## 5 Link-Based Stability Notions  
Until now, we have generalized the classical notion of stability by considering separately the preferences of the men and the preferences of the women. We now intend to define new notions of stability that take into account, simultaneously, the preferences of the men and the women. Such a new notion will depend on the strength of the link of the married people,_i.e._, how much a man and a woman want to be married with each other. This is useful to obtain a new notion of stable marriage that looks at the happiness of the pairs (man,woman) rather than at the happiness of the members of a single sex. A way to define the strength of the link of two people is the following.  
**Definition 4** (link additive-strength): _Given a man, \(m\), and a woman, \(w\), the link additive-strength of the pair \((m,w)\), denoted by \(la(m,w)\), is the value obtained by summing the preference value that \(m\) gives to \(w\) and the preference value that \(w\) gives to \(m\),i.e., \(la(m,w)=p(m,w)+p(w,m)\)._  
Notice that we can use other operators beside the sum to define the link strength of a pair, such as, for example, the maximum, the minimum or the product.  
**Definition 5** (additive-weight): _Given a marriage, \(M\), the additive-weight of \(M\), denoted by \(w_{a}(M)\), is the sum of the links of all its pairs, i.e., \(\sum_{\{(m,w)\in M\}}la(m,w)\)._  
We now give a notion of stability that exploits the definition of the link additive-strength given above.  
**Definition 6** (link-additive-stability): _Given a marriage, \(M\), a man, \(m\), and a woman, \(w\), the pair \((m,w)\) is a link-additive-blocking pair for \(M\) if it does not belong to \(M\) and the following conditions hold:_* \(la(m,w)>la(m^{\prime},w)\)_,_
* \(la(m,w)>la(m,w^{\prime})\)_,_  
_where \(m^{\prime}\) is the partner of \(w\) in \(M\) and \(w^{\prime}\) is the partner of \(m\) in \(M\). A marriage is link-additive-stable if it does not contain link-additive-blocking pairs._  
**Example 4**: _Let \(\{w_{1},w_{2}\}\) and \(\{m_{1},m_{2}\}\) be, respectively, the set of women and men. Consider the following instance of an SMW, \(P\): \(\{m_{1}:w_{1}^{[30]}>w_{2}^{[3]}\), \(m_{2}:w_{1}^{[4]}>w_{2}^{[3]}\), \(w_{1}:m_{2}^{[6]}>m_{1}^{[5]}\), \(w_{2}:m_{1}^{[10]}>m_{2}^{[2]}\}\). In this example, there is a unique link-additive-stable marriage, that is \(M_{1}=\{(m_{1},w_{1}),(m_{2},w_{2})\}\), which has additive-weight \(w_{a}(M_{1})=35+5=40\). Notice that such a marriage has an additive-weight higher than the male-optimal stable marriage of \(c(P)\), that is \(M_{2}=\{(m_{1},w_{2}),(m_{2},w_{1})\}\), which has additive-weight \(w_{a}(M_{2})=13+10=23\). _  
Link-stability applies anytime there is an incentive to break a partnership to create a new one with an overall larger cumulative worth. This is certainly desirable as it leads to stronger positions in investments or it applies in the case where the benefit will depend on the overall contribution of both partners, but will, for example, benefit a third party. Another motivation is that of considering a man and a women as different sources of resources. For example, assume members of two different groups want to pair up for a project (for example, a grant proposal that allows only two PIs). Each gives a preference to the project by saying how many resources it would be willing to use for a project involving that person. However, the actual number of resources needed for the project to actually happen will be known later, only when the partners sit down and discuss the project. The success of the project strongly depends on the number of resources available to it, and one may prefer to participate in a project that he/she likes less and where he/she will put less resources, but that has a higher overall number of resources available.