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

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 $i$.

💡 Research Summary

The paper studies the Shapley network design game on undirected graphs, focusing on the single‑sink setting where every player’s destination vertex is the same. In this game each player chooses a path from a private source $s_i$ to the common sink $t$, and the cost of every edge $e$ with weight $c_e$ is split equally among all players whose paths contain $e$. The central efficiency measure is the price of stability (PoS), defined as the ratio between the total cost of the best Nash equilibrium and the cost of a socially optimal solution.

The authors first recall that the game is a potential game: the Rosenthal potential $\Phi$ is given by $\Phi = \sum_{e\in E} c_e H_{k_e}$ where $k_e$ is the number of players using edge $e$ and $H_{k}$ denotes the $k$‑th harmonic number. Any state minimizing $\Phi$ is a Nash equilibrium, and the equilibrium with minimum total cost is precisely the minimum‑potential state. In the general (multi‑sink) case the best known upper bound on PoS is $O(\log n)$, and no tighter bound was known.

The contribution of the paper is a proof that, when all players share the same destination, the PoS improves dramatically to $O(\log n / \log\log n)$. The proof proceeds in several steps.

1. Structural reduction to a tree. The optimal solution can be transformed into a Steiner tree rooted at $t$ that connects all sources. Because the graph is undirected, the union of the optimal $s_i$‑$t$ paths forms a tree $T^$; each edge $e$ of $T^$ is used by a set $P(e)$ of players, and we denote $k_e^* = |P(e)|$.

2. Bounding edge congestion in any minimum‑potential equilibrium. The authors show that for any edge $e$ in a minimum‑potential equilibrium $N$, the number of players $k_e$ using $e$ cannot exceed $O(\log n / \log\log n)$. The argument exploits the single‑sink topology: if an edge were heavily congested, the harmonic term $H_{k_e}$ would become large, inflating the potential far beyond the potential of $T^*$. By a careful counting of how many source‑sink paths can be forced to share a given edge without creating cycles, they derive the logarithmic‑logarithmic bound.

3. Relating potential to actual cost. Because $c_e H_{k_e} \le c_e (1 + \ln k_e)$, the total potential of any state is at most a constant factor times its total cost. Conversely, the potential of $T^$ is at most a constant factor times its cost, since $k_e^ \ge 1$ for every edge. Hence
   \