Atomic Selfish Routing
Presentation by Nir Aviv
for the Price of Anarchy Seminar
Professor: Michal Feldman
5/3/2014
Introduction
• In this presentation we will discuss atomic
unsplittable selfish routing games.
• In these games, each player controls a non-negligible
amount of traffic, and must route it through a single
path.
• This model is relevant, for example, to the Internet,
where traffic between a sender and receiver is
usually sent along a single path. This is done to avoid
many packets arriving out-of-order and needing
reassembly at the destination.
Basic Definitions
• Let 𝐺 = (𝑉, 𝐸) be a directed graph, which may
contain parallel edges.
• The goal of each player 𝑗 ∈ {1,2, … 𝑛} is to transport
𝑤𝑗 ∈ ℝ+ units, over a single (simple) path, from a
source 𝑠𝑗 ∈ 𝑉 to a target 𝑡𝑗 ∈ 𝑉.
• The (pure) strategy set for each player 𝑗 is 𝑄𝑗 , the set
of all 𝑠𝑗 → 𝑡𝑗 paths in 𝐺.
• The set of strategy profiles is 𝑄 = 𝑄1 × 𝑄2 × ⋯ × 𝑄𝑛
Basic Definitions
• Each edge 𝑒 ∈ 𝐸 has a latency function: 𝑓𝑒 : ℝ+ → ℝ+
• The load on 𝑒 ∈ 𝐸 under strategy profile
𝑞1 , 𝑞2 , … , 𝑞𝑛 = 𝑞 ∈ 𝑄 is the amount of units
transported: 𝑙𝑒 = 𝑙𝑒 (𝑞) = Σ𝑗|𝑒∈𝑞𝑗 𝑤𝑗 .
• The cost for player 𝑗 under 𝑞 is defined to be
𝑐𝑗 = 𝑐𝑗 𝑞 = 𝑒∈𝑞𝑗 𝑓𝑒 𝑙𝑒 𝑤𝑗 .
• The (utilitarian) social cost for 𝑞 is the sum of player
𝑛
costs: 𝐶 = 𝐶 𝑞 = Σ𝑗=1
𝑐𝑗 𝑞 = Σ𝑒∈𝐸 𝑓𝑒 𝑙𝑒 𝑙𝑒 .
Basic Definitions
• A strategy profile 𝑞 ∈ 𝑄 is a pure Nash Equilibrium
(NE) if for all 𝑞 ′ = 𝑞1 , … , 𝑞𝑗−1 , 𝑞𝑗′ , 𝑞𝑗+1 , … 𝑞𝑛 ∈ 𝑄,
𝑐𝑗 𝑞 ≤ 𝑐𝑗 (𝑞 ′ ).
• In games where pure NE exist, the Price of Anarchy
(PoA) with respect to pure NE is
𝐶 𝑞
max
∗
𝑞∈𝑝𝑢𝑟𝑒𝑁𝐸 𝐶 𝑞
,
where 𝑞 ∗ ∈ 𝑄 is a (global) minimum of the social cost
function 𝐶.
Basic Definitions
• An atomic selfish routing game is unweighted if there
is some 𝑤 ∈ ℝ+ such that for all players 𝑗, 𝑤𝑗 = 𝑤.
• An atomic selfish routing game is affine-cost if for
each edge 𝑒 ∈ 𝐸 there are nonnegative coefficients
𝑎𝑒 , 𝑏𝑒 ∈ ℝ+ such that 𝑓𝑒 𝑙𝑒 = 𝑎𝑒 𝑙𝑒 + 𝑏𝑒 .
• We will show that pure NE always exist in
unweighted and affine-cost instances, and show tight
bounds on the PoA.
• But first, a useful example…
Example 1
• Consider the following affine-cost unweighted
atomic routing game with four players. Each player
has one unit to transport.
• Each player has two options: to use either a path of
length 1 or a path of length 2.
Example 1
• The situation where each player uses the path of
length 1 is the social optimum and a pure Nash
Equilibrium – each player pays the minimum of 1 with social cost 4.
𝑙=1
𝑙=1
𝑙=1
𝑙=1
Example 1
• The situation where each player uses the path of
length 2 is the also a pure Nash Equilibrium, with
social cost 10. (Players 1 and 2 incur a cost of 3, and
Players 3 and 4 incur a cost of 2).
𝑙=2
𝑙=2
𝑙=1
𝑙=1
Example 1 - Conclusions
• Therefore we have that in atomic selfish routing
problems:
– Different pure NE can have different social costs, and
different costs per edge, in contrast with the nonatomic
case.
– The PoA for affine-cost atomic instances can be higher
than 4/3, again in contrast with the nonatomic case.
– We have a lower bound of 5/2 on the PoA in unweighted
affine-cost instances.
Nonexistence of Pure NE
• We have seen that in atomic routing games, pure NEs
are not necessarily “unique” – but do they always
exist?
• We will briefly describe an atomic routing game
– which is not unweighted,
– which is not affine-cost, but rather has polynomial edge
costs of degree at most 2,
– in which there is no pure NE.
Example 2
• Consider the following game with two players, both
with source 𝑠 and target 𝑡. Player 1 has to route one
unit and Player 2 has to route two.
• Each player has four options: 𝑃1 = 𝑠 → 𝑡, 𝑃2 = 𝑠
→ 𝑣 → 𝑡, 𝑃3 = 𝑠 → 𝑤 → 𝑡, 𝑃4 = 𝑠 → 𝑣 → 𝑤 → 𝑡.
Example 2
• It can be shown that:
– If Player 2 uses 𝑃1 or 𝑃2 , then the best response for Player
1 is 𝑃4 .
– If Player 2 uses 𝑃3 or 𝑃4 , then the best response for Player
1 is 𝑃1 .
– If Player 1 uses 𝑃4 , then the best response for Player 2 is
𝑃3 .
– If Player 1 uses 𝑃1 , then the best response for Player 2 is
𝑃2 .
• Therefore the game has no pure NE.
The Potential Method
• To show the existence of pure NE for unweighted and
affine-cost atomic routing games, we will show:
– That (finite) potential games always have at least one pure
NE.
– That unweighted and affine cost games are potential
games.
• An potential game is a game for which a function
exists that “keeps track” of the change in cost
experienced by a deviating player.
The Potential Method
• Formally, Φ: 𝑄 → ℝ is an exact potential function if
for each 𝑞, 𝑞 ′ ∈ 𝑄 where 𝑞 = 𝑞1 , … , 𝑞𝑛 and 𝑞 ′
= (𝑞1 , … . , 𝑞𝑗−1 , 𝑞𝑗′ , 𝑞𝑗+1 , … 𝑞𝑛 ), it holds that 𝚽 𝒒′
− 𝚽 𝒒 = 𝒄𝒋 𝒒′ − 𝐜 𝐣 𝒒 .
• 𝑞 = 𝑞1 , … , 𝑞𝑛 ∈ 𝑄 is a local minimum of Φ if for
each 𝑞 ′ = 𝑞1 , … , 𝑞𝑗−1 , 𝑞𝑗′ , 𝑞𝑗+1 , … , 𝑞𝑛 ∈ 𝑄, Φ 𝑞
≤ Φ(𝑞 ′ ).
The Potential Method
• If a game has an exact potential function Φ, the set
of local minima of Φ is exactly the set of pure NEs.
• In potential games where 𝑄 is finite, Φ has at least
one local minimum and thus a pure NE exists.
• Remark: in finite potential games one can efficiently
construct a pure NE. This can be done using bestresponse dynamics.
Unweighted Games: Existence of Pure
NE
• We’ll show that unweighted (∀𝑗. 𝑤𝑗 = 𝑤) atomic
routing games are exact potential games.
• Denote by 𝐽𝑒 𝑞 the set of players that use the edge
𝑒 ∈ 𝐸 under strategy profile 𝑞.
• Define: 𝚽(𝒒) =
|𝑱𝒆 𝒒 |
𝚺𝒆∈𝑬 𝚺𝒊=𝟏 𝒘
⋅ 𝒇𝒆 𝒘 ⋅ 𝒊 .
• To show that Φ is an exact potential function, let 𝑞
∈ 𝑄 be a strategy profile, and let 𝑞 ′ ∈ 𝑄 be a strategy
profile obtained from 𝑞 by the player 𝑗 deviating
from path 𝑞𝑗 to path 𝑞𝑗′ .
Unweighted Games: Existence of Pure
NE
|𝑱𝒆 |
𝜮𝒆∈𝑬 𝜮𝒊=𝟏 𝒘 ⋅
• Reminder: Φ(𝑞) =
𝒇𝒆 𝒘 ⋅ 𝒊 . Consider the
difference Φ 𝑞′ − Φ 𝑞 .
• For each 𝑒 ∈ 𝑞𝑗′ ∖ 𝑞𝑗 we gain the term 𝑤 ⋅ 𝑓𝑒 𝑤
Affine-Cost Games: Existence of Pure
NE
• The existence of pure NE in affine-cost games can
also be shown through the potential method, using
the function
𝚽 𝒒 = 𝚺𝐞∈𝑬 (𝒇𝒆 𝒍𝒆 𝒍𝒆 + 𝚺𝒋∈𝑱𝒆 𝒇𝒆 𝒘𝒋 𝒘𝒋 ).
• It can be shown that Φ is a potential function, in a
similar argument to what we’ve seen.
Affine-Cost Games: Upper Bound on
PoA
• In the following two lemmas, 𝑞 ∈ 𝑄 is a pure NE and
𝑞 ∗ ∈ 𝑄 is an optimum of the social cost function.
• Lemma 1: For each player 𝑗,
Σ𝑒∈𝑞𝑗 𝑎𝑒 𝑙𝑒 (𝑞) + 𝑏𝑒 ≤ Σ𝑒∈𝑞𝑗∗ (𝑎𝑒 (𝑙𝑒 𝑞 + 𝑤𝑖 ) + 𝑏𝑒 ).
• Proof: Immediately from the definition of NE, no
player can reduce their cost by switching to their
path in the optimal profile.
Affine-Cost Games: Upper Bound on
PoA
• Lemma 2: 𝐶 𝑞 ≤ 𝐶 𝑞 ∗ + Σ𝑒∈𝐸 𝑎𝑒 𝑙𝑒 𝑞 𝑙𝑒 (𝑞 ∗ ).
• Proof:
– Multiplying each inequality from Lemma 1 by 𝑤𝑗 and
𝑛
summing over all players, 𝐶 𝑞 ≤ Σ𝑗=1
𝑤𝑗 (Σ𝑒∈𝑞𝑗∗ 𝑎𝑒 𝑙𝑒 𝑞
Affine-Cost Games: Upper Bound on
PoA
• Theorem: for atomic selfish routing games with
affine edge costs, the PoA is at most
3+ 5
2
≈ 2.618.
• Proof: it remains to bound the “error term” from the
last lemma. Applying the Cauchy-Schwarz Inequality
to the vectors 𝑎𝑒 𝑙𝑒 𝑞 𝑒∈𝐸 and 𝑎𝑒 𝑙𝑒 𝑞 ∗ 𝑒∈𝐸 ,
we get:
Σ𝑒∈𝐸 𝑎𝑒 𝑙𝑒 𝑞 𝑙𝑒 𝑞 ∗ ≤
≤ 𝐶 𝑞 ⋅ 𝐶 𝑞∗ .
Σ𝑒∈𝐸 𝑎𝑒 𝑙𝑒2 𝑞 ⋅ Σ𝑒∈𝐸 𝑎𝑒 𝑙𝑒2 𝑞 ∗
Affine-Cost Games: Upper Bound on
PoA
• We have that 𝐶 𝑞 − 𝐶 𝑞 ∗ ≤
dividing by 𝐶
• Denote 𝑥 =
have
𝑥2
𝑞∗
𝐶 𝑞
𝐶 𝑞∗
we get
𝐶 𝑞
𝐶 𝑞∗
𝐶 𝑞 ⋅ 𝐶 𝑞∗ ,
−1≤
𝐶 𝑞
𝐶 𝑞∗
.
, then by squaring both sides we
− 3𝑥 + 1 ≤ 0 ⇒ 𝑥 ≤
5
2
3+ 5
.
2
• The upper bound of on the PoA in unweighted
affine-cost instance can be obtained through a
smoothness argument.
Affine-Cost Games: Lower Bound on
PoA
• The game from Example 1 can be used to give a
lower bound for weighted affine-edge instances.
1+ 5
2
• Players 1 and 2 now have to transport 𝜙 =
units. Players 3 and 4 still have one unit each.
Affine-Cost Games: Lower Bound on
PoA
• The profile where each player takes the path of
length 1 is still the social optimum, with cost 2𝜙 2
+ 2. (Players 1 and 2 pay 𝜙 2 and Players 3 and 4 pay
1, all players incur minimum cost)
𝑙=𝜙
𝑙=𝜙
𝑙=1
𝑙=1
Affine-Cost Games: Lower Bound on
PoA
• The profile where each player takes the path of
length 2 is still a NE, with cost 4𝜙 2 + 4𝜙 + 2.
(Players 1 and 2 each pay 2𝜙 2 + 𝜙, and Players 3
and 4 each pay 𝜙 + 1).
𝑙 =1+𝜙
𝑙=𝜙
𝑙 = 1+𝜙
𝑙=𝜙
Affine-Cost Games: Lower Bound on
PoA
• Therefore the PoA in this game is at least
2𝜙2 +2𝜙+1
.
2
𝜙 +1
• 𝜙 2 = 𝜙 + 1, therefore we have a lower bound of
𝜙2 +3𝜙+2
𝜙+2
=
1+𝜙 𝜙+2
𝜙+2
=1+𝜙 =
3+ 5
.
2
Further Results - Mixed NE in AffineCost Games
• When considering mixed NE (that are guaranteed to
3+ 5
2
exist by Nash’s Theorem) the upper bound of
still holds on the PoA in affine-cost games. This is a
stronger result that implies the bound for pure NE
(since pure strategies are also mixed).
Further Results – Bounds on
Polynomial-Cost Games
• When considering games with latency functions that
are polynomials of degree at most 𝑑, there is an
upper bound of 𝑂 2𝑑 𝑑𝑑+1 on the PoA with respect
to mixed NE.
• We will now show an exponential lower bound: an
example of a game with polynomial latency functions
of degree at most 𝑑, and PoA with respect to pure
NE of Ω 𝑑𝑑\2 .
Polynomial Cost Functions: Lower
Bound
• Let 𝑙 > 𝑑 be a large enough integer.
• There are 𝑙 + 1 sets of “links”, such that in link-set 𝑘
𝑙!
∈ {0,1, … , 𝑙} there are 𝑛𝑘 = links. Each link has the
𝑘!
𝑑
same latency function 𝑥 .
• There are 𝑙 groups of players, each with unit weight,
such that in player-set 𝑘 ∈ 1,2, … , 𝑙 there are 𝑘 ⋅ 𝑛𝑘
𝑙!
=
players.
𝑘−1 !
• Players from set 𝑘 choose links from link-sets {𝑘
− 1, 𝑘, … , 𝑙}.
𝑇
Zero costs
Latency
function
= 𝑥𝑑
𝑙!
0!
𝑙!
1!
links
𝑙!
2!
links
𝑙!
3!
links
links
𝑙 links
1 link
…
Zero costs
𝑆1
𝑙!
0!
players
𝑆2
𝑙!
1!
players
𝑆3
𝑙!
2!
players
𝑆𝑙
𝑙 players
Polynomial Cost Functions: Lower
Bound
• In an optimal solution a player from set 𝑘 chooses a
link from set 𝑘 − 1, one player per link. Each player
𝑙−1 𝑙!
then pays 1 and the social cost is OPT=Σ𝑘=0 ⋅ 1𝑑
=
𝑙−1 1
𝑙! Σ𝑘=0
𝑘!
𝑘!
≈ 𝑙! ⋅ 𝑒.
Polynomial Cost Functions: Lower
Bound
• Consider the pure strategy profile 𝑞 where players
from set 𝑘 choose links from set 𝑘, 𝑘 players per link.
A player from set 𝑘 then pays 𝑘 𝑑 .
• Upon deviating to a link in set 𝑘 + 𝑖 for 𝑖
∈ −1,0,1, … 𝑙 − 𝑘 , the new cost is 𝑘 + 𝑖 + 1 𝑑
≥ 𝑘 − 1 + 1 𝑑 = 𝑘 𝑑 , therefore this is a pure NE.
• The social cost is C 𝑞 =
= 𝑙! ⋅ Ω(𝑑𝑑\2 ).
• Therefore
𝐶 𝑞
𝑂𝑃𝑇
𝑙!
𝑙
Σ𝑘=1
𝑘!
⋅
𝑘𝑑
≥
= Ω(𝑑𝑑\2 ), as required.
𝑙!
𝑑\2 !
𝑑\2
𝑑
Summary
Game Type
Unweighted, affine-cost
Affine-cost
Polynomial-cost, of max
degree 𝑑
Lower Bound on PoA
Upper Bound on PoA
2.5
2.5
1 + 𝜙 ≈ 2.618
1 + 𝜙 ≈ 2.618
Ω(𝑑 𝑑\2 )
𝑂(2𝑑 𝑑 𝑑+1 )
• Note: as in the nonatomic case, we have upper
bounds that are independent of the network
topology.
Bibliography
• N. Nisan, T. Roughgarden, E. Tardos, V. Vazirani –
Algorithmic Game Theory, Cambridge University
Press. Chapter 18, “Routing Games”.
• B. Awerbuch, Y. Azar, A. Epstein – The Price of
Routing Unsplittable Flow, In Proc. 37th Symp. Theory
of Computing, 2005
• B. Awerbuch, Y. Azar, Y. Richter, D. Tsur – Tradeoffs in
worst-case equilibria (for understanding the
restricted assignment model, in which the last
example was originally defined)