Post on 22-Feb-2016
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A Mean-Risk Model for Stochastic Traffic Assignment
Evdokia Nikolova Nicolas Stier-Moses Columbia University,
New York, NY, &Universidad Di Tella,
Buenos Aires, Argentina
Texas A&M University College Station, TX
AustinHouston
Dallas
College Station
TX
Gaming the system…
Evdokia Nikolova Stochastic Traffic Assignment
Stochastic Traffic Assignment
… and uncertain traffic …Scatter-plot speed vs. time of day
Evdokia Nikolova Source: Arvind Thiagarajan, Paresh Malalur, CarTel.csail.mit.edu
Stochastic Traffic Assignment
…make route planning a challenge
Evdokia Nikolova
• Highway congestion costs were$115 billion in 2009.
• Avg. commuter travels 100 minutes a day.
Stochastic Traffic Assignment
Commuters pad travel times Worst case > double the average
Source: Texas Transportation Institute; ABC News Survey.Evdokia Nikolova
Stochastic Traffic Assignment
Our model
• Directed graph G = (V,E)Multiple source-dest. pairs (sk,tk), demand dk
• Players: nonatomic or atomic unsplittableStrategy set: paths Pk between (sk,tk) for all kPlayers’ decisions: flow vector
• Edge delay functions:
||Rx)()( eeee xxl
Expected delay
Random variable with standard
deviation e(xe)Evdokia Nikolova
Stochastic Traffic Assignment
User cost functions• Mean-standard deviation objective:
• Pros: – Widely used to incorporate uncertainty (transportation, finance)– Incorporates risk-aversion– Interpretation under normal distributions: Equal to percentile of delay
• Cons: – May result in stochastically dominated paths– Difficult to optimize
2)()()(
routee
eeroutee
eeroute xxlxQ
Evdokia Nikolova
Stochastic Wardrop Equilibrium• Users minimize mean-stdev objective
• Definition: A flow x such that for every source-dest. pair k and for every route with positive flow between this pair,
- Nonatomic:
- Atomic: )()( '' routerouterouteroute IIxQxQ
' allfor ),()( ' routexQxQ routeroute
)(xQroute
Evdokia Nikolova Stochastic Traffic Assignment
Related Work• Routing games: Wardrop ‘52, Beckmann et al.
’56, …, a lot of work in AGT community and others Surveys of recent work: • AGT Book Nisan et al. ‘07• Correa, Stier-Moses ’11
• Uncertainty: Dial ‘71 Stochastic User Equilibrium
• Risk-aversion: • In routing games: Ordóñez, Stier-Moses’10, Nie’11 • In routing: Nikolova ‘10
Evdokia Nikolova Stochastic Traffic Assignment
Stochastic Traffic Assignment
Player’s best responses
• Stochastic shortest path with fixed means and standard deviations on edges
• Nonconvex combinatorial problem of unknown complexity: – best exact algorithm runs in time nO(log n) [n = #vertices] – admits Fully-Polynomial Approximation Scheme (Nikolova
’10)
Evdokia Nikolova
Stochastic Traffic Assignment
Talk outline
• Equilibrium existence and characterization• Contrast with deterministic game
• Succinct representation
• Inefficiency of equilibria
Evdokia Nikolova
Results I: Equilibrium existence & characterization
Equilibrium characterization
Exogenous noise Endogenous noise
Atomic users Eq. exists Potential game
No equilibrium! (in pure strategies)
Nonatomic users Eq. exists Solves convex program (expon. large)
Eq. exists (under general continuous objectives!) Solves variational inequality
Evdokia Nikolova Stochastic Traffic Assignment
Equilibria in nonatomic games ITheorem: Equilibria in nonatomic games with
exogenous noise exist.
Proof:
Corollary 1: Uniqueness; computation via column generation.
. allfor 0
, allfor
, allfor s.t.
)(min
:
2
0
Ppf
Kkfd
Eefx
fdzzl
p
Pppk
pePppe
Pp peep
Ee
x
e
k
e
Evdokia Nikolova Stochastic Traffic Assignment
Lemma: Flow vector f is locally optimal if for each path p with positive flow and each path p’,
( marginal benefit of ( marginal cost of reducing traffic on p ) increasing traffic on p’ )
2
''
2 )()(
pe
epe
eepe
epe
ee xlxl
Equilibria in nonatomic games ITheorem: Equilibria in nonatomic games with exogenous
noise exist.
Proof:
Corollary 2: If mean delays are constant: then, the equilibrium can be found in time solving
• Computational complexity of subproblem open.
Pp pe
epEe
x
e fdzzle
2
0
)(min
Evdokia Nikolova Stochastic Traffic Assignment
2min
pe
epe
ep
ee xl )()(lognOn
Equilibria in atomic games Theorem: The atomic routing game with exogenous noise
is a potential game, hence pure strategy equilibria exist.
Proof: We can devise a potential function similar to non-atomic setting. Or, verify the 4-cycle condition of Monderer & Shapley (1996):
Game is potential ifftotal change in players’ utilities along every cycle of length 4 is 0.
Evdokia Nikolova Stochastic Traffic Assignment
(p1’,p2’,p)
(p1,p2’,p)
Player 1:Path p1 p1’
Player 2:Path p2 p2’
Player 1:Path p1’ p1
Player 2:Path p2’ p2
(p1,p2,p)
(p1’,p2,p)
Equilibria in atomic games Theorem: The atomic routing game with exogenous
noise is a potential game, hence pure strategy equilibria exist.
Evdokia Nikolova Stochastic Traffic Assignment
0)',(),(
)','()',(
),'()','(
),(),'(
2
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'21'
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peep
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ppmeanppmean
ppmeanppmean
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ppmeanppmean
Equilibria in atomic games Theorem: The atomic routing game with exogenous noise is
a potential game, hence pure strategy equilibria exist.
• Not true when noise in endogenous.
• Can exhibit examples with no pure strategy equilibria. • Note correspondence to nonatomic game (convex
objective is a potential function.)
Evdokia Nikolova Stochastic Traffic Assignment
Equilibria in nonatomic games IITheorem: Equilibria in nonatomic games with endogenous
noise exist. Proof: Equilibrium is solution to Variational Inequality (VI)
where
VI Solution exists over compact convex set with Q(f) continuous [Hartman, Stampacchia ‘66]. ∎
• VI Solution unique if Q(f) is monotone: (Q(f)-Q(f’))(f-f’) ≥ 0. [not true here].
' flows feasible allfor 0)'()( ffffQ
operator]cost [path ))(),...,(()(
vector]flow[path ),...,(
||1
||1
fQfQfQ
fff
P
P
Evdokia Nikolova Stochastic Traffic Assignment
Claim: Flow f is an equilibrium if and only if Q(f).f <= Q(f).f’ .
Proof: (=>) Equilibrium flow routes along minimum-cost paths Q(f). Fixing path costs at Q(f), any other flow f’ that assigns flow to higher-cost paths will result in higher overall cost Q(f).f’.
(<=) Suppose f is not an eq. Then there is a flow-carrying path p with Qp(f) > Qp’(f). Shifting flow from p to p’ will obtain Q(f).f’ < Q(f).f, contradiction.
Stochastic Traffic Assignment
Talk outline
• Equilibrium existence and characterization• Contrast with deterministic game
• Succinct representation
• Inefficiency of equilibria
Evdokia Nikolova
Results II: Succinct representation of equilibria and social optima
• Proposition: Not every path flow decomposition of an equilibrium edge-flow vector is at equilibrium. (in contrast to deterministic routing games!)
a, 8
a+1, 3
b, 1
b-1, 8
S T
mean, variance
Evdokia Nikolova Stochastic Traffic Assignment 11
3
ncepath variameanpath
,
,,,
baQ
baQQQ
Q
bottombottom
topbottombottomtoptoptop
path
Results II: Succinct representation of equilibria and social optima
• Proposition: Not every path flow decomposition of an equilibrium edge-flow vector is at equilibrium. (in contrast to deterministic routing games!)
• Theorem 1: For every equilibrium given as edge flow, there exists a succinct flow decomposition that uses at most |E|+|K| paths.
• Theorem 2: For a social optimum given as edge flow, there exists a succinct flow decomposition that uses at most |E|+|K| paths.
Evdokia Nikolova Stochastic Traffic Assignment
Stochastic Traffic Assignment
Talk outline
• Equilibrium existence and characterization• Contrast with deterministic game
• Succinct representation
• Inefficiency of equilibria (price of anarchy)
Evdokia Nikolova
Example: Inefficiency of equilibria
Town A Town B
Suppose 100 drivers leave from town A towards town B.
What is the traffic on the network?Every driver wants to minimize her own travel time.
50
50
In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path.
Delay is 1.5 hours for everybody at the unique Nash equilibriumx/100 hours
x/100 hours
1 hour
1 hour
Example: Inefficiency of equilibria
Town A Town B
A benevolent mayor builds a superhighway connecting the fast highways of the network.
What is now the traffic on the network?
100
No matter what the other drivers are doing it is always better for me to follow the zig-zag path.
Delay is 2 hours for everybody at the unique Nash equilibriumx/100 hours
x/100 hours
1 hour
1 hour
0 hours
Example: Inefficiency of equilibria
A B
100
A B
50
50
vs
Adding a fast road on a road-network is not always a good idea! Braess’s paradox
In the RHS network there exists a traffic pattern where all players have delay 1.5 hours.
Price of Anarchy: measures the loss in system performance due to free-will
x/100 hours
x/100 hours
1 hour
1 hour
x/100 hours
x/100 hours
1 hour
1 hour
Price of Anarchy
• Cost of Flow: total user cost
• Social optimum: flow minimizing total user cost
• Price of anarchy: (Koutsoupias, Papadimitriou ’99)
Generalizes stochastic shortest path problem
Cost Optimum SocialCost mEquilibriusup
instancesproblem
Evdokia Nikolova Stochastic Traffic Assignment
Stochastic Traffic Assignment
Nonconvexity of Social Cost
Evdokia Nikolova
Results III: Price of Anarchy • Exogenous noise: The price of anarchy in the
stochastic routing game with exogenous noise is the same as in deterministic routing games: - 4/3 for linear expected delays- for general expected delays in class L
• Endogenous noise: Identify special setting with POA = 1; open if techniques extend to more general settingsOther results: - Social cost is convex when path costs are convex &
monotone.- Path costs are convex when means, stdevs are [but not
always monotone, so social cost is not always convex.]
1))(1( L
Deterministic related work: Roughgarden, Tardos ’02; Correa, Schulz, Stier-Moses ‘04, ‘08
Evdokia Nikolova Stochastic Traffic Assignment
Summary• Agenda: extension of classical theory of
routing games to stochastic settings (edge delays) and risk-aversion
• Equilibrium existence & characterization
• Succinct decomposition of equilibria and social opt.
• Price of anarchy: Same for exogenous noise. Open for endogenous (need new bounding methods).
Evdokia Nikolova Stochastic Traffic Assignment
Open questions• What is complexity of computing equilibrium?
• What is complexity of computing social optimum?
• Can there be multiple equilibria in nonatomic game with endogenous noise?
• What is Price of Anarchy for endogenous noise?
• Heterogeneous risk attitudes; other risk functions?
Evdokia Nikolova Stochastic Traffic Assignment