# A Game-Theoretic Rent-Seeking Framework for Improving Multipath TCP Performance +
## 3 Modeling Framework
### Diagonal Strict Concavity  
Consider a game with \(K\) subflows of an MPTCP connection. The strategy space is \(S\subset\mathbb{R}\), where \(S\) is a bounded set, and subflow \(i\)'s utility function is \(U_{i}:S^{K}\to\mathbb{R}\). [PERSON]'s condition [18] for uniqueness of the Nash equilibrium in a \(K\)-player game states that the equilibrium is unique when: (i) \(U_{i}(\lambda)\) is concave in subflow \(i^{\prime}\)s own action; (ii) There exists a vector \(\mathbf{z}\) of non-negative numbers such that the function \(\rho(\mathbf{\lambda},\mathbf{z}):=\sum_{i=1}^{K}\mathbf{z}_{i}U_{i}(\mathbf{\lambda})\) is diagonally strictly concave.  
To define the concept of _diagonal strict concavity_, we first compute the pseudo-gradient of \(\rho(.)\):  
\[\begin{split} g(\mathbf{\lambda},\mathbf{z})&=\begin{bmatrix} z_{1}\frac{\partial U_{1}(\lambda_{1}\lambda_{-1})}{\partial\lambda_{1}}\\ z_{2}\frac{\partial U_{2}(\lambda_{2}\lambda_{-2})}{\partial\lambda_{1}}\\ \vdots\\ z_{K}\frac{\partial U_{K}(\lambda_{K}\lambda_{-K})}{\partial\lambda_{1}} \end{bmatrix}\end{split} \tag{10}\]  
where \(\lambda_{-i}=\sum_{j=1(\
eq i)}^{K}\lambda_{j}\). Then, the function \(\rho(.)\) is said to be _diagonally strictly dominant_ in \(\mathbf{z}\in S\) for fixed \(\mathbf{z}\leq 0\) if for every \(\lambda_{0},\lambda_{1}\in S\)  
\[(\lambda_{0}-\lambda_{1})^{\prime}g(\lambda_{1},\mathbf{z})+(\lambda_{1}-\lambda_ {0})^{\prime}g(\lambda_{0},\mathbf{z})>0. \tag{11}\]  
Condition C\({}_{0}\): A sufficient condition for the function \(\rho(.)\) to be diagonally strictly concave is that the matrix \([G^{\prime}(\mathbf{\lambda},\mathbf{z})+G(\mathbf{\lambda},\mathbf{z}))]\) is negative definite for \(\mathbf{z}\in S\): here \(G(\mathbf{\lambda},\mathbf{z})\) is the Jacobian of the pseudo-gradient \(g(\mathbf{\lambda},\mathbf{z})\) with respect to \(\mathbf{\lambda}\).  
Condition C\({}_{1}\): A sufficient condition for the existence of a unique global equilibrium is that \(\rho(.)\) is diagonally strictly concave for some \(\mathbf{z}\) (i.e., C\({}_{0}\) holds).  
Note that Condition C\({}_{1}\) is essential for the stability of RSF MPTCP. Observe that C\({}_{1}\) is obtained directly from Theorem 4 in Rosen (Theorem 4 in [18]). We acknowledge that both prior works [13; 18] nicely established the existence, uniqueness, and stability of the underlying dynamical system for the Tullock _rent-seeking_ game. We implement these ideas in our work to design a cooperative framework for MPTCP. Furthermore, we consider a practical operational model, in which the existence of the utility function, and its interaction with multipath network constraints is one of the building blocks.  
**Lemma 2** (Stability).: _Assume that \(C_{0}\) and \(C_{1}\) both hold, the utility function \(U(.)\) exists, and the existence and uniqueness of the equilibrium \((\mathbf{\lambda^{\star}},\mathbf{p^{\star}})\) is guaranteed. Then, starting from any initial point \((\mathbf{\lambda}(0),\mathbf{p}(0))\), the trajectory \((\mathbf{\lambda}(t),\mathbf{p}(t))\) generated by the cooperative MPTCP algorithm converges to a unique solution, and Equation (1) has a globally stable solution given by \((\mathbf{\lambda^{\star}},\mathbf{p^{\star}})\)._  
First, we provide a proof that a Tullock _rent-seeking_ game with a finite capacity constraint has a diagonally strictly concave utility function and, thus, a unique _normalized_ equilibrium.  
Observe in Equation (15) that F is positive for \(\lambda>0\); therefore, the matrix \([G^{\prime}(\lambda,\mathbf{z})+G(\lambda,\mathbf{z}))]\) is negative definite. As a result, \(U(\lambda)\) is diagonally strictly concave. See [13] for more background.  
In the following, we provide further details on stability and our interpretation of utility maximization. Given the normalized equilibrium and the concave utility, the remainder of the proof of Lemma 2 is provided in Section 3.3.3.