Allocating energy in a first-to-n match

This paper models a decision where a player must allocate limited energy over a contest of uncertain length. The motivating example is a squash match where one of the players is not as fit as the other. Should a player's energy be concentrated in the early games of the match? Should it be spread evenly over all possible games? Or should it be conserved for the end of the match? We model this as a decision problem where, in each game, the decision-maker must determine how much energy to expend. We assume that there are only a small number of discrete energy choices for each game and that the more energy the decision-maker expends, the more likely he is to win that game. We solve for the optimal decision with dynamic programming. With only two possible energy choices for each game, we show that it does not matter how energy is expended. In the case where there are three or more energy choices, we show how to take advantage of the structure of the problem to determine the optimal sequence of decisions. As for practical advice, the model suggests that when the decision-maker falls behind in a match, he ought to switch to a more conservative approach by dividing his remaining energy evenly among all the possible remaining games.