Meaning of tilde-greater-than (~>) in version requirement?

The RubyGems manual calls this a
pessimistic version constraint.

Assume you have specified an n-part version number, e.g. 1.3 (2-part) or
3.5.6.2 (4-part) as the constraint. Then, in order to fulfill the constraint,
a version number must satisfy both of the following conditions

1. The first n-1 parts of the version number must be identical to the first n-1
   parts of the constraint
   (e.g. 1.x or 3.5.6.x match, but 0.x or 3.5.7.x don’t) and

2. The last part of the version number must be greater than or equal to the last
   part of the constraint
   (e.g. 1.9999 and 3.5.6.2 match, but 1.2 or 3.5.6.1 don’t).

In other words

~> x1.x2.x3. … .xn-2.xn-1.xn

matches

x1.x2.x3. … .xn-2.xn-1.y, y >= xn

The reason this is called a “pessimistic” constraint, and also the use case for
it, is that when you just say > x.y.z, you are being optimistic: you assume
that from here on out, until all eternity, the API will never ever change. This
is of course a pretty bold assumption. However, most projects have rules about
when they are allowed to
break backwards compatibility,
and how they have to change their version number when they do break backwards
compatibility. You can encode those version numbering rules using a pessimistic
constraint, and thus you can be sure that your code will always continue to work
(assuming that the author of the other project actually adheres to his own
rules, which unfortunately isn’t always the case).