Controlling the losing probability in a monotone game

We deal with a complex game between Alice and Bob where each contender’s probability of victory grows monotonically by unknown amounts with the resources employed. For a fixed effort on Alice’s part, Bob increases his resources on the basis of the results for each round (victory, tie or defeat) with the aim of reducing the probability of defeat to below a given threshold. We read this goal in terms of computing a confidence interval for the probability of losing and realize that the moves in some contests may bring in an indeterminacy trap: in certain games Bob cannot simultaneously have both a low probability-of-defeat measure and a narrow confidence interval. We use the inferential mechanism called twisting argument to compute the above interval on the basis of two joint statistics. Careful use of such statistics allows us to avoid indeterminacy.