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There exists a leavable, Borel gambling problem with a goal, where at most three gambles are available at each fortune, where each gamble has at most two points in its support, but for which stationary plans are not uniformly adequate. Introduction. In the theory of gambling formulated by Dubins and Savage [5], a gambling problem is a triple (F,T,u), where (i) the set of fortunes F is a nonempty set, (ii) the gambling house T is a subset of F x P(F), where P(F) denotes the collection of all finitely additive probabilities defined on all subsets of F, and for each f G F, the set of gambles available at f, T(f) = {7: (/, 7) G T} is assumed to be nonempty, and (iii) the utility function u is a real-valued function on F. For f G F, a strategy o, available at f specifies (a) a gamble oq S T(/), and (b) for each positive integer n and every sequence (/i,...,/n) of elements of F, a gamble an(fi, ■ ■ ■, fn) £ r(/„). Roughly speaking, such a strategy, a, available at /, specifies for a gambler with initial fortune /, (a) what he should do initially, and (b) what he should do after n days, given that his fortunes on the first n days were (/1,..., /„). A strategy a available at / induces a probability, which we shall denote once again by a, defined on the Borel subsets of the history space H = FN, where N denotes the set of positive integers and H is equipped with the product of discrete topologies. Let £(/) = {a: available at /}. The value of a gambling problem V is a real-valued function on F, defined by V(f) — sup / limsupw(a;„) dcr(h), f G F, c€^(f) J n where h = (x\,..., xn,_) is an element of H. (We only consider gambling problems for which / limsupn u(xn) da(h) is defined for all / G F and a G £(/).) A stationary plan S is a function defined on F into P(F) such that S(f) G T(f) for all f G F. A stationary plan defines, for each / G F, a strategy Sf available at /. The initial gamble of Sf is S(f) and, given that the fortunes on the first n days were (/1,..., fn), the gamble specified by Sf is S(/„). The utility of a stationary plan S is a real-valued function Us on F, defined by Us(f) = /limsup u(xn) dSf(h), f G F, J n Received by the editors September 16, 1986 and, in revised form, January 15, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 60G40.

Stationary plans need not be uniformly adequate for leavable, Borel gambling problems