Improve the approximation guarantee for DORG

Derive an approximation guarantee for the descending-order reward greedy (DORG) algorithm that is strictly better than the established $1/\sqrt{e}$ bound by exploiting the fact that DORG processes players with activation probabilities greater than $1/2$ in decreasing order.

Background

The paper proves that DORG, which processes players in decreasing order of their activation probabilities, achieves a $1/\sqrt{e}$-approximation for the heterogeneous asynchronous multi-player bandit allocation problem. The proof relies particularly on DORG’s treatment of players whose activation probabilities exceed $1/2; for players below this threshold, the analysis applies the general greedy guarantee.

The authors observe that the $1/\sqrt{e}$$guarantee holds for any greedy procedure that processes players with activation probabilities above$$1/2$ in decreasing order, even if the remaining players are processed arbitrarily. They conjecture that incorporating this additional structural property more fully into the analysis could yield a stronger approximation ratio for DORG.