Where to stand when playing darts?
Bj\"orn G. Franzen, Jeffrey E. Steif, Johan W\"astlund

TL;DR
This paper investigates the optimal standing distance in darts, analyzing how the distribution of the dart's landing point affects whether standing closer always yields better expected outcomes, with results depending on distribution properties.
Contribution
It characterizes conditions under which standing closer is always optimal, focusing on selfdecomposable distributions and the behavior of characteristic functions.
Findings
Selfdecomposable distributions always favor standing closer.
For cosine payoff, decreasing characteristic function implies closer is better.
Distributions with point masses or compact support can favor standing farther.
Abstract
This paper analyzes the question of where one should stand when playing darts. If one stands at distance and aims at , then the dart (modelled by a random vector in ) hits a random point given by . Next, given a payoff function , one considers and asks if this is decreasing in ; 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 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 , then it is always better to stand closer if and only if the…
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