The random action of the host choosing to open one of the two loosing boxes is not distinguishable by the player: the resulting states are always part of the sa

The random action of the host choosing to open one of the two loosing boxes is not distinguishable by the player: the resulting states are always part of the same information-set in game-theory parlance. As player actions are always only depending on information-sets, the table actually contains information-sets as rows. It only looks weird because most of the information-sets of the game assign deterministic values to all the variables.