An $O({\log n\over \log\log n})$ Upper Bound on the Price of Stability for Undirected Shapley Network Design Games

TL;DR

This paper establishes an upper bound of O(log n / log log n) on the price of stability for undirected Shapley network design games with a single sink, improving understanding of the efficiency of equilibria in such games.

Contribution

The paper provides the first non-trivial upper bound on the price of stability for undirected Shapley network design games with a single sink.

Findings

Upper bound of O(log n / log log n) on the price of stability

Analysis specific to the single sink case

Improves bounds on efficiency of Nash equilibria in network design games

Abstract

In this paper, we consider the Shapley network design game on undirected networks. In this game, we have an edge weighted undirected network $G(V,E)$ and $n$ selfish players where player $i$ wants to choose a path from source vertex $s_i$ to destination vertex $t_i$. The cost of each edge is equally split among players who pass it. The price of stability is defined as the ratio of the cost of the best Nash equilibrium to that of the optimal solution. We present an $O(\log n/\log\log n)$ upper bound on price of stability for the single sink case, i.e, $t_i=t$ for all $i$.