A continuity question of Dubins and Savage

Abstract

Abstract Lester Dubins and Leonard Savage posed the question as to what extent the optimal reward function U of a leavable gambling problem varies continuously in the gambling house Γ, which specifies the stochastic processes available to a player, and the utility function u, which determines the payoff for each process. Here a distance is defined for measurable houses with a Borel state space and a bounded Borel measurable utility. A trivial example shows that the mapping Γ ↦ U is not always continuous for fixed u. However, it is lower semicontinuous in the sense that, if Γn converges to Γ, then lim inf Un ≥ U. The mapping u ↦ U is continuous in the supnorm topology for fixed Γ, but is not always continuous in the topology of uniform convergence on compact sets. Dubins and Savage observed that a failure of continuity occurs when a sequence of superfair casinos converges to a fair casino, and queried whether this is the only source of discontinuity for the special gambling problems called casinos. For the distance used here, an example shows that there can be discontinuity even when all the casinos are subfair.