lt ?(OPT)
• We also know for any S,
• cost(S) lt ?(S) lt Hk cost(S).
• So cost(NE) lt ?(NE) lt ?(OPT) lt Hk cost(OPT).

_
_
_
NE
_
_
_
48
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.

ce
E
sets
i
j
49
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 ?e(S) S ce(j)/j.
• Like before, ? tracks improvement,
• within log factor of cost gt
• Price of
  Stability lt log(k).

edge cost
ke
j1
users
50
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 S ce(ke)/ke fi(Si)
• PoS is still within log(k) if ce is concave

e ? Si
51
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

52
Thank you.
53
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.

54
Price of Anarchy in Multi-Source Games
s1
O(k)
s3sk
e
e
t2
s2
1
e
e
t3tk
O(k)
t1
OPT costs 1, but its not a NE. The only NE
costs O(k), so optimistic price of anarchy is
almost k.
55
Result for Multi Source Games
2
1
We know a NE may not exist, so settle for
approximate NE. How bad an approximation must we
have if we insist on buying OPT?
3
1
3
2

• 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.

56
Proof Idea

• 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.

2
1
3
1
3
2
57
Connection Sets
1

• 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

b
a
1
58
Connection Sets
1

• 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

b
a
1
59
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.

2
1
3
1
3
2
60
Tree Decomposition

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

4
1
3
2
2
3
1
5
5
4
61
Tree Decomposition

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

4
1
3
2
2
3
1
5
5
4
62
Tree Decomposition

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

4
1
3
2
2
3
1
5
5
4
63
Tree Decomposition

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

4
1
3
2
2
3
1
5
5
4
64
Payment Scheme

• Connection sets in each path P are paid for by
  terminals associated with paths entering P.

2
2
1
3
4
65
Payment Scheme

• Connection sets in each path P are paid for by
  terminals associated with paths entering P.

2
2
1
3
4
66
Payment Scheme

• Connection sets in each path P are paid for by
  terminals associated with paths entering P.

4
3
1
3
2
3
2
5
2
2
3
1
1
5
5
4
67
Approximation Algorithm
Theorem For multi-source 2-terminal games, can
find a (3e)-approx. NE in polytime on an
2-approx. to OPT. For gt2 terminals, above
approximation becomes (4.65e), since need to use
best known approx for Steiner tree.
68
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)/kede(ke).
• Total cost of edge is
• ce(ke) ke?de(ke).

69
Cost Latency

• From earlier proof, we know that if for all S,
• cost(S) lt A??(S) lt AB?cost(S),
• then the price of stability is lt AB.
• E.g. if ce is concave, de is polynomial with
  degree m,
• then Price of Stability is lt (m1)?log(k).
• With only latency and no edge costs,
• we have PoS lt m1 for polynomial delays

-
-
-
70
Only Latency

• Similar to routing games Roughgarden, Tardos
• Comparison between these two games
• atomic vs. non-atomic
• Price of Stability vs. Price of Anarchy

NE is unique
t
t
t
xm
xm
.5/0
.5/0
1/1
0/0
s
s
s
71
Latency

• In this case Nash Equilibria can be computed.

Convert all edges
d(1)
All edges capacity 1
d(x)
d(2)
d(3)

• 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

72
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.

73
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?

74
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 gtgt
  log(k)