We study rerouting policies in a dynamic round-based variant of a well known game theoretic traffic model due to Wardrop. Previous analyses (mostly in the context of selfish routing) based on Wardrop's model focus mostly on the static analysis of equilibria. In this paper, we ask the question whether the population of agents responsible for routing the traffic can jointly compute or better learn a Wardrop equilibrium efficiently. The rerouting policies that we study are of the following kind. In each round, each agent samples an alternative routing path and compares the latency on this path with its current latency. If the agent observes that it can improve its latency then it switches with some probability depending on the possible improvement to the better path.We can show various positive results based on a rerouting policy using an adaptive sampling rule that implicitly amplifies paths that carry a large amount of traffic in the Wardrop equilibrium. For general asymmetric games, we show that a simple replication protocol in which agents adopt strategies of more successful agents reaches a certain kind of bicriteria equilibrium within a time bound that is independent of the size and the structure of the network but only depends on a parameter of the latency functions, that we call the relative slope. For symmetric games, this result has an intuitive interpretation: Replication approximately satisfies almost everyone very quickly.In order to achieve convergence to a Wardrop equilibrium * besides replication one also needs an exploration component discovering possibly unused strategies. We present a sampling based replication-exploration protocol and analyze its convergence time for symmetric games. For example, if the latency functions are defined by positive polynomials in coefficient representation, the convergence time is polynomial in the representation length of the latency functions. To the best of our knowledge, all previous results on the speed of convergence towards Wardrop equilibria, even when restricted to linear latency functions, were pseudopolynomial. In addition to the upper bounds on the speed of convergence, we can also present a lower bound demonstrating the necessity of adaptive sampling by showing that static sampling methods result in a slowdown that is exponential in the size of the network. A further lower bound illustrates that the relative slope is, in fact, the relevant parameter that determines the speed of convergence.
We investigate adaptive routing policies for large networks in which agents reroute traffic based on old information. It is a well known and practically relevant problem that old information can lead to undesirable oscillation effects resulting in poor performance. We investigate how adaptive routing policies should be designed such that these effects can be avoided.The network is represented by a general graph with latency functions on the edges. Traffic is managed by a large number of agents each of which is responsible for a negligible amount of traffic. Initially the agents' routing paths are chosen in an arbitrary fashion. From time to time each agent revises her routing strategy by sampling another path and switching with positive probability to this path if it promises smaller latencies. As the information on which the agent bases her decision might be stale, however, this does not necessarily lead to an improvement. The points of time at which agents revise their strategy are generated by a Poisson distribution. Stale information is modelled in form of a bulletin board that is updated periodically and lists the latencies on all edges.We analyze such a distributed routing process in the socalled fluid limit, that is, we use differential equations describing the fractions of traffic on different paths over time. In our model, we can show the following effects. Simple routing policies that always switch to the better alternative lead to oscillation, regardless at which frequency the bulletin board is updated. Oscillation effects can be avoided, however, when using smooth adaption policies that do not always switch to better alternatives but only with a probability depending on the advantage in the latency. In fact, such policies have dynamics that converge to a fixed point corresponding to a Nash equilibrium for the underlying routing game, provided the update periods are not too large.In addition, we also analyze the speed of convergence towards approximate equilibria of two specific variants of smooth adaptive routing policies, e. g. , for a replication policy adopted from evolutionary game theory.
We consider a dynamic load balancing scenario in which users allocate resources in a non-cooperative and selfish fashion. The perceived performance of a resource for a user decreases with the number of users that allocate the resource. In our dynamic, concurrent model, users may reallocate resources in a round-based fashion. As opposed to various settings analyzed in the literature, we assume that users have quality of service (QoS) demands. A user has zero utility when falling short of a certain minimum performance threshold and having positive utility otherwise. Whereas various load-balancing protocols have been proposed for the setting without quality of service requirements, we consider protocols that satisfy an additional locality constraint: The behavior of a user depends merely on the state of the resource it currently allocates. This property is particularly useful in scenarios where the state of other resources is not readily accessible. For instance, if resources represent channels in a mobile network, then accessing channel information may require time-intensive measurements. We consider several variants of the model, where the quality of service demands may depend on the user, the resource, or both. For all cases we present protocols for which the dynamics converge to a state in which all users are satisfied. More importantly, the time to reach such a state scales nicely. It is only logarithmic in the number of users, which makes our protocols applicable in large-scale systems
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