In this supplementary file, we examine the network performance in terms of competitive ratio and convergence if selfish strategy is applied, and provide detailed proofs of theorems stated in the main manuscript.
SELFISH USER STRATEGYOne natural alternative to solving the association problem, with respect to our goal, is to let the clients behave myopically by applying in decentralized AP selection the best-reply policy. Explicitly, it means that every user keeps moving to associate with the AP that could offer it the best throughput until no user can gain higher throughput by unilaterally deviating from its current decision (Nash Equilibrium).To simplify the analysis for selfish users, we make two assumptions in this section. In the next section, we will use a more realistic assumptions.First, we assume that the interference between the communications of two APs is not considered, i.e., the nearby APs operate on orthogonal channels. Second, the association procedure of a user is considered as an atomic operation, so only one user performs the association at a time.The time at which a user makes a decision to change APs is marked as a decision step. However, we do not require users to follow a certain decision order, which means in each decision step the user who is picking a new AP could be any one.• F. Xu and Q. Li. are Under these assumptions, we will show that such selfish user game converges to a Nash Equilibrium often having non-optimal performance. More complicated scenarios even cannot grantee the existence of the Equi-We denote by U a the set of users connecting with AP a. So let n a = |U a | represent the cardinality of this set. We designate by st u the percentage of service time the user u gains from associated AP, and T u corresponds to the throughput of u. And for any user u and AP a, we use R ua to denote the transmission rate under the situation only u is associating with a. R ua varies even for the same user. For the rest of this section, unless otherwise specified, the transmission rate refers to the effective transmission rate, which considers the overhead caused by retransmissions, random backoff and so on.To examine the performance of this protocol, we consider two aspects: convergence and competitive ratio. The competitive ratio here is equivalent to the price of anarchy 1 (PoA) using minimum user throughput as social cost.The following subsections show first whether the selfish user protocol will eventually stabilize and how fast the protocol will achieve convergence in general, and after that give the competitive ratio of the protocol.
Convergence of the Selfish StrategyIn this subsection, we will show how to model this selfish throughput strategy as a special case of the weighted congestion game, where the weight of a user varies as the associated AP set, which is singleton, changes. This game is proved to be converged with not ideal speed by leveraging the technique similar to [1].Given the Lemma 1 and assumptions we made, we describe this game in the wireless LAN scenario. Consider a...