Why is the maximum sum of two proportions = 1 with the multinomial logit

Question

Suppose I have two numbers, actually two proportions, a and b, where:

a = e^x / (1 + e^x + e^y)

b = e^y / (1 + e^x + e^y)

I know that if x and y are very small the lower limit of a and b is 0. I also know that if x and y are both very big a and b both approach 0.50.

However, it seems if a = 0.80 then the upper limit of b = 0.20. More generally, a + b <= 1 and the upper limit of b is 1 - a. Why is that?

Thank you for any advice in understanding why a + b must be <= 1 and why the upper limit of b is 1 - a. Sorry if this is a duplicate.

Answer 1

Notice that a + b = 1 - 1/(1+e^x + e^y) < 1

and the sum will approach 1 (equivalently, 1/(1+e^x + e^y) will approach 0)

as x and y approach infinity

Answer 2

Let $c = \frac{1}{1+e^x + e^y}$, which is a positive real number.

Then $a+b+c=1$, and any positive solution of $a+b+c=1$ can be attained by some choice of $x$ and $y$. By taking $c$ to be small one gets $b$ as close as desired to $1-a$; and certainly $a+b \leq 1$.

The exponentials $e^x$ and $e^y$ are just a way of saying "a pair of positive numbers".