Multicast Network Design Game on a Ring

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 \frac{4}{3}\frac{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 \frac{4}{3}\frac{4}{3}, and provide an upper bound of \frac{26}{19}\frac{26}{19}. To the end, we conjecture and argue that the right answer is \frac{4}{3}\frac{4}{3}.