Near Optimal Network Design With Selfish Agents

1
Near Optimal Network Design With Selfish Agents

• Eliot Anshelevich Anirban Dasupta Eva Tardos
  Tom Wexler

Cornell University
Presented by Andrey Stolyarenko School of CS,
Tel-Aviv University
Some of the slides are taken from E.Anshelevich
and L.Kaiser presentations
2
Selfish Agents in Networks

• Traditional network design
  problems are centrally controlled
• What if network is instead built by many
  self-interested agents?
• As we saw on previous lectures, properties of
  resulting network may be very different from the
  globally optimum one

3
Connection Games
4
The Connection Game A Story
Think of sea transport companies or broadband
internet providers. These are our agents

• each company needs to connect a few ports or
  users
• every connection has a constant cost
• connection is bought if all together pay for it

5
The Connection Game Selfish as usual

• We do not consider negotiations, communication
• No external mechanism or regulation
• All desired users must be connected, no tradeoff
• Everyone will go for a cheaper price if possible

6
The Connection Game Model
7
The Connection Game Example
s1
t3
s2
t2
s3
t1
8
The Connection Game Example
9
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

TODAY
NEXT WEEK
10
What are we interested in?

• From Nashs Theorem (1950) we know that
  mixed-strategy (non deterministic) Nash
  Equilibria always exist
• There for We are interested in pure-strategy
  (deterministic) Nash Equilibria
• From now and on Nash Equilibria (NE) will
  mean
• Deterministic Nash Equealibira

11
What are we interested in?

• How bad can NE be? Price of Anarchy
• How good can NE be? Price of Stability
• (1 e)-approx. NE

12
Nash Equilibrium
t2

• A NE is a set of payments for players such that
  no player wants to deviate.
• A player must connect his terminals
• A player does not care whether other players
  connect.
• When considering deviations, a player assumes
  that other players payments are fixed.

t1
s3
t3
s1
s2
13
Nash Equilibrium

• A NE is a set of payments for players such that
  no player wants to deviate.
• A player must connect his terminals
• A player does not care whether other players
  connect.
• When considering deviations, a player assumes
  that other players payments are fixed.

14
Nash Equilibria - Formal
15
Three Observations
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Example 1 - Two Different NEs
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
17
Reminder The POA and POS
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
Question What were the POA and POS in Example 1 ?
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NE Doesnt have to Exits!
Dont forget NEpure-NE for now
19
Example 2 - No Nash
s1
t2
a
all edges cost 1
b
d
c
t1
s2
20
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.
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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.

22
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.
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When NE exist, how bad can it be?

• In The Connection Game the POA is at most N - The
  number of agents
• If the worst NE p const more than N times OPT
  then there must be a player i whose payments pi
  are strictly more then OPT
• Player i could deviate by purchasing the entire
  optimal solution by himself

24
When NE exist, how good can it be?

• In Exaple 1 we saw that POS was 1

NEXT!
25
Single Source Games
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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?

27
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
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Greedy Algorithm
In both examples, players were limited by
possible deviations.
e
Given OPT, pay for edges in OPT from the
bottom up, greedily (openhanded) , as constrained
by deviations. If we buy all edges, were done!
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Single Source Games
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Notation
e
31
The Greedy Algorithm
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Example
4
4
3
5
5
4
4
4
4
3
3
5
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We get NE!
If we buy all edges we are done!
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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
35
Proof
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Path Lemma
37
Path Lemma
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Proof Finale
e
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But, Wait!
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!

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Dont Worry, Everything is fine. Just,
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.
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(1 e)-approx. NE in Polytime

• Theorem 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.
• Proof Sketch
• Greedy algorithm from previous proof either
  finds a NE or a cheaper tree than it was given.
• Only take significant improvements.

42
Multi Source Games
43
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.
44
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.

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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
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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
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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
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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
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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
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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
51
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
52
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
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Payment Scheme

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

2
2
1
3
4
54
Payment Scheme

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

2
2
1
3
4
55
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
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Approximation Algorithm
Theorem For multi-source 2-terminal games, can
find a (3e)-approx. NE in polytime on an
1.55-approx. to OPT. For gt2 terminals, above
approximation becomes (4.65e), since need to use
best known approx for Steiner tree.
57
Results and More

• Single Source
• POS 1
• Polytime NE approx
• What happens in directed graphs?
• What happens if we add a maximum payment that a
  player is willing to may in order to stay
  connected?

58
Results and More

• Multi Source
• The existence of NE is NPC if the number of
  players is a part of the input. Show by 3-SAT
  reduction
• POS can be O(n)
• (3e)-NE approx. always exist
• (4.65e)-NE approx algorithm for 1.55OPT
• There are games which the best NE is 1.5-approx.
  Lower bound is 1.5.

59
THANK

• YOU!