# Mdpli Mitigation of Routing Congestion on Data Networks: A Quantum Game Theory Approach
## 3 Classical and Quantum Strategies  
It is well known that a game is defined by three elements: players, strategies, and rewards. In this case, our game has a non-cooperative nature, that is, it is a game with competition between individual players, the players being the packets that travel through the network. The strategies are to take or not a channel when there are enough packets to congest the channel. Lastly, the reward is the total time that is specified by adding the routing and traveling time (with a negative sign since the lower the time, the bigger the reward). If more than one packet is interested in a channel, because it is part of their currently shortest path, each packet has two possible strategies: choose this preferred channel (risking that other packets will also select it and then congest the channel) _or_ search for its following shortest path (longer but possibly idler). These two strategies will be called option 1 and 0, respectively. For example, in the first game of a player, he has two options: (1) Take the shortest path risking that the congestion increases their traveling time significantly. (0) Try  
Figure 1: Example of network model for \(n1=10\). with their second shortest path, where there might be no congestion. In Table 1, an example of a situation where two packets are interested in the same channel, is presented. The playing games may be of different kind, that is classical or quantum. Let us consider first classical games. The player's mixed classical strategies are probabilistic, that is, option 0 is chosen with probability \(p\) and option 1 with probability \((1-p)\). Then, probability \(p\) near zero corresponds to greedy players, since players will tend to always take the shortest path even when many players are competing for that channel. A value of \(p\) closer to one will create players more patiently looking for another idler route. To study the quantum game for channel choice we follow the _EWL_ protocol for 2 players [8] and then extended for \(N\) players [21]. The _first_ step is to assign a quantum state to each of the possible strategies. The quantum protocol is exactly the same as the classical one, the only difference being that strategies 0 and 1 were previously represented on a bit and now they are represented on a qubit. Strategy 0 (leave the preferred route) is mapped to the quantum state \(|0\rangle\) and strategy 1 (take the preferred route) to the quantum state \(|1\rangle\). The _second_ step is to create a quantum circuit where each player will be assigned a qubit that will start in state \(|0\rangle\). The _third_ step is to create a maximally entangled state between all the players. This is done by applying the entangling operator \(J=\frac{1}{\sqrt{2}}(\mathbb{I}^{\otimes N}+i\sigma_{x}^{\otimes N})\), as seen in Figure 2, where the number of players is \(N=2\). In the _fourth_ step, every player chooses her most suitable strategy individually and independently. This is done by modifying the state of her own qubit locally. To do this, every player applies one or more one-qubit gates, modifying the state of her qubit. A general one-qubit gate is a unitary matrix that can be represented as [22]:  
\[U(\theta,\phi,\lambda)= \begin{pmatrix}\cos(\frac{\theta}{2})&-e^{i\lambda}\sin (\frac{\theta}{2})\\ e^{i\phi}\sin(\frac{\theta}{2})&e^{i(\phi+\lambda)}\cos(\frac{\theta}{2})\end{pmatrix}  \tag{1}\]  
We can already highlight the fact that while classic players have only one parameter to choose their mixed strategy: \(p\), quantum players have three parameters to choose in order to select their own pure strategy: \(\theta\), \(\phi\) and \(\lambda\). The _fifth_ step is to apply the operator \(J^{\dagger}\) (\(J\) conjugate transpose) after the players' strategies. Furthermore, finally, the _sixth_ step consists of measuring the state of the qubits to know the output of the circuit and, therefore, the final action of each player.