This paper formulates a Stochastic Social Optimum (SSO) that relates to the Stochastic User Equilibrium (SUE) in the same way as the Social Optimum (SO) relates to the User Equilibrium (UE) in a deterministic environment. At the SSO solution, the total of the users' perceived costs is minimised. The formulation and analysis is carried out in a general utility-maximising framework, with the probit and logit models being special cases. Conditions for the SSO flow pattern are derived, from which it can be seen that the marginal social costs play the same role in the SSO as the standard costs play in SUE. In particular, it is shown that the SSO solution can be obtained through the use of an algorithm for SUE, but with the marginal costs replacing the standard costs in the stochastic loading and that optimal tolls are the differences between the marginal social costs and the standard costs. For the case of the logit model an explicit path-based objective function is obtained which is of a pleasing symmetrical form when compared with the objective functions for SUE, SO and UE. Additionally, a link-based objective function for the general utility-maximising case is formulated for SSO, which is similar in form to the SUE objective function of Sheffi and Powell.
