Adaptively Learning Tolls to Induce Target Flows Aaron Roth Joint work with Jon Ullman and Steven Wu

Non-Atomic Congestion Games A graph representing a road network

Non-Atomic Congestion Games

Equilibrium Flows

Routing games are potential games

Manipulating equilibrium flow (classic problem)

A Natural Problem [Bhaskar, Ligett, Schulman, Swamy FOCS 14]

The [BLSS] solution in a nutshell

Very neat! (Read the paper!) A couple of limitations: Latency functions must have a simple, known form: only unknowns are a small number of coefficients. Latency functions must be convex Heavy machinery Have to run Ellipsoid. Computationally intensive. Centralized.

Some desiderata Remove assumptions on latency functions No known parametric form Not necessarily convex Not necessarily Lipschitz Make update elementary Ideally decentralized.

Proposed algorithm: Tatonnemont.

Reverse engineering why it should work View the interaction as repeated play of a game between a toll player and a flow player. What game are they playing? Flow player’s strategies are feasible flows, toll player’s strategies are tolls in some bounded range What are their cost functions? How are they playing it?

Behavior of the flow player is clear.

Behavior of the toll player?

So algorithm is consistent with:

Questions Does the equilibrium of this game correspond to the target flow? Does this repeated play converge (quickly) to equilibrium?

Questions

Does this repeated play converge (quickly) to equilibrium? It does in zero sum games! [FreundSchapire96?] Actual strategy of GD player, empirical average of BR player.

So it converges in a zero sum game.

Strategic Equivalence Adding a strategy-independent term to a player’s cost function does not change that player’s best response function And so doesn’t change the equilibria of the game…

So it converges in a zero sum game.

So the dynamics converge!

Do approximate min-max tolls guarantee the approximate target flow?

Upshot

Questions Exact convergence without assumptions on latency functions? Extensions to other games? Seem to crucially use the fact that equilibrium is the solution to a convex optimization problem…

Questions Exact convergence without assumptions on latency functions? Extensions to other games? Seem to crucially use the fact that equilibrium is the solution to a convex optimization problem…