A Mean-Risk Model for Stochastic Traffic Assignment

1. A Mean-Risk Model for Stochastic Traffic Assignment Texas A&M University College Station, TX Columbia University, New York, NY, &Universidad Di Tella, Buenos Aires, Argentina EvdokiaNikolova Nicolas Stier-Moses TX Dallas College Station Austin Houston

2. Gaming the system… Stochastic Traffic Assignment

3. … and uncertain traffic … Scatter-plot speed vs. time of day Stochastic Traffic Assignment Source: ArvindThiagarajan,PareshMalalur,CarTel.csail.mit.edu

4. …make route planning a challenge • Highway congestion costs were$115 billion in 2009. • Avg. commuter travels 100 minutes a day. Stochastic Traffic Assignment

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

6. Our model • Directed graph G = (V,E) Multiple source-dest. pairs (sk,tk), demand dk • Players: nonatomicor atomic unsplittable Strategy set: paths Pk between (sk,tk) for all k Players’ decisions: flow vector • Edge delay functions: Expected delay Random variable with standard deviation e(xe) Stochastic Traffic Assignment

7. 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 Stochastic Traffic Assignment

8. 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: Stochastic Traffic Assignment

9. 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 Stochastic Traffic Assignment

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) Stochastic Traffic Assignment

11. Talk outline • Equilibrium existence and characterization • Contrast with deterministic game • Succinct representation • Inefficiency of equilibria Stochastic Traffic Assignment

12. Results I: Equilibrium existence & characterization Stochastic Traffic Assignment

13. Equilibria in nonatomic games I Theorem: Equilibria in nonatomic games with exogenous noise exist. Proof: Corollary 1: Uniqueness; computation via column generation. 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’ ) Stochastic Traffic Assignment

14. 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. Stochastic Traffic Assignment

15. 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. Player 2: Path p2’  p2 (p1,p2,p) (p1,p2’,p) Player 1: Path p1  p1’ Player 1: Path p1’  p1 (p1’,p2,p) (p1’,p2’,p) Player 2: Path p2  p2’ Stochastic Traffic Assignment

16. Equilibria in atomic games Theorem: The atomic routing game with exogenous noise is a potential game, hence pure strategy equilibria exist. Stochastic Traffic Assignment

17. 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.) Stochastic Traffic Assignment

18. 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]. 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

19. Talk outline • Equilibrium existence and characterization • Contrast with deterministic game • Succinct representation • Inefficiency of equilibria Stochastic Traffic Assignment

20. 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!) mean, variance a, 8 b, 1 S T a+1, 3 b-1, 8 Stochastic Traffic Assignment

21. 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. Stochastic Traffic Assignment

22. Talk outline • Equilibrium existence and characterization • Contrast with deterministic game • Succinct representation • Inefficiency of equilibria (price of anarchy) Stochastic Traffic Assignment

23. Example: Inefficiency of equilibria Delay is 1.5 hours for everybody at the unique Nash equilibrium 50 x/100 hours 1 hour Town B Town A 1 hour x/100 hours 50 Suppose 100 drivers leave from town A towards town B. Every driver wants to minimize her own travel time. What is the traffic on the network? In any unbalanced traffic pattern, all drivers on the most loaded path have incentive to switch their path.

24. Example: Inefficiency of equilibria Delay is 2 hours for everybody at the unique Nash equilibrium 100 x/100 hours 1 hour Town B Town A 0 hours 1 hour x/100 hours A benevolent mayor builds a superhighway connecting the fast highways of the network. What is now the traffic on the network? No matter what the other drivers are doing it is always better for me to follow the zig-zag path.

25. Example: Inefficiency of equilibria 100 x/100 hours x/100 hours 1 hour 1 hour vs B A 50 1 hour 1 hour x/100 hours x/100 hours Adding a fast road on a road-network is not always a good idea! Braess’s paradox B A In the RHS network there exists a traffic pattern where all players have delay 1.5 hours. 50 Price of Anarchy: measures the loss in system performance due to free-will

26. 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 Stochastic Traffic Assignment

27. Nonconvexity of Social Cost Stochastic Traffic Assignment

28. 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 Stochastic Traffic Assignment

29. Summary • Agenda: extension of classical theory of routinggames 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). Stochastic Traffic Assignment

30. 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? Stochastic Traffic Assignment