Optimal threshold in multi-stage competitions

The purpose of this article is to present various static and dynamic normative models of risk-averse (RA), risk-neutral (RN) and risk-prone (RP) decision makers (DM) in single- or multiple-stage competition decisions, where each decision concerns passing or not passing a threshold goal set by the DM. The DM can be an athlete or a coach having a goal to maximise the expectation of his or her achievement, or the expected utility of his or her ranking in the competition. These models assume that the DM's performance is based on an empirical, probabilistic evaluation of his or her ability to pass a given threshold, as well as on a static evaluation of success, considering the risk of not achieving the given threshold goal. The results compare the optimal levels for a risk-neutral DM with that of one who is risk-averse or risk-prone. Presented also are recursive equations providing optimal threshold levels in dynamic multi-stage models. Furthermore, multiple trials are considered. Sometimes a competitor in a multi-stage competition fails to get past the first stage. The purpose of our model is to understand this phenomenon and to suggest an optimal strategy. Examples are based on probability distributions such as the uniform, normal, exponential, double exponential, etc. Also examples for several utility functions are provided. The pole-vault event is discussed and finally a recommended optimal strategy considering the game conditions, regulations, and the opponent's achievements is indicated. Some psychological aspects are also discussed.