The re-opening of dubins and savage casino in the era of diversification

In Dynamic Programing, mixed strategies consist of randomizing the choice of actions. In some problems, such as portfolio management, it makes sense to diversify actions rather than choosing among them purely or randomly. Optimal betting in casinos and roulette by a gambler with fixed goal was studied by Dubins and Savage [9] and their school without the element of diversification (betting simultaneously on different holes of the roulette), once it was proved (Smith's theorem - Smith [16], Dubins [8] and Gilat and Weiss [10]) that diversification does not increase the probability of reaching the goal. We question the scope of this finding, that was based on the assumption that the holes on which gamblers can bet are disjoint, such as 1 and BLACK in regular roulette. A counter example is provided in which holes are nested, such as 1 and RED. Thus, it may be rational for gamblers with a fixed goal to place chips on more than one hole at the table. This note is related to a joint work with Michèle Cohen on the preference for safety in the Choquet Expected Utility model.