ORDINAL POTENTIAL FUNCTIONS FOR NETWORK SELECTION IN HETEROGENEOUS WIRELESS NETWORKS Liang Ze Wong * , Tony Q.S. Quek *† , and Michael Padilla † * Institute for Infocomm Research, A*STAR, 1 Fusionopolis Way, #21-01 Connexis, Singapore 138632 † Singapore University of Technology and Design, 20 Dover Drive, Singapore 138682 ABSTRACT We consider the distributed allocation of spectrum in a heterogeneous wireless network. We model this as a non- cooperative game, where multiple players decide which chan- nels to transmit on. When co-channel interference between players is symmetric and channel utilities are additive, we define a generalized ordinal potential function for the game which is motivated by the energy function in Ising models. This guarantees convergence of best-response dynamics, and does not depend on the level of interference between players or on the price imposed on each channel. Index Terms— Heterogeneous networks; network selec- tion; ordinal potential games; congestion games 1. INTRODUCTION In heterogeneous wireless networks, multiple radio access technologies (RATs) coexist in the same network and are collectively able to provide better coverage, quality of ser- vice (QoS), and mobility support than if they were operating in isolation. To capitalize on the benefits of heterogeneous networks one needs to be able to transmit on multiple RATs simultaneously. This capability is found in multi-mode ter- minals (MMTs), which are able to access multiple bands simultaneously. We refer to these bands abstractly as “re- source blocks (RBs)”[1]. MMTs interfere with other MMTs sharing the same RBs, causing degradation in the QoS. The utility of an RB to an MMT is modelled as a function that decreases as the con- gestion increases. We formulate this as a non-cooperative game with partial information, where each MMT acts in a self-interested manner in trying to maximize its own utility. We show that best-response updates can be expressed as sim- ple threshold policies in the RB selection game. When con- gestion between MMTs is symmetric, we define a generalized ordinal potential function for the RB selection game. We fi- nally show that the convergence of best-response dynamics to a pure strategy Nash equilibrium is independent of any (fixed) channel prices that may be imposed on each RB, allowing net- work managers to optimize the channel price without worry- ing about convergence issues. An analysis of the interactions between users on a shared wireless channel with partial information has been carried out in [2]. The paper gives a detailed characterization of the equi- libria achievable when 2 users share a channel, knowing only their signal-to-noise ratio. In this paper, we expand the scope of consideration to networks with larger numbers of users and channels. Congestion games [3] and its generalizations (such as graphical congestion games [4]) have been widely used to model such scenarios. Instead of approximating our game us- ing a congestion game or exact potential game, as in [5], we show that the RB selection game restricted to one channel is an ordinal potential game, which further generalizes conges- tion games and graphical congestion games, and can be in- terpreted as a graphical congestion game on a weighted com- plete graph. Ordinal potential games have been used to prove convergence in wireless collision channels [6]. In that paper, a rate-alignment condition was required for the derivation of the ordinal potential function, but their simulations suggested that this condition was not necessary for convergence. Un- der slightly different conditions, we derive a different ordinal potential function which does not require the rate-alignment condition. We also consider multiple channels, and show that the sum of each channel’s ordinal potential functions is a gen- eralized ordinal potential function, thus giving our game the Finite Improvement Property [7] and guaranteeing conver- gence. 2. SYSTEM MODEL We model our system as a non-cooperative game, which we call the RB selection game. The players are the MMTs in the network and they seek to maximize their individual util- ities by transmitting on the RBs in the network. In our RB selection game we have a set K of K MMTs that can transmit on a set R of R RBs. Each MMT can either transmit or not transmit on RB r, indicated by p r k :=  1 if MMT k transmits on RB r 0 otherwise. (1) The action of MMT k is the binary vector p k =(p 1 k ,...,p R k ) ∈P k = {0, 1} R , (2) 2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP) 978-1-4799-2893-4/14/$31.00 ©2014 IEEE 6163