# A Game-Theoretic Rent-Seeking Framework for Improving Multipath TCP Performance +
## 2 Model Overview
### Our Proposed Scheme  
We now summarize our proposed MPTCP algorithm. The key idea is to schedule the packets via multiple paths at different rates in such a way that they balance the delay across all paths and are, therefore, more likely to arrive at the receiver in order. More specifically, an MPTCP congestion control mechanism with heterogeneous network paths will perceive different utilities on each path. In essence, we strive to harmonize the performance achieved across multiple paths by opportunistically exploiting path dynamics to provide seamless MPTCP performance. The primary advantage of our method is that it implicitly addresses all known issues that limit MPTCP performance (i.e., HOL blocking, reordering latency, route heterogeneity). Furthermore, it enables the pooling of multipath resources in a way that emulates the performance of a single harmonized path. One way to achieve utility maximization is to formulate a Tullock rent-seeking game [8; 25] and use this to equalize the delays across all paths in order to ameliorate reordering delay. A plausible approach is to set equal delay for all subflows (across all paths) and determine the corresponding sending rates that jointly satisfy the delay constraints and load-balancing objectives. Delay-Balanced MPTCP (RSF-DB): In RSF-DB, each source has a set of paths and a predefined target delay \(d>0\), which is fixed and equal for all paths. The source maintains a congestion window \(W_{i}\) for each path \(i\) and measures its forward delay \(q_{i}\) and roundtrip time \(\tau_{i}\) for each path. Here, \(q_{i}\) is the queueing plus service delay from the source to the sink on path \(i\), where \(K\) is the number of paths. Then, the window adaptation in the congestion avoidance phase is:  
* For each packet loss on path \(i\), update \(W_{i}\gets W_{i}/2\);
* For each new acknowledgment received on path \(i\), increase \(W_{i}\) using \(W_{i}\gets W_{i}+\Delta(W_{i})\), where: \[\Delta(W_{i})=\frac{1}{W_{i}}\min\Big{\{}1,\frac{d-q_{i}}{q_{i}}\Big{\}}.\] (1)  
This algorithm is designed to adjust the window size dynamically so that \(q_{i}\) converges toward the target delay \(d\) in the steady state. Note that \(q_{i}\) is not necessarily strictly bounded by \(d\), so the \"window increase\" could, in some cases, be negative if the situation warrants. A concise summary of the window update operations for three well-known MPTCP algorithms (LIA, OLIA, and BALIA) is shown in Table 1.  
Similarly, we designed a complementary solution called loss-balanced MPTCP (RSF-LB), in which each MPTCP source has a set of paths and a predefined target loss rate. The key idea here is to balance the loss _rate_ (i.e., lost packets per second) on each path to ameliorate the potential effects of heterogeneity in the packet loss _probabilities_, \(l_{i}\), on different paths. For RSF-LB, we replace (1) with  
\[\Delta(W_{i})=\frac{1}{W_{i}}\min\Big{\{}1,\frac{\mu-\lambda_{i}l_{i}}{ \lambda_{i}l_{i}}\Big{\}} \tag{2}\]  
so that the instantaneous \(\lambda_{i}l_{i}\) converges toward the target loss rate \(\mu\) packets/sec in the steady state. We extend [PERSON]'s framework [6] by exploiting the well-known Tullock rent-seeking game [8] to design a cooperative MPTCP with common coupled constraints. We consider a non-linear dynamical system model for an MPTCP connection over \(K\) network paths, as seen in Equation (3):  
\[\dot{\mathbf{y}}=f(\mathbf{y}),\ \ \mathbf{y}(0)=\mathbf{y}_{0}, \tag{3}\]where \(f(\mathbf{y})=\mathbf{0}\) has a unique solution. In this equation,  
\[\mathbf{y}=(\mathbf{\lambda},\mathbf{p});\ \mathbf{\lambda}=(\lambda_{1},\lambda_{2},\ldots \lambda_{K})\]  
the \(\lambda_{i}\) values denote the packet-sending rates, and  
\[\mathbf{p}=(p_{1},\ldots p_{K})\]  
denote the congestion prices (based on the delay \(q_{i}\) or loss \(l_{i}\)) for the respective paths.