Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou's network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if $\ell_e(x)$ is the latency function of an edge $e$, we replace it by $\hat{\ell}_e(x)$ with $\ell_e(x) \le \hat{\ell}_e(x)$ for all $x$. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \CM(r) denotes the cost of the worst Nash flow in the modified network for rate $r$ and \Copt(r) denotes the cost of the optimal flow in the original network for the same rate then [\ePoA = \max_{r \ge 0} \frac{\CM(r)}{\Copt(r)}.] We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.Comment: 17 pages, 2 figures, preliminary version appeared at ESA 201

Nash Equilibria in Discrete Routing Games with Convex Latency Functions

In a discrete routing game, each of n selfish users employs a mixed strategy to ship her (unsplittable) traffic over m parallel links. The (expected) latency on a link is determined by an arbitrary non-decreasing, non-constant and convex latency function φ. In a Nash equilibrium, each user alone is minimizing her (Expected) Individual Cost, which is the (expected) latency on the link she chooses. To evaluate Nash equilibria, we formulate Social Cost as the sum of the users ’ (Expected) Individual Costs. The Price of Anarchy is the worst-case ratio of Social Cost for a Nash equilibrium over the least possible Social Cost. A Nash equilibrium is pure if each user deterministically chooses a single link; a Nash equilibrium is fully mixed if each user chooses each link with non-zero probability. We obtain: For the case of identical users, the Social Cost of any Nash equilibrium is no more than the Social Cost of the fully mixed Nash equilibrium, which may exist only uniquely. Moreover, instances admitting a fully mixed Nash equilibrium enjoy an efficient characterization. For the case of identical users, we derive two upper bounds on the Price of Anarchy: For the case of identical links with a monomial latency function φ(x) = x d, the Price of Anarchy is the Bell number of order d + 1. For pure Nash equilibria, a generic upper bound from the Wardrop model can be transfered to discrete routing games. For polynomial latency functions with non-negative coefficients and degree d, this yields an upper bound of d + 1. For th

A Metaheuristic Framework for Bi-level Programming Problems with Multi-disciplinary Applications

Bi-level programming problems arise in situations when the decision maker has to take into account the responses of the users to his decisions. Several problems arising in engineering and economics can be cast within the bi-level programming framework. The bi-level programming model is also known as a Stackleberg or leader-follower game in which the leader chooses his variables so as to optimise his objective function, taking into account the response of the follower(s) who separately optimise their own objectives, treating the leader’s decisions as exogenous. In this chapter, we present a unified framework fully consistent with the Stackleberg paradigm of bi-level programming that allows for the integration of meta-heuristic algorithms with traditional gradient based optimisation algorithms for the solution of bi-level programming problems. In particular we employ Differential Evolution as the main meta-heuristic in our proposal.We subsequently apply the proposed method (DEBLP) to a range of problems from many fields such as transportation systems management, parameter estimation and game theory. It is demonstrated that DEBLP is a robust and powerful search heuristic for this class of problems characterised by non smoothness and non convexity