## 1 Introduction  
The stable matching problem (SM) [1] is a well-known problem of matching the elements of two sets. It is also called the _stable marriage_ problem, since the standard formulation is in terms of men and women, and the matching is interpreted in terms of a set of marriages. Given \(n\) men and \(n\) women, where each person expresses a strict ordering over the members of the opposite sex, the problem is to match the men to the women, so that there are no two people of the opposite sex who would both rather be matched with each other than their current partners. If there are no such people, all the marriages are said to be _stable_. In [2], [PERSON] and [PERSON] proved that it is always possible to find a stable marriage and provided a polynomial time algorithm, which can be used to find one of two extreme stable marriages, the so-called _male-optimal_ or _female-optimal_ solutions. The Gale-Shapley algorithm has been used in many real-life scenarios [3], such as in matching hospitals to resident doctors [4], medical students to hospitals, sailors to ships [5], primary school students to secondary schools [6], as well as in market trading. In the classical stable marriage problem, both men and women express a strict preference order over the members of the other sex in a qualitative way. Here, we consider stable marriage problems with weighted preferences. In such problems, each man (resp., woman) provides a score for each woman (resp., man). Stable marriage problems with weighted preferences [7,8] are interesting, since they are more expressive than the classical stable marriage problems, since in the classical stable marriage problem, a man (resp., a woman) cannot express how much he (resp., she) prefers a certain woman (resp., man). Moreover, they are useful in some real-life situations, where it is more natural to express scores that can model notions, such as profit or cost, rather than a qualitative preference ordering. In this context, we define new notions of stability and optimality, we compare such notions with the classical ones and we show algorithms to find marriages that are stable and/or optimal according to these notions. While expressivity increases by adopting weighted preferences, we show that in most cases, the desired solutions can be found by adapting existing algorithms for the classical stable marriage problem. In this paper, we also study if the stable marriage procedures that return one of these new stable marriages are manipulable. In [9], [PERSON] has shown that, when there are at least three men and three women, every stable marriage procedure is manipulable,_i.e._, there is a profile in which an agent can mis-report his preferences and obtain a stable marriage that is better than or equal to the one obtained by telling the truth. In this setting, mis-reporting preferences means changing the preference ordering [9] or truncating the preference list [10]. In this paper, we consider a possible additional way of mis-reporting one's own preferences, which is by just modifying the weights, in a way such that the orderings are preserved and the lists remain complete. We show that it is actually possible to manipulate by just doing this. Thus, in some cases, adding weights makes stable marriage procedures less resistant to manipulation. Moreover, we show that, in these cases, there are no reasonable restrictions on the weights that can prevent such manipulation. As an example of an incentive to manipulate, we can think of an Extreme Programming scenario [11], where we have to match pairs of programmers that must work together. In this scenario, programmers may have an idea of their colleagues preferences and might try to manipulate by lying about their utility or scoring on colleagues to make sure they are paired with their feasible favorite. Moreover, an incentive to manipulate can also be present in the case of matching students to schools. The paper is organized as follows. In Section 2, we present related research.