Single Sample Prophet Inequality for Uniform Matroids of Rank 2

We study the prophet inequality when the gambler has an access only to a single sample from each distribution. Rubinstein, Wang and Weinberg showed that an optimal guarantee of 1/2 can be achieved when the underlying matroid has rank 1, i.e. in the case of a single choice. We show that this guarantee can be achieved also in the case of a uniform matroid of rank 2 by a deterministic mechanism, and we show that this is best possible among deterministic mechanisms. We also conjecture that a straightforward generalization of our policy achieves the guarantee of 1/2 for all uniform matroids.