Mixed Nash equilibria in selfish routing problems with dynamic constraints ?
We study the problem of routing traffic through a congested network consisting of m m parallel links, each having a certain speed. Moreover, we are given n n selfish (non-cooperative) agents, each of them willing to route her own piece of traffic on exactly one link. Agents are selfish in that they only pick a link which minimize the delay of their own piece of traffic. In this context much effort has been lavished in the framework of mixed  Nash equilibria where the agents routing choices are regulated by probability distributions, one for each agent, which let the system thus enter a steady state from which no agent is willing to unilaterally deviate. In this work we consider situations in which some agents have constraints on the routing choice: in a sense they are forbidden to route their traffic on some links. We show that at most one Nash equilibrium may exist and, in some cases with equal speed links and where each agent is forbidden to route on at most one link, we give necessary and sufficient conditions on its existence; these conditions correlate the traffic load of the agents. We consider also a dynamic behaviour of the network when the constraints may vary, in particular when a constraint is removed: we establish under which conditions the network is still in equilibrium. These conditions are all effective in the sense that, given a set of yes/no routing constraints on each link for each agent, we provide the probability distributions corresponding to the unique Nash equilibrium associated to the constraints (if it exists). Moreover these conditions and the possible Nash equilibrium are computed in time O(mn) O ( m n ) .
selfish routing nash equilibrium non-cooperative games