I assume you mean that I assume P(money in Bi | buyer chooses Bi )=0.25? Yes, I assume this, although really I assume that the seller’s prediction is accurate with probability 0.75 and that she fills the boxes according to the specified procedure. From this, it then follows that P(money in Bi | buyer chooses Bi )=0.25.
Yes, you are right. Sorry.
Why would it be a logical contradiction? Do you think Newcomb’s problem also requires a logical contradiction?
Okay, it probably isn’t a contradiction, because the situation “Buyer writes his decision and it is common knowledge that an hour later Seller sneaks a peek into this decision (with probability 0.75) or into a random false decision (0.25). After that Seller places money according to the decision he saw.” seems similar enough and can probably be formalized into a model of this situation.
You might wonder why am I spouting a bunch of wrong things in an unsuccessful attempt to attack your paper. I do that because it looks really suspicious to me for the following reasons:
You don’t use language developed by logicians to avoid mistakes and paradoxes in similar situations.
Even for something written in more or less basic English, your paper doesn’t seem to be rigorous enough for the kinds of problems it tries to tackle. For example, you don’t specify what exactly is considered common knowledge, and that can probably be really important.
You result looks similar to something you will try to prove as a stepping stone to proving that this whole situation with boxes is impossible. “It follows that in this situation two perfectly rational agents with the same information would make different deterministic decisions. Thus we arrived at contradiction and this situation is impossible.” In your paper agents are rational in a different ways (I think), but it still looks similar enough for me to become suspicious.
So, while my previous attempts at finding error in your paper failed pathetically, I’m still suspicious, so I’ll give it another shot.
When you argue that Buyer should buy one of the boxes, you assume that Buyer knows the probabilities that Seller assigned to Buyer’s actions. Are those probabilities also a part of common knowledge? How is that possible? If you try to do the same in Newcomb’s problem, you will get something like “Omniscient predictor predicts that player will pick the box A (with probability 1); player knows about that; player is free to pick between A and both boxes”, which seem to be a paradox.
I’ve skimmed over the beginning of your paper, and I think there might be several problems with it.
I don’t see where it is explicitly stated, but I think information “seller’s prediction is accurate with probability 0,75” is supposed to be common knowledge. Is it even possible for a non-trivial probabilistic prediction to be a common knowledge? Like, not as in some real-life situation, but as in this condition not being logical contradiction? I am not a specialist on this subject, but it looks like a logical contradiction. And you can prove absolutely anything if your premise contains contradiction.
A minor nitpick compared to the previous one, but you don’t specify what you mean by “prediction is accurate with probability 0.75”. What kinds of mistakes does seller make? For example, if buyer is going to buy the B1, then with probability 0.75 the prediction will be “B1”. What about the 0.25? Will it be 0.125 for “none” and 0.125 for “B2”? Will it be 0.25 for “none” and 0 for “B2”? (And does buyer knows about that? What about seller knowing about buyer knowing...)
When you write “$1−P (money in Bi | buyer chooses Bi ) · $3 = $1 − 0.25 · $3 = $0.25.”, you assume that P(money in Bi | buyer chooses Bi )=0.75. That is, if buyer chooses the first box, seller can’t possibly think that buyer will choose none of the boxes. And the same for the case of buyer choosing the second box. You can easily fix it by writing “$1−P (money in Bi | buyer chooses Bi ) · $3 >= $1 − 0.25 · $3 = $0.25″ instead. It is possible that you make some other implicit assumptions about mistakes that seller can make, so you might want to check it.