Originally Posted by Yakk

Alternatively, you could be in a situation with multiple optimal points.

Imagine four cities:
A> In one, everyone has a gun. 1,000 people die from gunshots every day.
B> In another, nobody has a gun. Nobody dies from gunshots.
C> In the third, half the people have a gun. 500 people with guns die from gunshots. 1000 people without guns die from gunshots.
D> In the forth, 1% of the population has a gun. 30 people with guns die from gunshots, and 100 people without guns die from gunshots.

Notice that B is the safest city (no guns). But, everyone having a gun is safer than half the population having a gun. And, if half the population has a gun, buying a gun is the easiest way to reduce your death rate from guns.

If someone made guns harder to get, and you moved to situation D, you would have a 10 fold drop in gun deaths.

Now, this is a hypothetical situation, but it does demonstrate how you could have two different points that are locally optimal. Getting from one to the other can't be done easily, without lots of effort.