Paradox of Random Natural Numbers

Question

I've got a question about a game taken from a book called Rachunek prawdopodobieństwa dla (prawie) każdego by Jacek Jakubowski and Rafał Sztencel.

Adam and Bolek have a machine that generates a pair of random natural numbers that are next to each other (like $1$ and $2$ or $55$ and $54$) and each boy gets a different number from the pair. They know the number of their opponent but they don't know their own numbers.

Here are the rules:

• The winner is the one with greatest number.
• The loser must give the winner as many coins as great is the loser's number (so if the loser has number $40$, he has to give 40 coins to the winner).
• If a player doesn't like his number he can decide to get a new pair of random numbers.

Both players will never want to generate a new pair, because they think like this: There is a $50\%$ chance that I am the winner. And if the number of my opponent is $k$ then I'll win $k$ coins or lose $k-1$ coins. So my overall score would be $\frac{k - (k-1)}{2} = 0.5$ of a coin.

This strategy is of course based on wrong assumptions - the set of numbers that can be generated by the machine cannot be infinite. Additionally, it can't be that both players are winners of the game.

My questions are:

• What would be their strategy if they knew that the set is finite?

• Is their strategy truly wrong even if the set is infinite? If their strategy turns out to be a winning one, how could it be that both of them are going to be the winners of the game?
  Apparently, this is not true because:

  "One can define a uniform distribution on some infinite sets (uniform on $[0,1][0,1]$ is totally legit). But one cannot define a uniform distribution on an infinite yet countable set, such as the natural numbers."
  ~ Clement C.

  Here is an additional reference with a good explaination.

Answer 1

In the original question, the lack of a uniform distribution on the naturals saves you. The probability of all the numbers has to sum to $1$, so the probability has to decrease with $n$ eventually.

If you limit the number to the range $[1,N]$, a player who sees $N$ will certainly want to change numbers as he knows he is losing. If either player can force a change, a player who sees $N-1$ should request a change, because either his opponent sees $N$ and will also request a change or his opponent sees $N-2$ and is winning. Then a player who sees $N-2 \dots$