Suppose we have a full Hilberts hotel … actually, lets make it a “Hilbert world” instead. This world consists of one building that have infinitely many rooms, numbered . In each room lives one person, and he is going to live there in his entire infinite life. In the beginning each person have 1$, but then they figure out a clever way to get richer. Each minute the person in room gives all his money to the person in room . So after one minute they will each have 2$ and after minutes they will each have dollars. (This is similar to the Banach-Tarski paradox, where you can “use infinity” to turn one ball into two balls). Of course, money would lose their value in such a world, but the inhabitants of such a world could also get an infinitely amount of utility by sending food, cake, bubble wrap, whatever, instead of sending money.
However, this is a very simplified model an infinite world. Presumably, the infinitely many rooms would be in . So let’s say there is a room for each point in , and that you can only send money to one of your 26 neighbors and it takes one minute to send the money. Then is it no longer possible for your wealth to grow exponentially fast. Even if everyone cooperated and tried to make the person in rich, he could have at most dollars after minutes. This is only polynomial, but still pretty good for the person in (0,0,0). But this protocol is unfair: we want everyone to earn at least some amount during the first minutes. However, we can easily see that the persons in rooms in will in total have at most dollars after minutes, and goes to as goes to infinity. This shows that we cannot use the infinite world to make everyone richer by an epsilon in finite time. (If you know about amenability, you may notice that the sequence of sets is a Følner sequence, so the comparison to the Banach-Tarski paradox goes deeper than just “doubling something using some infinite trick that is not possible in the real world”).
So, let’s say you are a God and you want to construct a world, with the following restrictions:
You first build the world and afterwards you can’t change anything.
Each person starts in a room (and doesn’t move) each person has a finite number of neighbors and he can send any amount of information and money to his neighbor. Each transaction takes one minute.
Each person starts with 1$ and all he care about is money.
Everyone cooperates as long as he knows that he will get rich by doing so (even if he could become rich faster by cheating).
When the world starts, you can explain a protocol to everyone in the world using loudspeakers (everyone will get the same message).
Your objective is to make everyone’s wealth grow exponentially fast.
One way to do this is to give the world the structure of the Cayley graph of the free group with two generators (if all the rooms have the same positive volume and the distance between neighboring rooms has to be bounded it is not possible to embed this in , but you are God, so you don’t care about ). Now each person has 4 neighbors. E.g. has , , , and and has and . The protocol is that everyone except should each minute send all his money to the neighbor that is closer to . This way everyone (besides ) will receive money from 3 persons, so at time they will have dollars. The person at will receive money from 4 person and won’t pay to anyone, so he will become rich even faster.
So you are a God, you have just created this fantastic world with infinitely many people and you are just about to announce your strategy for how everyone can become rich in no time without working, when you discover that you have made a terrible mistake: you forgot to label the doors. In fact, there is no way to distinguish any two rooms, so now the inhabitants of your world can’t know what direction they should send their money.
If only you could tell one person that he was a chosen one (=), he could then tell his neighbors and they would give him money each minute. They would then tell their neighbors, and so on (remember, no one would lie, since they know they will get rich even if they are honest). This way it would take some time for people far away from the chosen one before they starts getting richer, but when they do, their wealth will grow exponentially.
Unfortunately, you have no way to communicate to only one person: you can only use the loudspeakers. You did, however, remember to give them a dice each. Is there a protocol that would make everyone’s wealth grow exponentially with probability 1?
Edit: I asked this question here on mathoverflow . The answer is no. It turns out to be a special case of the “Mass Transport Principle”. It can be found on page 283 in this book which is available online (page 293 in the pdf-file).