A Generalized Cover's Problem

TL;DR

This paper generalizes Cover's problem by analyzing an adversarial permutation construction game with partial information, demonstrating that a permutation can be reconstructed with probability better than chance under certain visibility conditions.

Contribution

It extends Cover's original result from two elements to n elements, providing a new framework for permutation reconstruction with partial information.

Findings

Permutation reconstruction probability exceeds random chance

Visibility of certain elements improves reconstruction success

Generalizes Cover's problem to larger permutations

Abstract

Generalizing a problem posed by Cover, we propose an adversarial game in which a permutation is incrementally constructed in a setting of partial information. As in the secretary problem, this permutation is exposed in stages via the successive components of its Lehmer code. Extending Cover's result, which constitutes the case $n = 2$, we establish that a random permutation of $n$ adversarially constructed real numbers can be reconstructed with better-than-random probability, provided that certain among the numbers it permutes are made visible during the process.