Why is the Variance of Multi-line Play Less than the Variance of Single-Line Play?

I’ve known for years that Triple Play reduces the variance of video poker. And Five Play, Ten Play, Fifty Play, and Hundred Play even more so. When most of us had the software Video Poker for Winners, the variance for Triple Play and Five Play and Ten Play were listed. So, this information was readily available. Unfortunately, that software doesn’t work on modern machines, and it’s not going to be updated. So most of us no longer have access to this source of information.

Saying dollar Ten Play has a lower variance than the single-line version of the same game is simply not correct. The Ten Play version, at $50 per play, has a considerably higher variance than the single-line version at $5 per play. But variance-per-coin-bet is a different story. Dollar Ten Play and $10 single play both require $50 to play. Between these two games, the Ten Play version has a much lower variance.

I’ve had some theories as to why this is true, but my major certainty on this matter came from looking at the software, which I trusted, rather than understanding the actual math itself.

Recently, however, I attended a lecture by Dr. Stewart N. Ethier, Professor Emeritus at the department of mathematics at the University of Utah, which addressed this subject. Although Ethier’s presentation was far more mathematical than I can usefully present to a general audience, I understand his point well enough so I can simplify it for my readers. If you wish to see the original paper, which he released simultaneously with the lecture, you may view it at //arxiv.org/abs/2409.03607.

The key way of looking at it that Ethier presented is one that seems obvious once it’s pointed out, but I, for one, never noticed it. The variance of a video poker hand is the sum of two components; namely the variance of the deal and the variance of the draw, which I will refer to as var (total) = var (deal) + var (draw).

So far, so good, Captain Obvious. What does this buy us?

What it buys us is that the variance of the draw is MUCH larger than the variance of the deal. For example, the variance of 9/6 Jacks or Better is about 19.5, but this is the sum of the variance of the deal (about 2) and the variance of the draw (about 17.5). One of the major components of the variance is the frequency of hitting the royal flush, which is about 1/40,000 for the draw versus 1/650,000 for the deal.

The variance for Triple Play is the variance of the deal, plus the variance of the draw, divided by 3. For Five Play we divide by 5. For Hundred Play we divide by 100. Notice we’re dividing the larger of the components of the variance.

Although Ethier provides quite a few decimal places in his paper (which you can look at if you wish), these are the variance numbers for 9/6 Jacks or Better

Single Line – 19.5

Triple Play – 7.82

Five Play – 5.48

Ten Play – 3.72

Fifty Play – 2.32

Hundred Play – 2.15

The primary reason for preferring lower variance is the preservation of bankroll. That is, playing multi-line versions of the same game reduces your risk of ruin — i.e., your chance of going broke. It’s frequently not possible to have a choice between multi-line games and single-line games with the same pay schedule. But when it is possible, the safer play is to go with the multi-line game.

“Safer” isn’t everyone’s choice. Higher variance can be called “more fun,” and a number of players gamble for fun. I gamble for longevity and profit, and bankroll preservation is a primary concern.