A numbers-on-foreheads game

Abstract: Is there a joint distribution of $n$ random variables over the natural numbers, such that they always form an increasing sequence and whenever you take two subsets of the set of random variables of the same cardinality, their distribution is almost the same? We show that the answer is yes, but that the random variables will have to take values as large as $2^{2^{\dots ^{2^{\Theta\left(\frac{1}{\epsilon}\right)}}}}$, where $\epsilon\leq \epsilon_n$ measures how different the two distributions can be, the tower contains $n-2$ $2$'s and the constants in the $\Theta$ notation are allowed to depend on $n$. This result has an important consequence in game theory: It shows that even though you can define extensive form games that cannot be implemented on players who can tell the time, you can have implementations that approximate the game arbitrarily well.