Max Celebrity Games

Abstract

We introduce Max celebrity games a new variant of Celebrity games defined in [4]. In both models players have weights and there is a critical distance $\beta$ as well as a link cost $\alpha$. In the max celebrity model the cost of a player depends on the cost of establishing links to other players and on the maximum of the weights of those nodes that are farther away than $\beta$ (instead of the sum of weights as in celebrity games). The main results for $\beta >1$ are that: computing a best response for a player is NP-hard; the optimal social cost of a celebrity game depends on the relation between $\alpha$ and $w_{max}$; ne always exist and ne graphs are either connected or a set of $r\ge 1$ connected components where at most one of them is not an isolated node; for the class of connected ne graphs we obtain a general upper bound of $2 \beta +2$ for the diameter. We also analyze the price of anarchy (PoA) of connected ne graphs and we show that there exist games $\varGamma$ such that $\text {PoA} (\varGamma )=\varTheta (n/\beta )$; modifying the cost of a player we guarantee that all ne graphs are connected, but the diameter might be $n-1$. Finally, when $\beta =1$, computing a best response for a player is polynomial time solvable and the $\text {PoA} = O(w_{max}/w_{min})$.