A numbers-on-foreheads game

TL;DR

This paper constructs a joint distribution of increasing natural number variables with nearly identical distributions over subsets, requiring extremely large values, impacting the implementation of extensive form games in game theory.

Contribution

It demonstrates the existence of such distributions with values growing as a power tower, enabling approximate implementation of complex games.

Findings

Variables can be distributed with nearly identical subset distributions.

Values of variables grow as a tower of exponents related to 1/ε.

Approximate game implementations are possible despite timing constraints.

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.