It seems to me that your argument relies on the utility of having a probability p of gaining x being equal to p times the utility of gaining x. It’s not clear to me that this should be true.
The trouble with the “money pump” argument is that the choice one makes may well depend on how one got into the situation of having the choice in the first place. For example, let’s assume someone prefer 2B over 2A. It could be that if he were offered choice 1 “out of the blue” he would prefer 1A over 1B, yet if it were announced in advance that he would have a 2⁄3 chance of getting nothing and a 1⁄3 chance of being offered choice 1, he would decide beforehand that B is the better choice, and he would stick with that choice even if allowed to switch. This may seem odd, but I don’t see why it’s logically inconsistent.
It seems to me that your argument relies on the utility of having a probability p of gaining x being equal to p times the utility of gaining x. It’s not clear to me that this should be true.
The trouble with the “money pump” argument is that the choice one makes may well depend on how one got into the situation of having the choice in the first place. For example, let’s assume someone prefer 2B over 2A. It could be that if he were offered choice 1 “out of the blue” he would prefer 1A over 1B, yet if it were announced in advance that he would have a 2⁄3 chance of getting nothing and a 1⁄3 chance of being offered choice 1, he would decide beforehand that B is the better choice, and he would stick with that choice even if allowed to switch. This may seem odd, but I don’t see why it’s logically inconsistent.