Multicast Network Design Game on a Ring

TL;DR

This paper analyzes various efficiency measures of the multicast network design game on a ring, establishing tight bounds for price of stability, potential optimizer costs, and sequential price of anarchy and stability, thus advancing understanding of network formation dynamics.

Contribution

It provides the first tight bounds for the price of stability and explores sequential game variants, answering open questions and proposing conjectures for future research.

Findings

Price of stability is exactly 4/3.

Potential optimizer costs are bounded by 2, with a matching lower bound.

Sequential price of anarchy is exactly 2.

Abstract

In this paper we study quality measures of different solution concepts for the multicast network design game on a ring topology. We recall from the literature a lower bound of 4/3 and prove a matching upper bound for the price of stability, which is the ratio of the social costs of a best Nash equilibrium and of a general optimum. Therefore, we answer an open question posed by Fanelli et al. in [12]. We prove an upper bound of 2 for the ratio of the costs of a potential optimizer and of an optimum, provide a construction of a lower bound, and give a computer-assisted argument that it reaches $2$ for any precision. We then turn our attention to players arriving one by one and playing myopically their best response. We provide matching lower and upper bounds of 2 for the myopic sequential price of anarchy (achieved for a worst-case order of the arrival of the players). We then initiate the study of myopic sequential price of stability and for the multicast game on the ring we construct a lower bound of 4/3, and provide an upper bound of 26/19. To the end, we conjecture and argue that the right answer is 4/3.