Mixed Nash equilibria in selfish routing problems with dynamic constraints

AbstractWe study the problem of routing traffic through a congested network consisting of m parallel links, each having a certain speed. Moreover, we are given 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 agent’s 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)

Evacuation planning under selfish evacuation routing

In case of an evacuation a large number of evacuees must be routed through a street network to let them leave the endangered area and reach safe places. In such a situation a lot of evacuees use the street network in a short time span and so the network capacity will be insufficient. With an evacuation plan the traffic could be guided through the network for a better use of network capacity. But to implement the solution planned by a central decision maker, optimal routes must be communicated to all network users, which lead to a high communication effort. Furthermore, it must be ensured that the evacuees take the given routes. But a lot of people do not follow the instructions from authorities in a panic situation. They do what they assume is best for themselves. Such selfish behaviour leads to a suboptimal distribution of traffic and results in congestion. In this thesis we present a concept to guide the evacuees through the network without determining optimal routes for all network users. With the blockage of street sections we force the evacuees to use other routes than the preferred ones but give them the possibility to choose their routes on their own. The thesis presents different mathematical model formulations and heuristic for the described problem. In a comprehensive computational study, with real world examples, the functionality of the presented concept and methods are tested