Poorly Explained Solution
The solution for this problem is very poorly explained. You need to apply geometry and a game theory mindset to get this problem.
What you need to do here is draw graphs of the 4 equations you have here. The yaxis would be the total each could possibly have (max is 2) and the xaxis would be the value f could possibly have (max is 1).
M1 = f
J2 = 2f
M2 = 3/2  f
J1 = f + 1/2
M1 = Marie if she went first, J2 = Jeremy if he went second
M2 = Marie if she went second, J1 = Jeremy if he went first
Understand the graphs. Marie wants more than Jeremy and Jeremy wants more than Marie. Both want to maximize their respective shares as well. Look at the ranges in the graphs where this makes sense. You'll need to get the intersection points of the 4 lines with each other and understand the semantics of the ranges.
You'll find 3 values for f = 1/4, 3/4 and 1. Plug in these values into the 4 equations. Each f value for the 4 equations constitutes a set. Realize that the only control that Jeremy has here is the the fraction f he cuts the first cake in. And the only control that Marie has here is whether or not to go first.
Looking at each set of values, apply the logic that Jeremy wants to maximize his share so determine which f value he would want to play, and that Marie also wants to maximize her share so determine in which cases she would want to play first and which second.
You will soon see that the only scenario is the f=3/4 case. And in this case, it also so happens that Marie would get 3/4 total share no matter if she goes first or second (and that Jeremy would get 1 1/4).
