Abstract
It is shown that if a gambleγ stakes positive amounts on infinitely many holes of a subfair roulette-table, then for everyɛ>0, there is a gambleγ * with positive stakes on only a finite number of holes, such thatγQ≦γ*Q+e for every nondecreasing functionQ bounded above by 1 on [0, ∞]. It is deduced from this proposition that a gambler who wishes to maximize his chances to increase his current fortune by a specified amount, has no advantage in ever placing positive stakes on more than a finite number of holes on any single spin. This result settles a question left open in [1].
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