First: the definition of convexity is the exact opposite of what other sources call convexity. What you call convexity they call concavity and vice versa. You might want to flip that before other people are confused.
Second: your use of utility functions is wrong. The problem as stated gives a utility for the two balls to match or not match. That’s a utility function over a three-dimensional space over all lotteries between (R,R), (R,B), (B,R), (B,B). And it’s symmetric. You literally don’t care what the first ball is, as long as the second ball matches it. The important thing is to get knowledge of the first ball, which restricts the problem to one edge of the feasible tetrahedron. You can’t reduce that to a utility function over lotteries over the first ball.
Third: the demonstration is unconvincing. You claim that there are no Dutch books for any utility function over lotteries, convex or concave. You construct something that is not a Dutch book for one particular problem. Not only is this not a proof (as you say) but it doesn’t even give any reason to believe the proposition. The fact that you can construct one point in a very large space which is in X, does not give any evidence for the proposition “all points are X”. At least to this mathematician.
First: the definition of convexity is the exact opposite of what other sources call convexity. What you call convexity they call concavity and vice versa. You might want to flip that before other people are confused.
Second: your use of utility functions is wrong. The problem as stated gives a utility for the two balls to match or not match. That’s a utility function over a three-dimensional space over all lotteries between (R,R), (R,B), (B,R), (B,B). And it’s symmetric. You literally don’t care what the first ball is, as long as the second ball matches it. The important thing is to get knowledge of the first ball, which restricts the problem to one edge of the feasible tetrahedron. You can’t reduce that to a utility function over lotteries over the first ball.
Third: the demonstration is unconvincing. You claim that there are no Dutch books for any utility function over lotteries, convex or concave. You construct something that is not a Dutch book for one particular problem. Not only is this not a proof (as you say) but it doesn’t even give any reason to believe the proposition. The fact that you can construct one point in a very large space which is in X, does not give any evidence for the proposition “all points are X”. At least to this mathematician.