Split circle in pairs parts of equal area

Birthday cake: Two fiends A, B are going to share a circular cake, as follows: They alternate turns doing a straight cut each, starting by A and doing 4 cuts in total. Then they take turns picking one piece at a time, again starting from A. Can you find a way so that B picks the pieces with minimum half the area of the cake (in total)?

My idea would be for B to imitate A's cuts, so as to create pieces of equal area in pairs. For example, I am thinking: 1st cut: any random chord, which splits the circle into two parts of area P and Q. Then B draws the vertical bisector of the chord, so he splits each part into two smaller parts of area P1=P2 and Q1=Q2. Then I guess we draw some other chord by a 3rd cut and the 4th cut must somehow split all the smaller pieces into two parts each. Is it possible? Then I guess it is easy for B to pick exactly the same parts that A picks.

Why not cut with 4 diameters, at angles $0, \pi/4, \pi/2, 3\pi/4$? That gives you $8$ (which is even) identical slices of cake.