Bottleneck routing with elastic demands

doi:10.1016/j.orl.2017.11.007

Operations Research Letters

Volume 46, Issue 1, January 2018, Pages 93-98

https://doi.org/10.1016/j.orl.2017.11.007 Get rights and content

Abstract

Bottleneck routing games are a well-studied model to investigate the impact of selfish behavior in communication networks. In this model, each user selects a path in a network for routing her fixed demand. The disutility of a user only depends on the most congested link visited. We extend this model by allowing users to continuously vary the demand rate at which data is sent along the chosen path. As our main result we establish tight conditions for the existence of pure strategy Nash equilibria.

Introduction

Bottleneck routing games are a theoretical model to study the effects of resource allocation in distributed communication networks [[1], [4]]. Every user of the network is associated with a non-negative demand that she wants to send from her source to the respective destination, and her goal is to find a path that minimizes the congestion of the most congested link. It has been argued (cf. [[5], [28]]) that in the context of packet-switched communication networks, the performance of a path is more closely related to the most congested link than the classical sum-aggregation of costs (as in [[18], [29], [34]]), and there are several proposals (cf. [[26], [36]]) for replacing the sum-aggregation of congestion costs with the max-aggregation, primary, because the max-aggregation leads to favorable properties of protocols in terms of their stability in presence of communication delays [36].

While bottleneck routing games are an important step in terms of integrating routing decisions with bottleneck objectives, they lack one fundamental tradeoff inherent in packet-switched communication networks: once a path is selected, a user increases the sending rate in case of low congestion and decreases it in case of high congestion. In this paper, we address this tradeoff by introducing bottleneck congestion games with elastic demands, where users can continuously vary their demands. Formally, there is a finite set of resources and a strategy of a player is a tuple consisting of a subset of resources and a demand. Resources have player-specific cost functions that are non-decreasing and strictly convex. Every user is associated with a non-decreasing strictly concave utility function measuring the received utility from sending at a certain demand rate (cf. [[18], [31]]). The goal of a user is to select both a subset of resources and a demand rate that maximizes the utility (from the demand rate) minus the congestion cost on the most expensive resource contained in the chosen resource set. Our model thus integrates as a special case ( $i$ ) single-path routing (which is up to date standard as splitting packets over several routes leads to different packet inter-arrival times and synchronization problems) and ( $i i$ ) congestion control via data rate adaption based on the maximum congestion experienced.

We derive conditions for the cost functions that ensure that the resulting bottleneck congestion games with elastic demands admit a pure Nash equilibrium (PNE). The existence of pure Nash equilibria is favorable for large communication network as they provide a deterministic steady state from which no player has an incentive to deviate. Mixed Nash equilibria, on the other hand are less desirable as they may lead to oscillating route choices which degrade the performance of the system [21]. Our condition requires that for every player the player-specific resource cost functions are non-decreasing, strictly convex and equal up to resource specific shifts in their argument. While monotonicity and convexity are natural conditions, the last assumption seems limiting. We can show, however, that without it there are examples without any PNE. Our proof is constructive, i.e., we devise an algorithm that computes a PNE.

Our results thus give further indication that the max-aggregation of congestion costs has desirable properties in terms of equilibrium existence. Specifically, our results imply that networks with $M ∕ M ∕ 1$ functions (cf. [23]) possess a PNE, if congestion costs are aggregated with the maximum. This stands in contrast to the classical sum-aggregation of congestion costs, where PNE need not exist [12].

An extended abstract of the results presented in this paper appeared in Harks et al. [[14]].

Bottleneck routing games with fixed demands admit strong equilibria [[13], [24]], a strengthening of PNE that are resilient to coordinated deviations of groups of players. The complexity of computing PNE and strong equilibria in these games was investigated in [9]. For works on the price of anarchy of PNE and the worst-case quality of strong equilibria we refer to [[2], [3], [5], [7], [16], [17]]. Further related is the model of [25] who studied generalizations of congestion games in which the sum-aggregation is replaced by an arbitrary aggregation function.

In previous work [11], we established the existence of an equilibrium for a class of aggregative location games. This result implies the existence of a PNE for the present model when the allowable sets of resource of players contain singletons only. Congestion games with variable demands coincide with the present model except that the traditional sum-aggregation of costs is used. For these games only affine and certain exponential cost functions lead to the existence of PNE [12]. These results imply that for $M ∕ M ∕ 1$ delay functions a PNE does not always exist.

Integrated routing and congestion control has been studied in [[8], [19], [20], [32], [33]], where the existence of a PNE is proved by relating it to optimal solutions of an associated convex utility maximization problem. These models require that every user possibly splits the flow among a number of paths that may even be exponential in the size of the underlying graph. This issue has been addressed in [[6], [27]], where controllable route splitting at routers is assumed which can effectively limit the resulting number of used routes. For all the above models, however, the end-to-end applications may suffer in service quality due to packet jitter caused by different path delays. Partly because of this issue, the standard TCP/IP protocol suite still uses single path routing. Also in contrast to our model, all these models assume that congestion feedback is aggregated via the sum instead of the max operator.

Another model for resource allocation in telecommunication networks isthe class of MAXBAR-games. Here, players select a single path in a network with fixed edge capacities. Then, all players synchronously increase their rate until the capacity of an edge is reached. After such an event all rates of users using this tight edge are fixed. MAXBAR games possess a PNE [35], and a strong equilibrium [10] even when the rate increase is non-homogeneous.

Section snippets

The model

Let $R = {1, \dots, m}$ be a nonempty and finite set of $m \in N$ resources, and let $N = {1, \dots, n}$ be a nonempty and finite set of $n \in N$ players. For every $i \in N,$ let $X_{i} \subset 2^{R} ∖ {\emptyset}$ be a nonempty set of nonempty subsets of resources available to player $i$ and let $X = \times_{i \in N} X_{i}$ denote their Cartesian product. We call $x_{i} \in X_{i}$ an allocation of player $i$ and we denote by $x = {(x_{i})}_{i \in N} \in X$ the overall allocation vector. For every player $i$ and every resource $r \in R$ we are given a player-specific cost function $c_{i, r} : R_{\geq 0} \to R_{\geq 0}$ . Every player $i \in N$ has a

A characterization of pure Nash equilibria

In this section, we give a complete characterization of PNE in bottleneck congestion games. Our characterization relies on the notion of a demand equilibrium which we define as a strategy profile with the property that no player can increase her payoff by unilaterally changing her demand only.

Definition 3.1 Demand Equilibrium

A strategy profile $(x, d)$ is called a demand equilibrium if $π_{i} (x, d) \geq π_{i} (x, \tilde{d})$ for all $i \in N$ and $\tilde{d} = (d_{- i}, {\tilde{d}}_{i})$ with ${\tilde{d}}_{i} \in [σ_{i}, τ_{i}]$ .

Every PNE is a demand equilibrium, but not vice versa.

Computing demand equilibria

Corollary 3.5 suggests that for computing a PNE we should be able to compute a demand equilibrium. In this section, we describe an algorithm for this purpose. First, we need the notion of a distributed equilibrium. Let $G = (N, S, π)$ be a bottleneck congestion game, $M \subseteq N$ , and let $r$ be a resource. We define the restriction of $G$ on $M$ and $r$ as the bottleneck congestion game $G |_{(M, r)} = (M, S^{'}, π^{'})$ with $S_{i}^{'} = {{r}} \times [σ_{i}, τ_{i}]$ for all $i \in M$ and $π_{i}^{'} (x, d) = U_{i} (d_{i}) - c_{i} (ℓ_{r} (x, d) + υ_{r})$ .

Definition 4.1 Distributed Equilibrium

Let $x \in X$ and $N_{r} (x) ≔ {i \in N : r \in x_{i}}$

An algorithm for computing PNE

In this section, we give an algorithm that computes a PNE. The algorithm starts with an arbitrary strategy profile and computes a demand equilibrium. Then, if a player can improve, we let this player play a best reply (where necessarily the player’s resource set changes), and recompute a demand equilibrium. The technically involved part is to show that the algorithm terminates. To prove this, we first derive several properties of intermediate strategy profiles during the execution of the

A counterexample for general convex costs

In this section, we show that the assumption that the cost function of each player does not depend on the resource (up to a resource-specific shift) is a necessary assumption that cannot be dispensed with for the existence of a PNE. This is true, even if we impose the additional assumption that for each of the resources the cost functions of all players coincide, and that the strategies of each player $i$ correspond to the set of $(u_{i}, v)$ -paths in a network.

Theorem 6.1

There are bottleneck

Conclusions

We studied bottleneck routing games where players vary both their rate and their path. As our main result, we derived an algorithm computing a PNE provided that cost functions are non-decreasing, convex and equal up to resource-specific shifts in the argument. This condition is met by the practically relevant $M ∕ M ∕ 1$ delay functions with heterogeneous service rates, thus, our result implies the existence of PNE for this model. Our algorithm is centralized but mimics a decentralized dynamic in the

Acknowledgments

We wish to thank two anonymous referees for their intensive work to improve the paper.

The second author’sresearch was carried out in the framework of Matheon supported by Einstein Foundation Berlin (MI8).

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View full text

Bottleneck routing with elastic demands

Abstract

Introduction

Section snippets

The model

A characterization of pure Nash equilibria

Computing demand equilibria

An algorithm for computing PNE

A counterexample for general convex costs

Conclusions

Acknowledgments

Theoret. Comput. Sci.

J. Math. Econom.

Bottleneck routing games in communication networks

IEEE J. Sel. Areas Commun.

Network congestion control with Markovian multipath routing

Math. Program. A

On the inefficiency of equilibria in linear bottleneck congestion games

Multi-path TCP: a joint congestion control and routing scheme to exploit path diversity in the Internet

IEEE/ACM Trans. Netw.

Computing pure Nash and strong equilibria in bottleneck congestion games

Math. Program. A

Routing games with progressive filling

IEEE/ACM Trans. Netw.

Congestion games with variable demands

Math. Oper. Res.

Strong equilibria in games with the lexicographical improvement property

Internat. J. Game Theory

A generalization of Brouwer’s fixed point theorem

Duke Math. J.

Rate control in communication networks: Shadow prices, proportional fairness, and stability

J. Oper. Res. Soc.