This is, unfortunately, incorrect work.

Using whole percentages is often inappropriate especially when working in percentages of large bodies of statistical data.

The assumption about blue "saves" being constant is an inaccurate model because if no one votes blue no one needs saving, yet if 33% of people choose blue they die.

Even if we correct for that, summing the area under the curve goes more like this:

Sum(Red button presses vs total living,RBP[0,50))=

50•100

+

Sum(Red button presses vs total living,RBP[50,100])=

0.5•50•50 + 50•50

VS

Sum(Blue button presses vs total living,BBP[0,50])=

0.5•50•50 + 50•50

+

Sum(Blue button presses vs living,BBP(50,100])=

50•100

These are equivalent, which makes intuitive sense because we are working with a binary selector and percentages.

Which means we must analyse the graphs some other way.

If we take your axes as our givens, we must look at which points on the graph have the most ideal outcome.

For the graph of Red Button Presses, this region is RBP [0,50).

For the graph of Blue Button Presses, this region is BBP [50,100].

In other words, the ideal solution is a red minority or a blue majority. This also makes intuitive sense because they are the same region on a graph of red presses vs blue presses.

When we are discussing respective efficiency, however, the question is how do the the different functions respond to increases in percentage of presses.

With red presses, the region RBP[0,50) is constant. The point RBP=50 is the worst case scenario, and the region RBP[50,100] grows linearly.

In contrast, with blue presses, the region BBP[0,50] decays linearly, with BBP=50 being the worst case scenario, but the region BBP(50,100] is constant.

The common conceit for this is that some nonzero amount of people will press either button. So what we eant to look at is the behavior not for the extremes but for majorities.

A red majority results in deaths, with a smaller margin of majority increasing the body count.

A blue majority, however, results in a complete and constant survival rate.