Martin Gairing

Computing Nash equilibria for scheduling on restricted parallel links

We consider the problem of routing n users on m parallel links, under the restriction that each user may only be routed on a link from a certain set of allowed links for the user. Thus, the problem is equivalent to the correspondingly restricted problem of assigning n jobs to m parallel machines. In a pure Nash equilibrium, no user may improve its own individual cost (delay) by unilaterally switching to another link from its set of allowed links. As our main result, we introduce a polynomial time algorithm to compute from any given assignment a pure Nash equilibrium with non-increased makespan. The algorithm gradually changes a given assignment by pushing unsplittable user traffics through a network that is defined by the users and the links. Here, we use ideas from blocking flows. Furthermore, we use similar techniques as in the generic Preflow-Push algorithm to approximate a schedule with minimum makespan, gaining an improved approximation factor of 2 - 1/w 1 for identical links, where w 1 is the largest user traffic. We extend this result to related links, gaining an approximation factor of 2. Our approximation algorithms run in polynomial time. We close with tight upper bounds on the coordination ratio for pure Nash equilibria.

Nashification and the coordination ratio for a selfish routing game

We study the problem of n users selfishly routing traffic through a network consisting of m parallel related links. Users route their traffic by choosing private probability distributions over the links with the aim of minimizing their private latency. In such an environment Nash equilibria represent stable states of the system: no user can improve its private latency by unilaterally changing its strategy. Nashification is the problem of converting any given non-equilibrium routing into a Nash equilibrium without increasing the social cost. Our first result is an O(nm2) time algorithm for Nashification. This algorithm can be used in combination with any approximation algorithm for the routing problem to compute a Nash equilibrium of the same quality. In particular, this approach yields a PTAS for the computation of a best Nash equilibrium. Furthermore, we prove a lower bound of Ω(2√n) and an upper bound of O(2n) for the number of greedy selfish steps for identical link capacities in the worst case. In the second part of the paper we introduce a new structural parameter which allows us to slightly improve the upper bound on the coordination ratio for pure Nash equilibria in [3]. The new bound holds for the individual coordination ratio and is asymptotically tight. Additionally, we prove that the known upper bound of 1+√4m-3/2 on the coordination ratio for pure Nash equilibria also holds for the individual coordination ratio in case of mixed Nash equilibria, and we determine the range of m for which this bound is tight.

Exact Price of Anarchy for Polynomial Congestion Games

We show exact values for the worst-case price of anarchy in weighted and unweighted (atomic unsplittable) congestion games, provided that all cost functions are bounded-degree polynomials with nonnegative coefficients. The given values also hold for weighted and unweighted network congestion games.

Exact price of anarchy for polynomial congestion games

We show exact values for the price of anarchy of weighted and unweighted congestion games with polynomial latency functions. The given values also hold for weighted and unweighted network congestion games.

Selfish Routing with Incomplete Information

Abstract In his seminal work, Harsanyi (Manag. Sci. 14, 159–182, 320–332, 468–502, 1967) introduced an elegant approach to study non-cooperative games with incomplete information. In our work, we use this approach to define a new selfish routing game with incomplete information that we call Bayesian routing game. Here, each of n selfish users wishes to assign its traffic to one of m parallel links. However, users do not know each other’s traffic. Following Harsanyi’s approach, we introduce, for each user, a set of possible types. In our model, each type of a user corresponds to some traffic and the players’ uncertainty about each other’s traffic is described by a probability distribution over all possible type profiles. We present a comprehensive collection of results about our Bayesian routing game. Our main findings are as follows: • Using a potential function, we prove that every Bayesian routing game has a pure Bayesian Nash equilibrium. More precisely, we show this existence for a more general class of games that we call weighted Bayesian congestion games. For Bayesian routing games with identical links and independent type distribution, we give a polynomial time algorithm to compute a pure Bayesian Nash equilibrium. • We study structural properties of fully mixed Bayesian Nash equilibria for the case of identical links and show that they maximize Individual Cost. In general, there is more than one fully mixed Bayesian Nash equilibrium. We characterize fully mixed Bayesian Nash equilibria for the case of independent type distribution. • We conclude with bounds on Coordination Ratio for the case of identical links and for three different Social Cost measures: Expected Maximum Latency, Sum of Individual Costs and Maximum Individual Cost. For the latter two, we are able to give (asymptotically) tight bounds using the properties of fully mixed Bayesian Nash equilibria we proved.

Routing (un-) splittable flow in games with player-specific affine latency functions

In this work we study weighted network congestion games with player-specific latency functions where selfish players wish to route their traffic through a shared network. We consider both the case of splittable and unsplittable traffic. Our main findings are as follows. For routing games on parallel links with linear latency functions, we introduce two new potential functions for unsplittable and for splittable traffic, respectively. We use these functions to derive results on the convergence to pure Nash equilibria and the computation of equilibria. For several generalizations of these routing games, we show that such potential functions do not exist. We prove tight upper and lower bounds on the price of anarchy for games with polynomial latency functions. All our results on the price of anarchy translate to general congestion games.

Weighted Congestion Games: The Price of Anarchy, Universal Worst-Case Examples, and Tightness

We characterize the Price of Anarchy (POA) in weighted congestion games, as a function of the allowable resource cost functions. Our results provide as thorough an understanding of this quantity as is already known for nonatomic and unweighted congestion games, and take the form of universal (cost function-independent) worst-case examples. One noteworthy by-product of our proofs is the fact that weighted congestion games are “tight,” which implies that the worst-case price of anarchy with respect to pure Nash equilibria, mixed Nash equilibria, correlated equilibria, and coarse correlated equilibria are always equal (under mild conditions on the allowable cost functions). Another is the fact that, like nonatomic but unlike atomic (unweighted) congestion games, weighted congestion games with trivial structure already realize the worst-case POA, at least for polynomial cost functions. We also prove a new result about unweighted congestion games: the worst-case price of anarchy in symmetric games is as large as in their more general asymmetric counterparts.

Selfish Routing in Non-cooperative Networks: A Survey

We study the problem of n users selfishly routing traffics through a shared network. Users route their traffics by choosing a path from their source to their destination of the traffic with the aim of minimizing their private latency. In such an environment Nash equilibria represent stable states of the system: no user can improve its private latency by unilaterally changing its strategy.

Computing stable outcomes in hedonic games

We study the computational complexity of finding stable outcomes in symmetric additively-separable hedonic games. These coalition formation games are specified by an undirected edge-weighted graph: nodes are players, an outcome of the game is a partition of the nodes into coalitions, and the utility of a node is the sum of incident edge weights in the same coalition. We consider several natural stability requirements defined in the economics literature. For all of them the existence of a stable outcome is guaranteed by a potential function argument, so local improvements will converge to a stable outcome and all these problems are in PLS. The different stability requirements correspond to different local search neighbourhoods. For different neighbourhood structures, our findings comprise positive results in the form of polynomial-time algorithms for finding stable outcomes, and negative (PLS-completeness) results.

The price of anarchy for polynomial social cost

In this work, we consider an interesting variant of the well studied KP model for selfish routing on parallel links, which reflects some influence from the much older Wardrop model [J.G. Wardrop, Some theoretical aspects of road traffic research, Proc. Inst. of Civil Eng. Part II 1 (1956) 325-378]. In the new model, user traffics are still unsplittable and links are identical. Social cost is now the expectation of the sum, over all links, of latency costs; each latency cost is modeled as a certain polynomial latency cost function evaluated at the latency incurred by all users choosing the link. The resulting social cost is called polynomial social cost, or monomial social cost when the latency cost function is a monomial. All considered polynomials are of degree d, where d ≥ 2, and have non-negative coefficients. We are interested in evaluating Nash equilibria in this model, and we use the monomial price of anarchy (MPoA) and the polynomial price of anarchy (PPoA) as our evaluation measures. Through establishing some remarkable relations of these costs and measures to some classical combinatorial numbers such as the Stirling numbers of the second kind and the Bell numbers, we obtain a multitude of results: • For the special case of identical users: The fully mixed Nash equilibrium, where all probabilities are strictly positive, maximizes polynomial social cost. The MPoA is no more than Bd, the Bell number of order d. This immediately implies that the PPoA is no more than Σ1 ≤ t ≤ dBt. For the special case of two links, the MPoA is no more than 2d-2(1 + (1/n)d-1), and this bound is tight for n = 2. • The MPoA is exactly ((2d - 1)d/(d - 1)(2d - 2)d-1)((d - 1)/d)d for pure Nash equilibria. This immediately implies that the PPoA is no more than Σ2 ≤ t ≤ d ((2t - 1)t/(t - 1)(2t - 2)t-1)((t - 1)/t)t.