@Christopher2222 

Any progress on this?

Thank you all for the tips! They are very helpful.

I would be nice, of course, if the Quantile function just worked - in the next version?

@Kitonum 

Thank you! And I guess the label of the x-axis could be moved anywhere with textplot.

Does this mean there is no "automatic" way?

Dear Jed,
Thank you for the answer.

The problem is that it does not work for other s. For example, for s=0.1 it gives

Warning, no iterations performed as initial point satisfies first-order conditions
                  

even though the correct answer is b=0.

I guess the difficulty comes from the fact that the optimal value of b jumps from 0 to 1 at s=0.18. For this reason I had the option "" so that Maximize looks for the global solution.

Dear Jed,
Thank you for the answer.

The problem is that it does not work for other s. For example, for s=0.1 it gives

Warning, no iterations performed as initial point satisfies first-order conditions
                  

even though the correct answer is b=0.

I guess the difficulty comes from the fact that the optimal value of b jumps from 0 to 1 at s=0.18. For this reason I had the option "" so that Maximize looks for the global solution.

Sorry, pagan, my example should have been {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}, as hirnuk says.

And yes, there is no partition {{1,2},{3,4}} of {1,2,3,4}. Actually, when I run G({1,2,3,4,5}); I get this:
{{{1, 2, 3, 4, 5}}, {1, {2, 3, 4, 5}}, {2, {1, 3, 4, 5}}, {3, {1, 2, 4, 5}},

  {4, {1, 2, 3, 5}}, {5, {1, 2, 3, 4}}, {1, 2, {3, 4, 5}}, {1, 3, {2, 4, 5}},

  {1, 4, {2, 3, 5}}, {1, 5, {2, 3, 4}}, {2, 3, {1, 4, 5}}, {2, 4, {1, 3, 5}},

  {2, 5, {1, 3, 4}}, {3, 4, {1, 2, 5}}, {3, 5, {1, 2, 4}}, {4, 5, {1, 2, 3}},

  {1, 2, 3, {4, 5}}, {1, 2, 4, {3, 5}}, {1, 2, 5, {3, 4}}, {1, 3, 4, {2, 5}},

  {1, 3, 5, {2, 4}}, {1, 4, 5, {2, 3}}, {2, 3, 4, {1, 5}}, {2, 3, 5, {1, 4}},

  {2, 4, 5, {1, 3}}, {3, 4, 5, {1, 2}}, {1, 2, 3, 4, {5}}, {1, 2, 3, 5, {4}},

  {1, 2, 4, 5, {3}}, {1, 3, 4, 5, {2}}, {2, 3, 4, 5, {1}}, {1, 2, 3, 4, 5, {}}

  }
I don't see any partition involving two sets of two and three elements like {{1,2},{3,4,5}} or two, two and element like {{1,2},{3,4},5}. The problem seems to be that in any generated partition only one set (at most) has more than one element. The same if the set has 6 elements, etc.

Sorry, pagan, my example should have been {{1},{2},{3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2,3}}, as hirnuk says.

And yes, there is no partition {{1,2},{3,4}} of {1,2,3,4}. Actually, when I run G({1,2,3,4,5}); I get this:
{{{1, 2, 3, 4, 5}}, {1, {2, 3, 4, 5}}, {2, {1, 3, 4, 5}}, {3, {1, 2, 4, 5}},

  {4, {1, 2, 3, 5}}, {5, {1, 2, 3, 4}}, {1, 2, {3, 4, 5}}, {1, 3, {2, 4, 5}},

  {1, 4, {2, 3, 5}}, {1, 5, {2, 3, 4}}, {2, 3, {1, 4, 5}}, {2, 4, {1, 3, 5}},

  {2, 5, {1, 3, 4}}, {3, 4, {1, 2, 5}}, {3, 5, {1, 2, 4}}, {4, 5, {1, 2, 3}},

  {1, 2, 3, {4, 5}}, {1, 2, 4, {3, 5}}, {1, 2, 5, {3, 4}}, {1, 3, 4, {2, 5}},

  {1, 3, 5, {2, 4}}, {1, 4, 5, {2, 3}}, {2, 3, 4, {1, 5}}, {2, 3, 5, {1, 4}},

  {2, 4, 5, {1, 3}}, {3, 4, 5, {1, 2}}, {1, 2, 3, 4, {5}}, {1, 2, 3, 5, {4}},

  {1, 2, 4, 5, {3}}, {1, 3, 4, 5, {2}}, {2, 3, 4, 5, {1}}, {1, 2, 3, 4, 5, {}}

  }
I don't see any partition involving two sets of two and three elements like {{1,2},{3,4,5}} or two, two and element like {{1,2},{3,4},5}. The problem seems to be that in any generated partition only one set (at most) has more than one element. The same if the set has 6 elements, etc.