A Mean-Risk Model for Stochastic Traffic Assignment Evdokia Nikolova Nicolas Stier-Moses Columbia University, New York, NY, & Universidad Di Tella, Buenos

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


…make route planning a challenge

Evdokia Nikolova

• Highway congestion costs were$115 billion in 2009.

• Avg. commuter travels 100 minutes a day.


Commuters pad travel times Worst case > double the average

Source: Texas Transportation Institute; ABC News Survey.Evdokia Nikolova


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


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


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



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


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


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


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


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.


(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.


0)',(),(

)','()',(

),'()','(

),(),'(

2

'21'

221

2

'21'

221

221

2

'21'

221

2

'21'

2

2

2

2

1

1

1

1

2

2

2

2

1

1

1

1

peep

peep

peep

peep

peep

peep

peep

peep

ppmeanppmean

ppmeanppmean

ppmeanppmean

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.)


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


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.


Talk outline



• 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.



Talk outline



• 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


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


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



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


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).


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?


Documents

A Mean-Risk Model for Stochastic Traffic Assignment Evdokia Nikolova Nicolas Stier-Moses Columbia University, New York, NY, & Universidad Di Tella, Buenos