Multicast Network Design Game on a Ring

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 ofand 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

The price of anarchy and stability in general noisy best-response dynamics

“…The study of other problems considered in [AS09, KM13,MM15] is an interesting direction for future research. Some of these bounds on potential minimizers are already quite sophisticated, and stochastic stability in independent learning may require nontrivial analysis.…”

“…Some of these bounds on potential minimizers are already quite sophisticated, and stochastic stability in independent learning may require nontrivial analysis. Estimating independent-logit PoA and PoS is still open even for ring networks [MM15]. where the δ-jobs sum up to ∆.…”

“…We then consider broadcast network design games studied in [KM13,MM15] for potential minimizers (logit dynamics), and show that in this case synchronization is not essential to obtain optimal configurations. First, for the case of parallel links between a common source and a common destination (links represent resources of different costs) players will select the optimum if there is a unique shortest link (Theorem 3.2).…”

“…The work [KM13] extends this approach to the PoS and shows similar results for broadcast network design games (where logit PoS = logit PoA up to a constant factor). The restriction to the ring topology is considered in [MM15] (together with other variants of PoS and PoA). The authors in [AFPP13] suggest to consider the expected social cost in logit dynamics and therefore the stationary distribution as the equilibrium concept.…”

Game-theoretic analysis of transportation problems

“…We next present our result about the MSPoS according to social function E, in which the ideas behind the proof comes from one of the results from Mamageishvili and Mihalák [26].…”