With What Probability Should a Change be Selected?

A change is a modification of the plan. Suppose we have $n$ candidate changes $c_1, c_2, ..., c_n$. Each change $c_i$ has an associated measure of distance $d_i$ from the ideal fit.

The question is to design a function $f(x)$ such that the selection of change $c_i$ with probability weight $f(d_i)$ is a "good selection strategy" [1]. An example of a "not-so-good strategy" is $f(x) = 1/x$ -- if  $n$ is large and there are several "poor" $c_i$s each with large $d_i$, the cumulative sum of $f(d_i)$ corresponding to these poor $c_i$s is still large. This means that there is a large probability that a poor change will be selected. 

$f(x) = 1/x^a$, $a = 3$ for example, is expected to perform better. Another candidate is $f(x) = 1/(1+a)^x$, $a = 0.25$ for example. [2] Empirically the last $f(x)$ seems to perform well in practice. Still better we would like that when some $d_i = 0$ then one of such $c_i$ is definitely selected. For this we can modify $f(x)$ to be $1/(x*(1+a)^x)$.

However more work is need to precisely discover the invariants of the good selection strategy.

[1] Due to stochastic nature of selection and the complexity of changes it is difficult to define "good selection strategy". One way to define it is to state specific examples, and expected weight ranges from the $c_i$. More work is needed to define it.

[2] The appropriate value of a depends on the scaling of distances and how greedy we want the selection to be.