# A Game-Theoretic Rent-Seeking Framework for Improving Multipath TCP Performance +
## 3 Modeling Framework
### Rent-Seeking Game Framework  
Rent-seeking is a concept from game theory in economics. Unlike profit-seeking, which tries to increase total wealth, rent-seeking tries to increase one's own share of existing wealth without increasing the total wealth. Specifically, it is an attempt to obtain a reward R (rent) by manipulating the environment in which the activities occur, rather than by creating new wealth. One example is lobbying and/or bribery of government officials, in which the efforts (e.g., time and/or money) spent may result in favorable outcomes (e.g., tax laws, corporate rebates), albeit with potentially reduced economic efficiency. Other examples include taxi-licensing fees, bridge tolls, and bidding in loss-load curves [26], all of which result in reallocation of resources.  
Strategies for _rent-seeking_ games were studied by [PERSON] in 1967 and form a good basis for studying MPTCP flows. We use this rent-seeking approach to design our utility function below.  
We consider an MPTCP connection in a network setting with \(K\) paths and one subflow on each path. Let \(\lambda_{i}=W_{i}/\tau_{i}\) (strictly positive \(\lambda_{i}>0\)) represent the sending rate for the \(i\)th subflow (\(1\leq i\leq K\)), and let \(p_{i}\) represent the corresponding price for that subflow. Then, assuming that the sending rates on each path are the actions to be chosen and that the connection throughput is concave, the individual throughput (payoff) obtained from path \(i\) can be represented as [13]:  
\[\theta_{i}=V\left(\lambda_{i}\middle/\sum_{j=1}^{K}\lambda_{j}\right), \tag{5}\]  
where \(V(.)\) is a concave function. Observe that the throughput \(\theta_{i}\) comes at the cost of \(\hat{p}\lambda_{i}\) (where \(\hat{p}\) is a constant). With this formulation, the utility for the MPTCP subflow over path \(i\) is structurally similar to that of a [PERSON] rent-seeking game [13], for which:  
\[U_{i}(\lambda)=\lambda_{i}\middle/\sum_{j=1}^{K}\lambda_{j}-\hat{p}\lambda_{i}. \tag{6}\]  
**Lemma 1** (Existence and Uniqueness).: _The utility \(U_{i}(\lambda)\) of an MPTCP subflow \(i\) is concave in its action \(\lambda_{i}\) and is continuous in the corresponding actions of other subflows. Further, for \(\hat{p}>0\) (strictly positive pricing), the trajectory \((\mathbf{\lambda}(t),\mathbf{p}(t))\) generated by the cooperative MPTCP algorithm has a unique Nash equilibrium in pure policies._  
The proof of the concavity and continuity of \(U_{i}(\lambda)\) follows from direct calculation. The existence and uniqueness can be established along the lines of [PERSON] [18] and [PERSON] [13], respectively.  
In contrast to [PERSON]'s approach [6], the subflows of an MPTCP connection can have an action on one path (such as an increase or decrease in \(\lambda_{i}\)) that is not dependent on the actions over other paths. Further, similarly to [6], the capacity of the system is limited in real communication networks. Therefore, the aggregate network bandwidth can be bounded by a finite constant, say \(B\). As a result,  
\[\sum_{j=1}^{K}\lambda_{j}\leq B. \tag{7}\]  
However, with this additional capacity constraint, _the uniqueness of the Nash equilibrium_ given by Lemma 1 collapses, and there can be (infinitely) many equilibria [27].