Variance reduction in Texas hold'em and in video poker

Abstract

In Texas hold'em, after an all-in bet is made and called before the flop, the turn, or the river, the two players sometimes agree to run it $n$ times, meaning that the remaining five, two, or one cards are dealt out not just once but $n$ times successively without replacement, with $1/n$ of the pot attached to each run. In $n$-play video poker, five cards are dealt exactly as in the conventional single-play game. After the player chooses which cards to hold, new cards are drawn to replace the discards, not just once but $n$ times independently, with $1/n$ of the bet attached to each draw. In both scenarios the players are attempting to reduce the variance of the return without changing the mean. We quantify the extent to which the variance is reduced.