The Price Of Stability for Network Design with Fair Cost Allocation

1
The Price Of Stability for Network Design with
Fair Cost Allocation

• Elliot Anshelevich, Anirban Dasgupta, Jon
  Kleinberg,
• Eva Tardos, Tom Wexler, Tim Roughgarden
• Presented by Kobi Yablonka
• (most of the slides are taken from Elliot
  Anshelevichs presentation The price of
  Stability for Network Design
  www.cs.princeton.edu/eanshele)

2
The context

• A network Design Game
• Number of independent agents
• Each agent try to minimize the cost

3
The Model

• Directed garph G(V,E) , with each edge e having
  a nonegative cost
• Each player i has a set of terminal nodes
  that he wants to connect
• Strategy for player i is s.t
  connects all node in
• All players using an edge split up the cost of
  the edge equally

4
The Model cont.

• vector of the
  players strategies
• - the number of players using edge e
• The cost to player i is
• The total edge cost of the network is
• The cost to a player is affected from the
  strategies of other players

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Nash Equilibrium

• A Nash Equilibrium (NE) is a set of payments for
  players such that no player wants to deviate.
• When considering deviations, player i assumes
  that other player payments are fixed.
• Given a solution consisting of a vector of
  strategies S there is no strategy for
  player i s.t

6
Price of stability

• The best Nash equilibrium relative to the global
  optimum
• Stands in contrasts to the price of anarchy
  which is the ratio of the worst Nash equilibrium
  to the optimum

7
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.

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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

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Example High Price of Stability
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Example High Price of Stability
cost(OPT) 1e
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. . .
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Example High Price of Stability
cost(OPT) 1e but not a NE player k
pays (1e)/k, could pay 1/k
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. . .
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Example High Price of Stability
so player k would deviate
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. . .
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Example High Price of Stability
now player k-1 pays (1e)/(k-1),
could pay 1/(k-1)
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. . .
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Example High Price of Stability
so player k-1 deviates too
t
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. . .
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Example High Price of Stability
Continuing this process, all players defect.
This is a NE! (the only Nash) cost 1

t
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. . .
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1 1
2 k
Price of Stability is Hk T(log k)!
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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.

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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.

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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 (set of edges for each
  player).
• Suppose player i is unhappy and decides to
  deviate.
• What happens to ?(S)?

e ? S
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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
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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
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Bounding Price of Stability

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

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Bounding Price of Stability

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

_
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Bounding Price of Stability

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

_
_
_
_
_
_
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Extensions

• Take a fair connection game with each edge having
  a nondecreasing concave cost function ce(x),where
  x is the number of players using edge e. Then the
  price of stability is at most Hk
• The proof is analogous to the previous proof.

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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
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More Questions

• Cost and Latency
• Only Latency
• Nash exist (same potential argument)
• Best NE costs at most OPT w/ twice as many
  players.
• Best Response Dynamics
• Can construct games with k players so that a
  certain ordering of moves takes 2O(k) time.
• Weighted Game

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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).

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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.
• if ce is concave, de is nondecreasing for all e,
  and
• for all e and xe then the price of stability is
  at most AHk (separate cost and latency)
• E.g. if ce is concave, de is polynomial with
  degree m,
• then Price of Stability is lt (m1)?log(k).

-
-
-
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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

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Latency

• Results (with single source)
• Nash exist (same potential argument)
• Best NE costs at most OPT w/ twice as many
  players.

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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?

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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)

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Games with at most 2 players per edge have NE
For edge e used by players i and j

• For edge with one player F(s) wici if I uses e
  0 otherwise
• When player i moves the change in F(S) is equal
  to the change in player's i payment scaled up by
  wi

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Best Response converges for single commodity

• All players have the same source and sink
• For every s-t path P the marginal weight is
• Order the paths of the players in tuple in
  according their marginal weight(lexicographic
  order)
• Show that in every step the lexicographic size of
  the tuple decrees
• If the move is P1 -gt P2
• P group of paths that have edges in either P1
  or P2
• C(P2) the cost of P2 after the move
• show that

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If NE do exist, Price of Stability will be gtgt
log(k)

• wi 2i-1

t
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1/2
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. . .
1
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k-1
0
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0
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Thank you.