 The source of the trouble is that possibilities can be subdivided into further subpossibilities. The possibility of Athena losing can be subdivided into two subpossibilities, one where Beatrice wins and another where Cecil wins. Because we lack any relevant evidence, the available evidence doesn’t seem to favor the coarser possibilities over the finer subpossibilities, leading to contradictory probability assignments. What we need, it seems, is some way of choosing a single, privileged way of dividing up the space of possibilities so that we can apply the PoI consistently.