An H n/2 Upper Bound on the Price of Stability of Undirected Network Design Games

Abstract

In the network design game with n players, every player chooses a path in an edgeweighted graph to connect her pair of terminals, sharing costs of the edges on her path with all other players fairly. We study the price of stability of the game, i.e., the ratio of the social costs of a best Nash equilibrium (with respect to the social cost) and of an optimal play. It has been shown that the price of stability of any network design game is at most H n , the n-th harmonic number. This bound is tight for directed graphs. For undirected graphs, the situation is dramatically different, and tight bounds are not known. It has only recently been shown that the price of stability is at most H n 1 − 1 Θ(n 4 ) , while the worst-case known example has price of stability around 2.25. In this paper we improve the upper bound considerably by showing that the price of stability is at most H n/2 + for any starting from some suitable n ≥ n( ).

Uma introdução à teoria dos jogos algorítmica

“…Para o protocolo de Shapley, os resultados para o PoA em ambas as redes, ND e D, e os resultados para o PoS das redes D foram provados por Tardos e Wexler (2007). Para esse mesmo protocolo, o resultado para o PoS das redes ND-UD e ND-MD foram provados por Mamageishvili et al (2014) e Li (2009), respectivamente. Todos os outros resultados foram demonstrados por Chen et al (2010).…”

Multicast Network Design Game on a Ring

“…Our goal is to investigate the maximum possible cost c of the resulting network, where c = a 0 + · · · + a i + a i+2 + · · · + a n . Take the first inequality (2) with weight 2 19 , the second inequality (3) with weight 24 19 , and the normalization equation a 0 + · · · + a o−1 + a o+1 + · · · + a n = 1 with weight 26 19 . We obtain that the sum on the left hand side s satisfies c ≤ s ≤ 26 19 , which gives that c ≤ 26 19 ≈ 1.368.…”

“…Take the first inequality (2) with weight 2 19 , the second inequality (3) with weight 24 19 , and the normalization equation a 0 + · · · + a o−1 + a o+1 + · · · + a n = 1 with weight 26 19 . We obtain that the sum on the left hand side s satisfies c ≤ s ≤ 26 19 , which gives that c ≤ 26 19 ≈ 1.368. The permutation from the proof of Theorem 5.4 cannot be used to provide a better bound, as there exists an example of a game, where the permutation results in a network of cost 26 19 times larger than the cost of optimum.…”

“…The price of stability of the game is much less understood for undirected graphs. While it is known to be strictly smaller than H n [11,19], namely, at most H n/2 [19], the largest known example has price of stability equal to roughly 2.245 [6]. Closing this gap is a major open problem in the field of congestion games and in the computational game theory in general.…”

“…In many results, the potential function, and the equilibria minimizing it, play an important role. Actually, equilibria minimizing potential function are regarded as stable (against noise) by Asadpour and Saberi [4], and accordingly, some authors studied the price of stability restricted to these kind of equilibria [16,19], the so-called potential-optimum price of stability.…”