# Improving the H_k Bound on the Price of Stability in Undirected Shapley Network Design Games

Yann Disser, Andreas Emil Feldmann, Max Klimm, Matus Mihalak

Research output: Chapter in Book/Report/Conference proceedingConference article in proceedingAcademicpeer-review

## Abstract

In this paper we show that the price of stability of shapley network design games on undirected graphs with k players is at most $$\smash{\frac{k^3(k+1)/2-k^2}{1+k^3(k+1)/2-k^2}h_k}$$ $$\smash{= \bigl(1 - \theta(1/k^4)\bigr)h_k}$$, where h k denotes the k-th harmonic number. This improves on the known upper bound of h k , which is also valid for directed graphs but for these, in contrast, is tight. Hence, we give the first non-trivial upper bound on the price of stability for undirected shapley network design games that is valid for an arbitrary number of players. Our bound is proved by analyzing the price of stability restricted to nash equilibria that minimize the potential function of the game. We also present a game with k = 3 players in which such a restricted price of stability is 1.634. This shows that the analysis of bilò and bove (journal of interconnection networks, volume 12, 2011) is tight. In addition, we give an example for three players that improves the lower bound on the (unrestricted) price of stability to 1.571.
Original language English Proc. Algorithms and Complexity, 8th International Conference (CIAC) Springer Verlag 158-169 12 https://doi.org/10.1007/978-3-642-38233-8_14 Published - 2013 Yes

### Publication series

Series Lecture Notes in Computer Science 7878