[Paper Review] On Dynamics in Selfish Network Creation

Bernd Kawald, Pascal Lenzner|arXiv (Cornell University)|Dec 19, 2012

Game Theory and Applications18 references4 citations

TL;DR

This paper investigates the dynamic convergence of selfish network creation games, focusing on whether uncoordinated, local improving moves by agents lead to stable equilibria. It proves that while convergence is guaranteed for Swap Games on tree-structured networks, it fails entirely when cycles are present—even a single non-tree edge breaks convergence, and no move policy ensures convergence on non-tree networks.

ABSTRACT

We consider the dynamic behavior of several variants of the Network Creation Game, introduced by Fabrikant et al. [PODC'03]. Equilibrium networks in these models have desirable properties like low social cost and small diameter, which makes them attractive for the decentralized creation of overlay-networks. Unfortunately, due to the non-constructiveness of the Nash equilibrium, no distributed algorithm for finding such networks is known. We treat these games as sequential-move games and analyze whether (uncoordinated) selfish play eventually converges to an equilibrium state. Thus, we shed light on one of the most natural algorithms for this problem: distributed local search, where in each step some agent performs a myopic selfish improving move. We show that fast convergence is guaranteed for all versions of Swap Games, introduced by Alon et al. [SPAA'10], if the initial network is a tree, and show that this process can be sped up to an almost optimal number of moves. For non-tree networks we show the surprising result that even one non-tree edge suffices to destroy the convergence guarantee and no move policy can enforce convergence. This answers an open problem from Ehsani et al. [SPAA'11] in the negative. We extend our negative results to the well-studied original version and prove that there is no convergence guarantee -- even if all agents play optimally. Furthermore, we show the quite surprising result that employing cost-sharing yields even worse dynamic behavior. Finally, we contrast our mostly negative theoretical results by a careful empirical study. Our simulations indicate two positive facts: (1) The non-convergent behavior seems to be confined to a small set of pathological instances and is unlikely to show up in practice. (2) In all our simulations we observed a remarkably fast convergence towards a stable network in O(n) steps, where n is the number of agents.

Motivation & Objective

• To understand whether uncoordinated, selfish local search in network creation games converges to a stable equilibrium.
• To analyze the impact of network topology—particularly the presence of cycles—on convergence guarantees.
• To evaluate whether cost-sharing mechanisms improve dynamic behavior in network creation games.
• To determine whether optimal play or specific move policies can enforce convergence in various network creation game variants.
• To contrast theoretical negative results with empirical observations on practical convergence behavior.

Proposed method

• Models network creation as a sequential-move game where agents perform myopic, selfish improving moves.
• Analyzes the Swap Game variant where agents exchange edges, focusing on convergence from tree initial states.
• Introduces and evaluates a natural move policy to accelerate convergence in tree-based networks.
• Employs best-response dynamics to simulate agent behavior and identify cycles or convergence.
• Constructs pathological examples to prove non-convergence on non-tree networks, even with optimal play.
• Extends analysis to the original network creation game (buy/delete edges) and a cost-sharing variant requiring bilateral consent.

Experimental results

Research questions

• RQ1 Does uncoordinated selfish play in the Swap Game converge to a Nash equilibrium when starting from a tree?
• RQ2 Can any move policy enforce convergence on non-tree networks in the Swap Game?
• RQ3 Is convergence guaranteed in the original network creation game with edge buying and deletion, even under optimal play?
• RQ4 Does introducing cost-sharing with bilateral consent improve the convergence behavior of network creation games?
• RQ5 How do empirical simulations compare to theoretical convergence guarantees in practice?

Key findings

• Fast convergence to equilibrium is guaranteed for the Swap Game when the initial network is a tree, with convergence in O(n) steps.
• A natural move policy can speed up convergence to near-optimal performance in tree-based Swap Games.
• The presence of even one non-tree edge in the initial network destroys convergence guarantees for the Swap Game.
• No move policy can enforce convergence on non-tree networks in the Swap Game, as shown by pathological counterexamples.
• In the original network creation game, convergence is not guaranteed even with optimal play, and some instances have no improving sequence leading to stability.
• Surprisingly, the cost-sharing variant of the game exhibits worse dynamic behavior than the original, with no convergence guarantees.