Where to stand when playing darts?
Preprint, 2020

This paper analyzes the question of where one should stand when playing darts.
If one stands at distance d>0 and aims at a in R^n, then the dart (modelled by a random 
vector X in R^n hits a random point given by a+dX. Next, given a payoff function f, one considers
sup_a Ef(a+dX) and asks if this is decreasing in d;  i.e., whether it is better to stand closer rather
than farther from the target.  Perhaps surprisingly, this is not always the case and understanding 
when this does or does not occur is the purpose of this paper.

We show that if X has a so-called selfdecomposable distribution, then it is always better
to stand closer for any payoff function. This class includes all stable distributions as well as many more.

On the other hand, if the payoff function is cos(x), then it is always better to stand closer
if and only if the characteristic function |phi_X(t)| is decreasing on [0,infty). We will then show that if there
are at least two point masses, then it is not always better to stand closer using cos(x). If there is a single
point mass, one can find a different payoff function to obtain this phenomenon.

Another large class of darts X for which there are bounded continuous payoff functions for which
it is not always better to stand closer are distributions with compact support. This will be obtained
by using the fact that the Fourier transform of such distributions has a zero in the complex plane.
This argument will work whenever there is a complex zero of the Fourier transform.

Finally, we analyze if the property of it being better to stand closer is closed under convolution and/or limits.

Fourier transforms

selfdecomposable distributions

darts

Author

Bjorn Franzen

Jeffrey Steif

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Johan Wästlund

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Färgning av slumpmässiga ekvivalensrelationer, slumpvandringar på dynamisk perkolation och bruskänslighet för gränsgrafen i den Erdös-Renyi-slumpgrafsmodellen

Swedish Research Council (VR), 2017-01-01 -- 2020-12-31.

Subject Categories

Mathematics

Probability Theory and Statistics

Mathematical Analysis

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4/7/2021 1