I'm trying to understand why the softmax function is defined as such:

$\frac{e^{z_{j}}} {\Sigma^{K}_{k=1}{e^{z_{k}}}} = \sigma(z)$

I understand how this normalizes the data and properly maps to some range (0, 1) but the different between weight probabilities varies exponentially rather than linearly. Is there a reason why we want this behaviour?

Also this equation seems rather arbitrary and I feel that it a large family of equations could satisfy our requirements. I have not seen any derivations online so I'm assuming it is merely a definition. Why not choose any other definition that satisfies the same requirements?