Karsten Tiemann

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.

Congestion games with player-specific constants

We consider a special case of weighted congestion games with playerspecific latency functions where each player uses for each particular resource a fixed (non-decreasing) delay function together with a player-specific constant. For each particular resource, the resource-specific delay function and the playerspecific constant (for that resource) are composed by means of a group operation (such as addition or multiplication) into a player-specific latency function. We assume that the underlying group is a totally ordered abelian group. In this way, we obtain the class of weighted congestion games with player-specific constants; we observe that this class is contained in the new intuitive class of dominance weighted congestion games. We obtain the following results: Games on parallel links: - Every unweighted congestion game has a generalized ordinal potential. - There is a weighted congestion game with 3 players on 3 parallel links that does not have the Finite Best-Improvement Property. - There is a particular best-improvement cycle for general weighted congestion games with player-specific latency functions and 3 players whose outlaw implies the existence of a pure Nash equilibrium. This cycle is indeed outlawed for dominance weighted congestion games with 3 players - and hence for weighted congestion games with player-specific constants and 3 players. Network congestion games: For unweighted symmetric network congestion games with player-specific additive constants, it is PLS-complete to find a pure Nash equilibrium. Arbitrary (non-network) congestion games: Every weighted congestion game with linear delay functions and player-specific additive constants has a weighted potential.

Nash equilibria, the price of anarchy and the fully mixed nash equilibrium conjecture

Motivation-Framework. Apparently, it is in human’s nature to act selfishly. Game Theory, founded by von Neumann and Morgenstern [39, 40], provides us with strategic games, an important mathematical model to describe and analyze such a selfish behavior and its resulting conflicts. In a strategic game, each of a finite set of players aims for an optimal value of its private objective function by choosing either a pure strategy (a single strategy) or a mixed strategy (a probability distribution over all pure strategies) from its strategy set. Strategic games in which the strategy sets are finite are called finite strategic games. Each player chooses its strategy once and for all, and all players’ choices are made non-cooperatively and simultaneously (that is, when choosing a strategy each player is not informed of the strategies chosen by any other player). One of the basic assumption in strategic games is that the players act rational, that is, consistently in pursuit of their private objective function. For a concise introduction to contemporary Game Theory we recommend [25].

Routing (un-) splittable flow in games with player-specific linear 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 without a constant term 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. We also show for several generalizations of these routing games that such potential functions do not exist. We prove upper and lower bounds on the price of anarchy for games with linear latency functions. For the case of unsplittable traffic the upper and lower bound are asymptotically tight.

Wardrop equilibria and price of stability for bottleneck games with splittable traffic

We look at the scenario of having to route a continuous rate of traffic from a source node to a sink node in a network, where the objective is to maximize throughput. This is of interest, e.g., for providers of streaming content in communication networks. The overall path latency, which was relevant in other non-cooperative network routing games such as the classic Wardrop model, is of lesser concern here. To that end, we define bottleneck games with splittable traffic where the throughput on a path is inversely proportional to the maximum latency of an edge on that very path-the bottleneck latency. Therefore, we define a Wardrop equilibrium as a traffic distribution where this bottleneck latency is at minimum on all used paths. As a measure for the overall system well-being-called social cost-we take the weighted sum of the bottleneck latencies of all paths. Our main findings are as follows: First, we prove social cost of Wardrop equilibria on series parallel graphs to be unique. Even more, for any graph whose subgraph induced by all simple start-destination paths is not series parallel, there exist games having equilibria with different social cost. For the price of stability, we give an independence result with regard to the network topology. Finally, our main result is giving a new exact price of stability for Wardrop/bottleneck games on parallel links with M/M/1 latency functions. This result is at the same time the exact price of stability for bottleneck games on general graphs.

A faster branch-and-bound algorithm for the test-cover problem based on set-covering techniques

The test-cover problem asks for the minimal number of tests needed to uniquely identify a disease, infection, etc. A collection of branch-and-bound algorithms was proposed by De Bontridder et al. [2002]. Based on their work, we introduce several improvements that are compatible with all techniques described in De Bontridder et al. [2002] and the more general setting of weighted test-cover problems. We present a faster data structure, cost-based variable fixing, and adapt well-known set-covering techniques, including Lagrangian relaxation and upper-bound heuristics. The resulting algorithm solves benchmark instances up to 10 times faster than the former approach and up to 100 times faster than a general MIP solver.

MultiProcessor Scheduling is PLS-Complete

We consider two natural models of local improvement. We show that the MULTIPROCESSOR SCHEDULING problem, i.e., the problem of scheduling weighted jobs on identical machines with the objective to minimize the makespan, is PLS-complete for a sufficiently large neighborhood. In the first model, in an improvement step, either the makespan decreases or the makespan remains unchanged and the number of makespan machines decreases. In the second model, we consider the selfish version of the problem, where the jobs are viewed as selfish agents. The cost of an agent is the load of the machine to which it is assigned. Agents may form arbitrary, non-fixed coalitions. The cost of a coalition is defined to be the maximum cost of its members. In an improvement step, the cost of the coalition of reallocating agents decreases. Both these problems are PLS-complete for local improvement algorithms with steps which include reallocating up to 33 jobs/agents. We show these results by reduction from the MAXCONSTRAINTASSIGNMENT problem (p,q,r)-MCA, which

Routing and scheduling with incomplete information

In many large-scale distributed systems the users have only incomplete information about the system. We outline game theoretic approaches that are used to model such incomplete information settings.