## 1 Introduction
The choice of the parameters taken into account in the definition of the utility metric and, in particular, the resulting rate of variation over time of the metric, will have a strong impact on the performance of network selection strategy, and should be considered in the decision on the network selection approach to be adopted. Moving to algorithms and mathematical approaches proposed for network selection, game theory was widely proposed as a solution in cases where no central processing point is available, and cooperation between users may or may not be available. In homogeneous scenarios, game theory was proposed in [8], among others, as a way to select the best cell in two-tier network, and in a broader scheme, in combination with a machine learning algorithm, to solve the problem of best provider selection and power allocation within the selected network [9]. In heterogeneous scenarios,game theory was adopted in [10] to jointly tackle the network selection and resource allocation problems in a multi-operator heterogeneous network in which users take simultaneous decisions; the paper also provided an extensive analysis of the use of game theory for network selection, including the issue of the definition of utility metrics to be adopted in the problem. In [11], the problem was extended by considering a game in which users take their network selection decisions sequentially, taking into account decisions of other users. A reasonable assumption in the network selection problem is, however, that no a priori information about the networks is available, except for their presence, requiring thus the acquisition of measurements of relevant networks parameters in order to enable an informed decision on the network to be selected. The Multi-Armed Bandit (MAB) framework, in which a player chooses among different arms with the goal of maximizing a reward, can be used to model the above problem [12; 13; 14]. In fact, by using [PERSON]'s words [12], MAB problems embody in essential form a conflict evident in all human actions. This is the conflict between taking those actions that yield immediate reward and those (such as acquiring information or skill, or preparing the ground) whose benefit will come only later. In the considered scenario, the action with future benefit is to measure network parameters, while connecting to a network and exploiting it for transmitting and receiving is the action with immediate reward. The idea of adopting an online learning algorithm is indeed in line with recent proposals for best network selection in 5G, foreseeing complex information acquisition phases involving both devices and network infrastructure for an effective network selection [15]. Current MAB models focus, however, mainly on actions that bring an immediate reward: existing algorithms built on the basis of such models mainly address thus how to decide which arm, i.e., which wireless network in the considered case, to select at each time step [16; 17; 18]. This is also the case for online learning algorithms based on the MAB model proposed to address the network selection problem, as for example the one in [19], where a continuous time MAB problem is solved in order to select the network providing the best QoE to the user. This work proposes a a new MAB model that takes into account the need for a measurement phase before the network selection decision can take place, introducing thus a trade-off between the time \(T_{M}\) dedicated to measuring the performance of each network, and the time \(T_{U}\) spent using the selected network before updating again the information on available networks. On the one hand, \(T_{M}\) should be long enough to guarantee an effective decision; on the other, it should be significantly shorter than \(T_{U}\), in order to ensure that the overhead related to measuring, expressed by the ratio \(T_{M}/\left(T_{M}+T_{U}\right)\), is kept at a reasonable level. The new MAB model, referred to as measure-use-MAB (_muMAB_), allows the player to select between two distinct strategic actions: _measure_ and _use_.