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 Austin Houston Dallas College Station TX

Gaming the system… Evdokia NikolovaStochastic Traffic Assignment

… and uncertain traffic … Scatter-plot speed vs. time of day Evdokia NikolovaStochastic Traffic Assignment Source: Arvind Thiagarajan, Paresh Malalur, CarTel.csail.mit.edu

…make route planning a challenge Evdokia NikolovaStochastic Traffic Assignment Highway congestion costs were $115 billion in Avg. commuter travels 100 minutes a day.

Commuters pad travel times Worst case > double the average Source: Texas Transportation Institute; ABC News Survey. Evdokia NikolovaStochastic Traffic Assignment

Our model Directed graph G = (V,E) Multiple source-dest. pairs (s k,t k ), demand d k Players: nonatomic or atomic unsplittable Strategy set: paths P k between (s k,t k ) for all k Players’ decisions: flow vector Edge delay functions: Expected delay Random variable with standard deviation  e (x e ) Evdokia NikolovaStochastic 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 Evdokia NikolovaStochastic Traffic Assignment

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: Evdokia NikolovaStochastic 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 NikolovaStochastic 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 n O(log n) [n = #vertices] – admits Fully-Polynomial Approximation Scheme (Nikolova ’10) Evdokia NikolovaStochastic Traffic Assignment

Talk outline Equilibrium existence and characterization Contrast with deterministic game Succinct representation Inefficiency of equilibria Evdokia NikolovaStochastic Traffic Assignment

Results I: Equilibrium existence & characterization Evdokia NikolovaStochastic Traffic Assignment

Equilibria in nonatomic games I Theorem: Equilibria in nonatomic games with exogenous noise exist. Proof: Corollary 1: Uniqueness; computation via column generation. Evdokia NikolovaStochastic 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’ )

Equilibria in nonatomic games I Theorem: 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. Evdokia NikolovaStochastic Traffic Assignment

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 iff total change in players’ utilities along every cycle of length 4 is 0. Evdokia NikolovaStochastic 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 NikolovaStochastic Traffic Assignment

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 NikolovaStochastic Traffic Assignment

Equilibria in nonatomic games II Theorem: 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]. Evdokia NikolovaStochastic 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’. ( Qp’(f). Shifting flow from p to p’ will obtain Q(f).f’ < Q(f).f, contradiction.

Talk outline Equilibrium existence and characterization Contrast with deterministic game Succinct representation Inefficiency of equilibria Evdokia NikolovaStochastic Traffic Assignment

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 ST mean, variance Evdokia NikolovaStochastic Traffic Assignment

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 NikolovaStochastic Traffic Assignment

Talk outline Equilibrium existence and characterization Contrast with deterministic game Succinct representation Inefficiency of equilibria (price of anarchy) Evdokia NikolovaStochastic Traffic Assignment

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 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 equilibrium x/100 hours 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 equilibrium x/100 hours 1 hour 0 hours

Example: Inefficiency of equilibria A B 100 A B 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 1 hour x/100 hours 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 Evdokia NikolovaStochastic Traffic Assignment

Nonconvexity of Social Cost Evdokia NikolovaStochastic Traffic Assignment

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 settings Other 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.] Deterministic related work: Roughgarden, Tardos ’02; Correa, Schulz, Stier-Moses ‘04, ‘08 Evdokia NikolovaStochastic 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 NikolovaStochastic 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 NikolovaStochastic Traffic Assignment