Incentives and Coordination in Bottleneck Models

TL;DR

This paper analyzes a strategic queueing model where agents choose when to join a first-come-first-served line, examining how strategic behavior and lack of coordination contribute to inefficiency under different parameter regimes.

Contribution

It introduces a variant of the bottleneck model with symmetric strategies and quantifies the sources of inefficiency in different parameter regimes.

Findings

When waiting cost w is fixed and n grows, the price of anarchy is tightly bounded by 2, mainly due to selfish behavior.

When n is fixed and w grows, the price of anarchy is on the order of sqrt(w/n), mainly due to lack of coordination.

Symmetric equilibria exhibit different inefficiency sources depending on parameter regimes.

Abstract

We study a variant of Vickrey's classic bottleneck model. In our model there are $n$ agents and each agent strategically chooses when to join a first-come-first-served observable queue. Agents dislike standing in line and they take actions in discrete time steps: we assume that each agent has a cost of $1$ for every time step he waits before joining the queue and a cost of $w>1$ for every time step he waits in the queue. At each time step a single agent can be processed. Before each time step, every agent observes the queue and strategically decides whether or not to join, with the goal of minimizing his expected cost. In this paper we focus on symmetric strategies which are arguably more natural as they require less coordination. This brings up the following twist to the usual price of anarchy question: what is the main source for the inefficiency of symmetric equilibria? is it the players' strategic behavior or the lack of coordination? We present results for two different parameter regimes that are qualitatively very different: (i) when $w$ is fixed and $n$ grows, we prove a tight bound of $2$ and show that the entire loss is due to the players' selfish behavior (ii) when $n$ is fixed and $w$ grows, we prove a tight bound of $\Theta \left(\sqrt{\frac{w}{n}}\right)$ and show that it is mainly due to lack of coordination: the same order of magnitude of loss is suffered by any symmetric profile.