On Nash Equilibria for a Network Creation Game

Abstract

We study a network creation game recently proposed by Fabrikant, Luthra, Maneva, Papadimitriou and Shenker. In this game, each player (vertex) can create links (edges) to other players at a cost of α per edge. The goal of every player is to minimize the sum consisting of (a) the cost of the links he has created and (b) the sum of the distances to all other players.Fabrikant et al. conjectured that there exists a constant A such that, for any α > A, all non-transient Nash equilibria graphs are trees. They showed that if a Nash equilibrium is a tree, the price of anarchy is constant. In this paper we disprove the tree conjecture. More precisely, we show that for any positive integer n 0 , there exists a graph built by n ≥ n 0 players which contains cycles and forms a nontransient Nash equilibrium, for any α with 1 < α ≤ n/2. Our construction makes use of some interesting results on finite affine planes. On the other hand we show that, for α ≥ 12n log n , every Nash equilibrium forms a tree.Without relying on the tree conjecture, Fabrikant et al. ). Additionally, we develop characterizations of Nash equilibria and extend our results to a weighted network creation game as well as to scenarios with cost sharing.

Intrinsic robustness of the price of anarchy

“…holds for every bid profile b. 3 As mentioned in Remark 2.3, this relaxed property is sufficient for all of the applications of smoothness arguments discussed in this paper. This smoothness argument implies a lower bound of 1/2 on the POA of pure Nash equilibria, which is tight in the worst case [38].…”

“…where (3) follows from the definition of the objective function; inequality (4) follows from the Nash equilibrium condition (1), applied once to each player i with the hypothetical deviation s * i ; and inequality (5) follows from the defining condition (2) of a smooth game. Rearranging terms yields the claimed bound.…”

“…This completes the verification of (11). 3 To see that such games are not always (1,1)-smooth in the earlier stronger sense, consider an example with two bidders and one good, with v1({1}) = 1, v2({1}) = 2, b * 11 = 0, b * 21 = 1 3 , b11 = 1, and b21 = 2 3 . The right-hand side of (11) is 1 while the left-hand side is 0.…”

Joint Operator Pricing and Network Selection Game in Cognitive Radio Networks: Equilibrium, System Dynamics and Price of Anarchy

“…The authors set the player's cost for using an edge e in the network as a combination of cost function c e (x) and latency function d e (x); the goal of each user in this game is to minimize the sum of his cost and latency. The same model is also used in [30]. Finally, note that, in [19], two components, namely, throughput (the allocated capacity to a player, which is obviously related to the delay experienced by such user) and the corresponding price (see [19,Eqs.…”

Celebrity games

“…Instead of buying links unilaterally, Corbo and Parkes (2005) proposed the possibility of having links formed by bilateral contracting: both endpoints must agree before creating a link between them and the two players share (half-half) the cost of establishing the link. NCG models can be cooperative -a possibility introduced by Albers et al (2006)-and therefore any node can purchase any amount of any link in the resulting graph, and a link can be created when its cost is covered by a set of players. The model studied in Bilò et al (2015b) (see also Bilò et al (2012)) considers the notion of bounded distance per player and propose two variants: the MaxBD game and the SumBD game, corresponding to the original Max and Sum cost models respectively.…”