Formula is here: $$ p(i)=\frac{e^\frac{f(i)}{T}}{\displaystyle \sum_j e^\frac{f(j)}{T}} $$
Prove:
1) Each $p(i)$ is a number between $0$ and $1$, no matter what the fitness is (positive or negative). This scheme does not require that fitness has to be positive.
2) The sum of all the $p(i)$'s is $1$, i.e. this is a probability distribution.
3) No matter what $T$ is:
- If two items have same fitness, they have same probability of being picked.
- If all fitnesses are the same, we pick random item.
4) No matter what the fitnesses are:
- As $T\to\infty$ we tend to pick random item.
- As $T\to0$ we tend to pick only the best item. That is, its probability is $1$, the probability of all others is $0$. If there are $m$ joint best items, we pick them with probability $1/m$, and all others with probability $0$.