A continuity question of Dubins and Savage

Lester Dubins and Leonard Savage posed the question as to what extent the optimal reward function U of a gambling problem varies continuously in the gambling house Γ and utility function u. 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. However, it is lower semicontinuous in the sense that, if Γ n converges to Γ, then lim inf U n ≥ U. 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.