Even if we just restrict to a tiny, non-zero area, around X7, we get arbitrarily high OP—it’s not a fluke or a calculation error. Which means that if we followed the distributive law, Q=(1-10-1000) X0 + 10-1000X7 must have a much larger OP than X6 - despite the fact that nearly every possible outcome is better than Q.
It’s not weird that the discrete probability distribution X_7 has infinite OP, because in the real world we never get absolute certainty.
And anyway, that’s obviously not how probabilities on the simplex are supposed to work, right? Just define a probability density on the simplex of probabilities (for example, uniform on the simplex), and for any given p in the simplex, its OP(p) is the reciprocal of the integral (using that measure) of the set {q | EV(q)>= EV(p)} (we do need utility for this, not only a preference ordering). Drawing a simplex for U(X_n)=n shows why we wouldn’t even naturally expect distributivity to hold.
It’s not weird that the discrete probability distribution X_7 has infinite OP, because in the real world we never get absolute certainty.
And anyway, that’s obviously not how probabilities on the simplex are supposed to work, right? Just define a probability density on the simplex of probabilities (for example, uniform on the simplex), and for any given p in the simplex, its OP(p) is the reciprocal of the integral (using that measure) of the set {q | EV(q)>= EV(p)} (we do need utility for this, not only a preference ordering). Drawing a simplex for U(X_n)=n shows why we wouldn’t even naturally expect distributivity to hold.