Cooperation in Bilateral Generalized Network Creation

TL;DR

The paper tackles how cooperation affects efficiency in a bilateral generalized network creation game (BGNCG) under weighted host graphs. It shows that without metric edge weights, even strong cooperation cannot improve the PoA, while in the metric-weight setting, substantial PoA improvements are achievable through bilateral cooperation, especially in the stronger BSE concept. The authors derive novel distance-cost and spanner-based techniques to obtain asymptotically tight bounds: in the metric case, the PoA in BSE is $O\bigl(\min\{\frac{\alpha\sqrt{\alpha}}{n}, 2n\}\bigr)$, with a matching lower bound revealing $\Theta(\sqrt{\alpha})$ behavior for $\alpha=O(n)$ and $PoA=\Theta(n)$ when $\alpha\ge n^2$, indicating that cooperation and metric geometry together are necessary for significant improvement. These results illuminate policy and design implications for decentralized network formation, suggesting that fostering large coalitions and leveraging metric edge lengths can meaningfully reduce inefficiency in large-scale networks.

Abstract

Studying the impact of cooperation in strategic settings is one of the cornerstones of algorithmic game theory. Intuitively, allowing more cooperation yields equilibria that are more beneficial for the society of agents. However, for many games it is still an open question how much cooperation is actually needed to ensure socially good equilibria. We contribute to this research endeavor by analyzing the benefits of cooperation in a network formation game that models the creation of communication networks via the interaction of selfish agents. In our game, agents that correspond to nodes of a network can buy incident edges of a given weighted host graph to increase their centrality in the formed network. The cost of an edge is proportional to its length, and both endpoints must agree and pay for an edge to be created. This setting is known for having a high price of anarchy. To uncover the impact of cooperation, we investigate the price of anarchy of our network formation game with respect to multiple solution concepts that allow for varying amounts of cooperation. On the negative side, we show that on host graphs with arbitrary edge weights even the strongest form of cooperation cannot improve the price of anarchy. In contrast to this, as our main result, we show that cooperation has a significant positive impact if the given host graph has metric edge weights. For this, we prove asymptotically tight bounds on the price of anarchy via a novel proof technique that might be of independent interest and can be applied in other models with metric weights.

Figures (4)

• Figure 1: Lower bound construction for the PoA of BGNCG networks in BSE. Left: BGNCG network $G_n^\ast$ with a low social cost. Right: BGNCG network $G_n$ in BSE. For any edge $\{u_i,u_j\}$, it holds that $w(u_i,u_j) = 0$. The remaining edges that are not depicted have weight $1$.
• Figure 2: (a) and (b): lower bound constructions for the PoA for M-BGNCG networks in PS, BNE, and BSE. The values of $a$ and $b$ differ depending on the solution concept used. The edge weight of an arbitrary edge $\{u,v\}$ of the host graph of $S_n^\ast$ and $S_n$ is defined as $d_{S_n^\ast}(u,v)$. (c) and (d): another lower bound construction for the PoA for M-BGNCG networks in BSE. For both graphs holds that $x = \lfloor \frac{\sqrt{\alpha}}{2} \rfloor$ and $z = n - \lfloor \frac{\sqrt{\alpha}}{2} \rfloor$. For any edge $\{v_i,v_j\}$, it holds that $w(v_i,v_j) = 1$, if $|j-i|=1$, and $w(v_i,v_j) = 2$ otherwise. All other edge weights can be inferred from the triangle inequality.
• Figure 3: A M-BGNCG instance containing a pair of nodes $u,v \in V$ with value of $\sigma$ of $\alpha + 1$.
• Figure 4: The general structure of the proof of \ref{['M-BGNCG_dist_cost_ratio_BSE']}. The edges between $v^\ast$ and the three sets show edge weight bounds for the edges between $v^\ast$ and any node of that set, respectively. The dashed lines represent the improving move in $G$. Every node $v_i \in M'$ buys exactly one edge to a node $u_i \in N'$, so that every node in $N'$ has to buy at most two edges.