Where to stand when playing darts? Questions in Probability Theory inspired by dart throwing

Abstract

This thesis investigates questions in probability theory inspired by a model of dart throwing. The deviation from where you aim is modelled as the distance to the dart board times a random vector, and in this thesis we refer to such a random vector as a dart. Points are then assigned by some bounded payoff function, and we study how the choice of the payoff function as well as the distribution of the dart affect the properties of the expected score when aiming in an optimal way.

Interestingly, it turns out that sometimes it can be better to move further away from the dartboard before throwing, and the main focus of this thesis is to characterise under what circumstances this is or isn't the case. For a dart X and payoff function f we call the pair (X,f) reasonable if it is always better to stand closer to the dartboard, and a dart X is called reasonable if (X,f) is reasonable for all payoff functions f.

We have found a large class of darts which are reasonable, namely those which have so-called selfdecomposable distributions, which includes many well known distributions, such as the exponential distribution, the logistic distribution, and also all stable distributions. Whether there exist reasonable darts that are not selfdecomposable remains an open question.

It turns out that when the payoff function is cosine, then reasonableness can be characterised in terms of the characteristic function of the dart, from which many different results follow. Furthermore, by studying functions of the form e^{cx}cos(omega x), we have found that no dart with compact support is reasonable.

We have also found several sufficient conditions for a dart to be non-reasonable with respect to some continuous payoff function, one of which is the following. If a dart has a point mass, but is not a constant, then it is non-reasonable with respect to some continuous function.

Finally, we have investigated under what types of operations a set of reasonable darts may be closed, and have found two results of this nature. Firstly any independent sum of reasonable darts is reasonable. Secondly, for any f in C_0(R^n), the set of darts which are reasonable with respect to f is closed with respect to convergence in distribution.