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
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
Connection Games
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
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
The Connection Game – Model
The Connection Game – Example
s1
t3
s2
t2
s3
t1
The Connection Game – Example
s1
t3
s2
t2
s3
t1
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.
TODAY
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
NEXT WEEK…
What are we interested in?
From Nash’s Theorem (1950) we know
that mixed-strategy (non deterministic)
Nash Equilibria always exist
There for We are interested in purestrategy (deterministic) Nash Equilibria
From now and on “Nash Equilibria”
(“NE”) will mean:
Deterministic Nash Equealibira
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
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
t1
s3
 A player does not care
whether other players
connect.
 When considering deviations, s1
a player assumes that other
players’ payments are fixed.
t3
s2
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
t1
s3
 A player does not care
whether other players
connect.
 When considering deviations, s1
a player assumes that other
players’ payments are fixed.
t3
s2
Nash Equilibria - Formal
Three Observations
Example 1 - Two Different NEs
t1, t2, … tk
t
t
1
k
s
s1 , s2 , … sk
1
t
k
s
One NE:
each player
pays 1/k
1
k
s
Another NE:
each player
pays 1
Reminder: The POA and POS
Price of Anarchy =
[Koutsoupias, Papadimitriou]
[Roughgarden, Tardos]
cost(worst NE)
s1…sk
cost(OPT)
(Min cost Steiner forest)
1
Price of Stability =
k
cost(best NE)
cost(OPT)
t1…tk
Question: What were the POA and POS in
Example 1 ?
NE Doesn’t have to Exits!
Don’t forget NE=pure-NE for now
Example 2 - No Nash
s1
t2
a
all edges
cost 1
d
s2
b
c
t1
Example 2 - No Nash
s1
t2
a
all edges
cost 1
d
s2
b
c
t1
We know that any NE must be a tree: WLOG
assume the tree is a,b,c.
Example 2 - No Nash
s1
t2
a
all edges
cost 1
d
s2
b
c
t1
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.
Example 2 - No Nash
s1
t2
a
all edges
cost 1
d
s2
b
c
t1
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.
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
When NE exist, how good can it be?
In Exaple 1 we saw that POS was 1
NEXT!
Single Source Games
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
1) Can we get a single player to pay?
Both players must
help buy top edge.
3
5
3
5
3
2) Can we split edge costs evenly?
4
4
4
4
4
5
4
Second node
won’t pay more
than 5 in total.
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, we’re done!
Single Source Games
Notation
e
The Greedy Algorithm
Example
4
4
3
5
4
4
4
5
4
3
5
3
We get NE!
If we buy all edges we are done!
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
Proof
Path Lemma
Path Lemma
Proof Finale
e
But, Wait!
Suppose greedy algorithm cannot pay for e
e
e’
1
4




2
3
Further, suppose 1 & 2 share cost(e’)
Consider 1 & 2 both deviating…
Player 1 stops contributing to e’
Danger: 2 still needs this edge!
Don’t Worry, Everything is fine. Just,
e
e’
1
2
3
4
Shouldn’t allow player 1 to deviate:
If only 2 deviates, all players reach the
source.
Idea: should use the “highest”
deviating paths first.
(1+ e)-approx. NE in Polytime
Theorem: 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.
Proof Sketch:
• Greedy algorithm from previous proof either finds a
NE or a cheaper tree than it was given.
• Only take significant improvements.
Multi Source Games
Price of Anarchy in Multi-Source Games
s1
ε
O(k)
ε
s2
t2
ε
ε
t1
O(k)
s3…sk
1
t3…tk
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
We know a NE may not exist, so
settle for approximate NE.
2
3
1
How bad an approximation must we
have if we insist on buying OPT?
1
2
3
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.
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.
2
3
1
• Each chunk is paid for by a
single player.
• Each player pays for at
most 3 chunks.
1
2
3
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.
b
a
 Connection set = chunk
1
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.
b
a
 Connection set = chunk
1
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
2
3
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
4
5
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
4
5
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
4
5
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
4
5
Payment Scheme
 Connection sets in each path P are paid for by
terminals associated with paths entering P.
2
2
1
3
4
Payment Scheme
 Connection sets in each path P are paid for by
terminals associated with paths entering P.
2
2
1
3
4
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
3
2
1
1
5
4
5
Approximation Algorithm
Theorem: For multi-source 2-terminal games,
can find a (3+ε)-approx. NE in polytime on an
1.55-approx. to OPT.
For >2 terminals, above approximation becomes
(4.65+ε), since need to use best known approx
for Steiner tree.
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?
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)
 (3+ε)-NE approx. always exist
 (4.65+ε)-NE approx algorithm for 1.55OPT
 There are games which the best NE is 1.5approx. Lower bound is 1.5.
THANK
YOU!