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 \(\varTheta \left( \sqrt{\frac{w}{n}}\right) \) and show that it is mainly due to lack of coordination: the same order of magnitude of loss is suffered by any symmetric profile.
We thank Refael Hassin, Moshe Haviv, Ella Segev and the participants of the “Queuing and Games” Seminar at TAU for useful comments on this manuscript. The full version of the paper (including all proofs) can be found at https://arxiv.org/abs/1808.00034.
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Notes
- 1.
Our minor simplifications of Vickrey’s model include an assumption that the agents can only join the queue after some starting time.
- 2.
Similar argument in favor of symmetric strategies is made in [26].
- 3.
Doing so alleviates the need to precisely compute symmetric equilibria and the need to determine if the game has a unique symmetric equilibrium or not.
- 4.
- 5.
Similarly to the “Fully Mixed Nash Equilibrium Conjecture” [9] we suspect that in our game symmetric equilibria are in fact the worst equilibria.
- 6.
Note that when \(w>2\) the two players game admits exactly three equilibria: the two optimal equilibria in which one player enters after the other, and the symmetric random equilibrium we discussed. Thus, our result is both a price of anarchy result for unrestricted equilibria and a price of stability result for symmetric equilibria.
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Babaioff, M., Oren, S. (2018). Incentives and Coordination in Bottleneck Models. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_3
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