Celebrity games

We introduce Celebrity games, a new model of network creation games. In this model players have weights (W being the sum of all the player's weights) and there is a critical distance ß as well as a link cost a. The cost incurred by a player depends on the cost of establishing links to other players and on the sum of the weights of those players that remain farther than the critical distance. Intuitively, the aim of any player is to be relatively close (at a distance less than ß ) from the rest of players, mainly of those having high weights. The main features of celebrity games are that: computing the best response of a player is NP-hard if ß>1 and polynomial time solvable otherwise; they always have a pure Nash equilibrium; the family of celebrity games having a connected Nash equilibrium is characterized (the so called star celebrity games) and bounds on the diameter of the resulting equilibrium graphs are given; a special case of star celebrity games shares its set of Nash equilibrium profiles with the MaxBD games with uniform bounded distance ß introduced in Bilò et al. [6]. Moreover, we analyze the Price of Anarchy (PoA) and of Stability (PoS) of celebrity games and give several bounds. These are that: for non-star celebrity games PoA=PoS=max{1,W/a}; for star celebrity games PoS=1 and PoA=O(min{n/ß,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when ß=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for ß=2.Peer ReviewedPostprint (author's final draft

The Max-Distance Network Creation Game on General Host Graphs

In this paper we study a generalization of the classic \emph{network creation game} in the scenario in which the $n$ players sit on a given arbitrary \emph{host graph}, which constrains the set of edges a player can activate at a cost of $\alpha \geq 0$ each. This finds its motivations in the physical limitations one can have in constructing links in practice, and it has been studied in the past only when the routing cost component of a player is given by the sum of distances to all the other nodes. Here, we focus on another popular routing cost, namely that which takes into account for each player its \emph{maximum} distance to any other player. For this version of the game, we first analyze some of its computational and dynamic aspects, and then we address the problem of understanding the structure of associated pure Nash equilibria. In this respect, we show that the corresponding price of anarchy (PoA) is fairly bad, even for several basic classes of host graphs. More precisely, we first exhibit a lower bound of $\Omega (\sqrt{ n / (1+\alpha)})$ for any $\alpha = o(n)$. Notice that this implies a counter-intuitive lower bound of $\Omega(\sqrt{n})$ for very small values of $\alpha$ (i.e., edges can be activated almost for free). Then, we show that when the host graph is restricted to be either $k$-regular (for any constant $k \geq 3$), or a 2-dimensional grid, the PoA is still $\Omega(1+\min\{\alpha, \frac{n}{\alpha}\})$, which is proven to be tight for $\alpha=\Omega(\sqrt{n})$. On the positive side, if $\alpha \geq n$, we show the PoA is $O(1)$. Finally, in the case in which the host graph is very sparse (i.e., $|E(H)|=n-1+k$, with $k=O(1)$), we prove that the PoA is $O(1)$, for any $\alpha$.Comment: 17 pages, 4 figure

Distance-uniform graphs with large diameter

An ϵ-distance-uniform graph is one with a critical distance d such that from every vertex, all but at most an ϵ-fraction of the remaining vertices are at distance exactly d. Motivated by the theory of network creation games, Alon, Demaine, Hajiaghayi, and Leighton made the follow- ing conjecture of independent interest: that every ϵ-distance-uniform graph (and, in fact, a broader class of ϵ-distance-almost-uniform graphs) has critical distance at most logarithmic in the number of vertices n. We disprove this conjecture and characterize the asymptotics of this extremal prob- lem. Speci-cally, for 1/n ≤ ϵ ≤ 1 /log n , we construct ϵ-distance-uniform graphs with critical distance 2ω(log n/log ϵ-1). We also prove an upper bound on the critical distance of the form 2O(log n/log ϵ-1) for all ϵ and n. Our lower bound construction introduces a novel method inspired by the Tower of Hanoi puzzle and may itself be of independent interest.Peer ReviewedPostprint (author's final draft

The Price of Anarchy in Network Creation Games Is (Mostly) Constant

We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1,2, ,n}, each identified with a vertex of a graph (network), where the strategy of player i, i=1, ,n, is to build some edges adjacent to i. The cost of building an edge is α>0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SumGame variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MaxGame variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α>273⋅n all equilibria in SumGame are trees and thus the price of anarchy is constant, and that for α>129 all equilibria in MaxGame are trees and the price of anarchy is constant. For SumGame this answers (almost completely) one of the fundamental open problems in the field—is price of anarchy of the network creation game constant for all values of α?—in an affirmative way, up to a tiny range of

Stars and celebrities: A network creation game

CoRRCelebrity games, a new model of network creation games is introduced. The specific features of this model are that players have different celebrity weights and that a critical distance is taken into consideration. The aim of any player is to be close (at distance less than critical) to the others, mainly to those with high celebrity weights. The cost of each player depends on the cost of establishing direct links to other players and on the sum of the weights of those players at a distance greater than the critical distance. We show that celebrity games always have pure Nash equilibria and we characterize the family of subgames having connected Nash equilibria, the so called star celebrity games. Exact bounds for the PoA of non star celebrity games and a bound of O(n/ß+ß) for star celebrity games are provided. The upper bound on the PoA can be tightened when restricted to particular classes of Nash equilibria graphs. We show that the upper bound is O(n/ß) in the case of 2-edge-connected graphs and 2 in the case of trees.Preprin

Bounded-Distance Network Creation Games

A \emph{network creation game} simulates a decentralized and non-cooperative building of a communication network. Informally, there are $n$ players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about its centrality in the network, in this paper we introduce a new setting in which if a player maintains its (either \emph{maximum} or \emph{average}) distance to the other nodes within a given associated \emph{bound}, then its cost is simply equal to the \emph{number} of activated edges, otherwise its cost is unbounded. We study the problem of understanding the structure of associated pure Nash equilibria of the resulting games, that we call \textsc{MaxBD} and \textsc{SumBD}, respectively. For both games, we show that when distance bounds associated with players are \emph{non-uniform}, then equilibria can be arbitrarily bad. On the other hand, for \textsc{MaxBD}, we show that when nodes have a \emph{uniform} bound $R$ on the maximum distance, then the \emph{Price of Anarchy} (PoA) is lower and upper bounded by $2$ and $O\left(n^{\frac{1}{\lfloor\log_3 R\rfloor+1}}\right)$ for $R \ge 3$ (i.e., the PoA is constant as soon as $R$ is $\Omega(n^{\epsilon})$, for some $\epsilon>0$), while for the interesting case $R=2$, we are able to prove that the PoA is $\Omega(\sqrt{n})$ and $O(\sqrt{n \log n} )$. For the uniform \textsc{SumBD} we obtain similar (asymptotically) results, and moreover we show that the PoA becomes constant as soon as the bound on the average distance is $n^{\omega\left(\frac{1}{\sqrt{\log n}}\right)}$