# Means To No End

The arithmetic mean is perhaps one of the oldest mathematical ideas, intimately tied to fairness in mutual endeavor. The arithmetic mean answers the question “what payout would we have gotten if we all got the same payout”, or in other words, which makes
p_1 + \cdots + p_n = A + \cdots + A$
true? Most people are familiar with the arithmetic mean, and a few math nerds might know about the geometric or harmonic means. But what is a *mean*, in general?