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We investigate the impact of random deviations in planned travel times using an extension of Vickrey's celebrated bottleneck model. The model is motivated by the fact that in real life, users can neither exactly plan the time at which they depart from home, nor the delay they experience before they join a particular congestion bottleneck under investigation. We show that the strategy advocated by the Nash equilibrium in Vickrey's model is not a user equilibrium in the model with travel time uncertainty. We then investigate the existence of a user equilibrium for the latter and show that in general such an equilibrium can neither be a pure Nash equilibrium, nor a mixed equilibrium with a continuous density. Our results imply that when random distortions influence user decisions, the dynamics of deterministic bottleneck models are inadequate to describe this more complex situation. We illustrate with numerical analysis how the mechanics of a bottleneck with delayed arrivals are unstable for any continuous strategy, thus shedding more light on the non-existence result. Alternatively, we formulate a logit model for the day-to-day dynamic adjustment in response to previous travel experiences. We further investigate the conditions for convergence of the dynamic choice model and numerically illustrate the resulting equilibrium strategy.

We investigate the impact of random deviations in planned arrival times on user equilibrium in an extension of Vickrey's celebrated bottleneck model. The model is motivated by the fact that in real life, users can not exactly plan the time at which they depart from home, nor the delay they experience before they join the congestion bottleneck under investigation. We show that the arrival density advocated by the Nash equilibrium in Vickrey's model, is not a user equilibrium in the model with random uncertainty. We then investigate the existence of a user equilibrium for the latter and show that in general such an equilibrium can neither be a pure Nash equilibrium, nor a mixed equilibrium with a continuous density. Our results imply that when random distortions influence user decisions, the dynamics of standard bottleneck models are inadequate to describe such more complex situations. We illustrate with numerical analysis how the mechanics of a bottleneck with delayed arrivals are unstable for any continuous arrival strategy, thus shedding more light on the non-existence result.

This paper introduces a class of games called the positive semidefinite games, for which we show the absence of mixed and nonstrict ESS's. As a result, a strategy is an ESS if and only if it is strict Nash. One famous example in this class of games is Rock{Paper{Scissors. For a smaller class of games called the positive definite games, we prove a similar result forThis paper introduces a class of games called the positive semidefinite games, for which we show the absence of mixed and nonstrict ESS's. As a result, a strategy is an ESS if and only if it is strict Nash. One famous example in this class of games is Rock{Paper{Scissors. For a smaller class of games called the positive definite games, we prove a similar result for NSS's. This result opens the door to a corollary: for doubly symmetric games, the existence of an ESS is assured. This is an interesting result because of the stronger dynamic stability properties of ESS's as compared to NSS's. The coordination games played on the identity matrix are an example of games in this latter class. NSS's. This result opens the door to a corollary: for doubly symmetricgames, the existence of an ESS is assured. This is an interesting result because of the stronger dynamic stability properties of ESS's as compared to NSS's. The coordination games played on the identity matrix are anexample of games in this latter class.

In a Self-Confirming Equilibrium (Fudenberg and Levine, 1993A) every player obtains partial information about other players' strategies and plays a best response to some conjecture which is consistent with his information. Two kinds of information structures are considered: In the first each player observes his own payoff while in the second the information is the distribution of players among the various actions. For each of these information structures we prove that pure Self-Confirming Equilibrium exists in some classes of games. Pure Nash equilibrium may fail to exist in these classes.