# Mdpli Mitigation of Routing Congestion on Data Networks: A Quantum Game Theory Approach
## 1 Introduction  
Congestion is a significant issue in the more diverse environments, from supermarket queuing, urban traffic, transport, local networks to 5G and LTE-A networks. Congestion problems arise when users compete over a set of limited resources causing an increase in the latency to all contenders. Latency, for its part, depends on the number of agents that use a resource. A common example of congestion is at rush hour when the city's vehicle flow is saturated due to excess demand for roads by drivers. When the number of vehicles surpass the capacity of the roads, there is a decrease in velocities that cause waste of time and excessive fuel consumption. Likewise, delays, loss of efficiency and increase of transmission time occur when networks are too demanded [1]. At present, the constant increase in the number of packets sent over networks evidences the congestion problems and suggests that more research is needed when designing efficient communication systems. By their nature, the congestion problems become appropriate for being modeled by Game Theory [2, 3], because selfish decisions of agents (packets on the network or vehicles in a city) cause the whole system performance to be adversely affected. In the case of Wireless Sensor Networks (WSN), a type of network that faces a more challenging environment compared to traditional networks, several works on the threat mitigation problem have recently been published. In [4], for instance, the authors present a survey of several game-theoretic defense strategies for Wireless Sensor Networks. Due to their dynamic nature, WSN are exposed to malicious intruders. Such a security situation involving an interaction between the defender(s) and attacker(s) is directly mapped to a game among players where each player strives to promote its own benefit. Besides, in [5], a game theory model to control congestion in wireless body sensor networks supported on intelligent drop packet mechanisms is proposed. Moreover, a repeated game approach is proposed to sensor nodes protection in a clustered WSN in [6], where the proposed model outperforms non-cooperative defense mechanism to prevent cluster members from dropping the high priority packets. Furthermore, today we know that if game theory models harness the capabilities of quantum computing [7, 8] better outcomes can emerge. Moreover, the authors of [9] present an interesting work that address the Nash equilibria and correlated equilibria of classical and quantum games in the context of their Pareto efficiency. They focus their study in three classical games: the prisoner's dilemma, battle of the sexes and the game of chicken and show that the Nash equilibria of these games in quantum mixed Pauli strategies are closer to Pareto optimal results than their classical counter-parts. Quantum routing games were first proposed in [10] where the Braess' paradox was studied along with an analysis of the flow in a network with quantum resources, more exactly, networks that provide quantum entangled particles to the players before they play their strategies. In addition, in [11] a quantum game is applied to diminish spectrum allocation times and power consumption in quantum communication networks [12]. In this work, we start by modeling a network that allows both classical and quantum packets and then propose a routing protocol designed using the Game Theory formalism. After that, different variables with information about the dynamics of the network were measured and we point out that the protocol that makes use of the quantum game theory rules [13] significantly outperforms its classical equivalent. We demonstrate how to avoid congestion in a network and decrease the traveling time per packet in a system with a high number of packets that selfishly decide which decision is the best for everyone of them.