On dynamics in selfish network creation

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 nonconstructiveness of the Nash equilibrium, no distributed algorithm for finding such networks is known. We treat these games as sequential-move games and analyze if (uncoordinated) selfish play eventually converges to an equilibrium. 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. Furthermore, we prove that this process can be sped up to an almost optimal number of moves by employing a very natural move policy. Unfortunately, these positive results are no longer true if the initial network has cycles and we show the surprising result that even one non-tree edge suffices to destroy the convergence guarantee. This answers an open problem from Ehsani et al. [SPAA'11] in the negative. Moreover, we show that on non-tree networks no move policy can enforce convergence. We extend our negative results to the well-studied original version, where agents are allowed to buy and delete edges as well. For this model we prove that there is no convergence guarantee -even if all agents play optimally. Even worse, if played on a non-complete host-graph, then there are instances where no sequence of improving moves leads to a stable network. Furthermore, we analyze whether cost-sharing has positive impact on the convergence behavior. For this we consider a version by Corbo and Parkes [PODC'05] where bilateral consent is needed for the creation of an edge and where edge-costs are shared among the involved agents. We show that employing such a cost-sharing rule yields even worse dynamic behavior.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. SPAA'13, 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...

Bounded-Distance Network Creation Games

“…More precisely, we plan to investigate whether best (or at least better, given the hardness result of Theorem 3.2) response dynamics might converge to a Nash equilibrium, and under which distance bounds this might possibly take place. Our preliminary analysis shows that the nonconvergence results provided by Kawald and Lenzner [2013] do not extend easily to our model, and so a new ad-hoc approach seems necessary.…”

The max-distance network creation game on general host graphs

“…To the best of our knowledge, this is the first result showing that an improving response dynamics on MaxNCG might not converge to an equilibrium. A deeper discussion about dynamics in NCGs can be found in [8,10]. Proof.…”

“…This implies that an improving response dynamic may not converge for the MaxNCG game as well (after relaxing such a liveness property). A deeper discussion on dynamics in NCGs can be found in [8,10].…”

The Efficiency of Best-Response Dynamics

“…[22] showed that using the Max-Cost deviator rule, tightly bounds the convergence rate to 1.5n (worst-case) in these games. The Max-Cost deviator rule was also considered in [30] for Swap-Games [4]. In Swap-Games with initial strategies corresponding to a tree it has been shown that the convergence time is O(n 3 ), but using the Max-Cost deviator rule improves the bound to O(n).…”