# Traffic assignment with uncertain travel times

As vehicles become smarter, increasingly automated and with access to massive amounts of data, they become better equipped to commucate, coordinate and compute optimal routes so as to minimize congestion. Such coordination is modeled by routing games, where a central goal is to understand and compute a socially optimal traffic assignment that minimizes overall congestion.

WNCG Prof. Nikolova and Prof. Stier-Moses from Universidad Di Tella, Argentina are laying the theoretical foundation of how risk aversion and uncertainty transform classic routing games. Routing games were one of the principal examples in the development of algorithmic game theory. In these games, multiple users need to route between different source-destination pairs and edges are congestible, namely, each edge delay is a non-decreasing function of the flow or number of users on the edge. A key motivation for studying routing games is understanding and mitigating congestion in transportation networks. However, unlike the classic routing game model where edge delays are known, in reality they are uncertain—heavy and uncertain traffic conditions exacerbate the commuting experience of millions of people across the globe. When planning important trips, commuters typically add an extra buffer to the expected trip duration to ensure on-time arrival.

Motivated by this, in a recent paper Nikolova and Stier-Moses propose and analyze a new traffic assignment model that takes into account uncertain travel times and risk-averse users. Users capture the tradeoff between travel times and their variability in a mean-standard deviation objective, defined as the mean travel time plus a risk-aversion factor times the standard deviation of travel time along a path. The paper provides a characterization of an equilibrium traffic assignment and provides conditions when it exists. The main challenge is posed by the users’ risk aversion, since the mean-standard deviation objective is nonconvex and nonseparable, meaning that a path cannot be split as a sum of edge costs. As a result, even an individual user’s subproblem—a stochastic shortest path problem—is a nonconvex optimization problem for which no polynomial time algorithms are known. In turn, the mathematical structure of the traffic assignment model with stochastic travel times is fundamentally different from the classic deterministic model. In particular, an equilibrium characterization requires exponentially many variables, one for each path in the network, since an edge-flow has multiple possible path-flow decompositions that are not equivalent. Because of this, characterizing the equilibrium and the socially-optimal traffic assignment, which minimizes the total user cost, is more challenging than in the traditional deterministic setting. Nevertheless, the paper proves that both can be encoded by a representation with just polynomially-many paths. Also, under the assumption that the standard deviations of travel times are independent from edge loads, it shows that the worst-case ratio between the total user cost in an equilibrium and that of an optimal solution is not higher than the analogous ratio in the deterministic setting. In other words, uncertainty does not further degrade the system performance in addition to strategic user behavior alone.

Read full journal paper: http://users.ece.utexas.edu/~nikolova/papers/stochWardropEqFinalOR.pdf

Earlier conference abstract: http://users.ece.utexas.edu/~nikolova/papers/SAGT2011-Paper39.pdf

Technical summary: http://www.sigecom.org/exchanges/volume_11/1/NIKOLOVA.pdf