On the relationship between Nash—Cournot and Wardrop equilibria

Abstract

A noncooperative game is formulated on a transportation network with congestion. The players are associated with origin-destination pairs, and are facing demand functions at their respective destination nodes. A Nash-Cournot equilibrium is defined and conditions for existence and uniqueness of this solution are provided. The asymptotic behavior of the Nash-Cournot equilibrium is then shown to yield (under appropriate assumptions) a total flow vector corresponding to a Wardrop equilibrium.

Spatio-temporal Control for Dynamic Routing Games

“…If player-1 chooses q 1 = 0, x * = 0 is the minimizer of the above equation when f (1) > d (1) p , hence (0, 0) is the SSNE. By similar arguments we can show that (1, 1) is the SSNE when (1 − p)f (2) + pf (1) < d (1) p .…”

Traffic Modeling and Variational Inequalities Using GAMS

“…The rst model is based on considering all the paths between the origin-destination pairs, and the second uses a multicommodity formulation, representing each origin or destination node as a di erent commodity. Both of these formulations use the Wardropian characterizations of equilibria 42], a special case of a Nash equilibrium (see 15,23] The Wardrop equilibrium principle 42] states that each driver will choose the minimum cost path between every origin destination pair and through this process, the paths used will all have equal cost; paths with costs higher than the minimum will have no ow. Mathematically, this principle can be phrased succinctly as 0 p ( ) w ?…”

“…We then considered the possibility of 2 failures in the network, on arcs (1,2) and arc (21,24). We incorporated capacities of 0.5 only on arcs (15,14) and (22,23) so that when a failure of arc (21,24) occurs, all the ow could not be rerouted through these arcs. The loop statement in Figure 4 is used to compute the steady state solution for each arc failure, required to enforce (28).…”

Efficiency Loss of Mixed Equilibrium Behaviors with Polynomial Cost Functions

“…The above two principles only consider the two extreme cases, i.e., there are many infinitesimal players in the UE principle and there is only one central player in the SO principle. Haurie and Marcotte investigated the network with some non-cooperative Cournot-Nash (CN) players, where the users belonging to the same player can fully cooperate with each other and different players will compete with each other 2 . The users of one CN player aim to minimize their own total cost while competing with the users of other players.…”