Let $n$ probabilities be given, $p_1,...,p_n$, $\sum p_i = 1$, $p_i \in (0,1)$. In period 1, $p_1 = \ ...\ = p_n$.
In each subsequent period, i want to decrease some of the probabilities, and increase others. If the probability is already high, I want an increase to be relatively small, and a decrease to be relatively high. Similarly, if the probability is low, a decrease should be small, but an increase should be bigger. In each period I already know which ones I want to see go up, and which ones I want to see go down, all I need is the formula with which to do it.
Does anybody know of such formula which satisfies that the marginal increase [decrease] is diminishing [rising] in the value of the probability [alternatively, make the size of this property depend on a parameter $\beta$, where for, say, higher $\beta$ means that there's a large difference in increases and decreases, while for, say, the lowest possible value of $\beta$ there is NO difference between increases and decreases, no matter how high the probability is], and which of course also satisfies that the probabilities remain probabilities that sum to 1?
I know it's rude to just ask for an answer, pero esto no es homeworko.
A proposal: You could replace each $p_i$ by $$p'_i=p_i+0.1*(rand-p_i)$$ where rand is a classical uniform semi random generator on [0,1].
Of course, you will have to normalize again at the end (i.e., divide every $p'_i$ by the sum of $p'_i$s.)
Why that ? Because $P((rand-p_i)<0)=p_i$ and, thus $P((rand-p_i)>0)=1-p_i$.
Thus, if $p_i$ is small, the probability will be much more likely to be positive, and vice versa. Coefficient $0.1$ is arbitrary and could be modified.
There is no risk that any $p'_i$ is not in $[0,1]$ as can be checked by taking extreme values for rand.