The fundamental theorem of video poker

As so often happens at the Palms, I’d ended up sitting next to a professional video poker player.  You can always spot them by that slapping action on the buttons, which is something that only comes with hours of training.

I’ve tried it, and it makes me look great for a few minutes but my arms get tired really quickly.  I’m such a noob.

Thanks to the slot card reader, I knew this particular chap was called Rick and that he had over 100,000 points on his card.  I’ve seen many players with more than that, but even so he has probably achieved the the lofty status of never having to pay to eat again.

Once you achieve the "MVP" tier of the Palms’ slot club (50,000 points earned in 90 days) you get half off at the buffet when you redeem points to eat, and if you make the "Hall of Famer" level (200,000 points in 90 days) you get a 75% discount.  So not only do you have gazillions of points to redeem, you just can’t spend them fast enough.

You can only eat so many $4.25 champagne brunches.

You apparently get other shit like movie passes and free car washes at those levels too.  Obviously, it’s part of my summer plan to achieve this status.  Who doesn’t love a free car wash?

Anyway, Rick was having a bad day on Deuces Wild.  He told anyone who cared to listen that he needed to take a break as he just made three mistakes in a row.

As he got up and wandered off, I glanced across at his screen and saw: 8h 7h 4h 5c 4s.  He’d held the first four cards, with the last 4 making a worthless low pair after the draw.

I know this one now without looking it up.  When you have a four card straight draw with a gap but three of those cards are the same suit, you should always hold for the long shot straight flush above the gutshot straight.  This holds true even if there are two gaps in the straight flush draw (as in this case: 8 7 4).

He’d done it wrong, but at least he’d realised and taking a break was a smart move.

Perhaps he could have taken a longer break than four minutes, but I guess it doesn’t take long to go to the ATM and fetch a coffee, and any break much longer than that is really going to eat into your hourly rate.

He sat back at the same machine and the woman next to him asked if he liked that one in particular.  He explained, "I usually play on the other side, but it’s not been hitting so I came this side today".

Well, game selection is important.

He continued, "This week it’s been full of tourists so it’s hard for the regulars to get a read on it".

"Tourists" are clearly the "internet players" of the professional video poker circuit.

He went on to boast how he can usually tell when – as well as at which machine – the jackpot is going to pay out.  "It should go early today, not like $1300 or $1400" (in reference to the progressive jackpot, which resets to $1000 when a royal flush is hit, then increases with each game played until the next one).

I’m quite surprised he wasn’t wearing shades so the machines couldn’t soulscan him right back.

It wasn’t long before I was dealt a hand I needed to think about: three aces with a three card royal flush draw and no wild cards.  With a pair and a three-to-the-royal option, you always go for glory (in deuces wild, any pair has to improve to three-of-a-kind or better for a payout).  With trips already on screen, I wasn’t quite so sure what to do.

My strategy card said to hold the aces, but I still hesitated because of the progressive jackpot.  At that time it was set to pay $1122 for a royal flush instead of the usual $1000.  I just couldn’t work out if it was enough to make a difference.

Rick saw me scrunching my face up at the screen, then my cheat sheets, then starting to type the hand into my phone so I could check it later after I finally made a decision, and he jumped to my rescue.

"You have to go for it", he offered.  "Three to the royal beats pretty much everything".

Well, he’s not far off.  The only hands you would normally hold over a three card royal are a made straight, a made flush or an open-ended four card straight flush draw.  Oh yes, and three of a kind.

But after he said that, how could I not draw to the royal flush?  You just know that it’s going to draw the two cards you’d have needed (or at the very least one of them and a wild card) if you hold the aces and nothing would have put me on tilt for the rest of the trip more than seeing his "I told you so face".

I was never so glad to see rags: a meaningless 7 and 3 was about as emphatically not a jackpot as you can get, nor would it have improved my three aces to a bigger win.

This hand was actually not as close as I thought it would be.  I plugged the numbers into some video poker software and it calculated the ER of the three aces as 10.09 (that’s an expected return of just over ten coins for a five coin bet) and the royal flush draw came in at 6.54, if you disregard the progressive.  So, in normal circumstances, holding the three of a kind is worth over 50% more!

Pumping the jackpot up to $1122 in the simulation didn’t make any difference to the strategy.  The value of trips doesn’t change (because there’s no way to redraw to a royal flush) but the three card royal draw’s ER rose to 7.05.  It’s extra free money when you hit it, but it’s still nowhere near enough to throw away trips for.

In fact, the jackpot has to reach a massive $2000 before you would consider breaking the three aces (ER=10.09) for a three-card royal flush draw that contains an ace (ER=10.11), and even then it would be an extremely volatile play for a tiny edge.

With the jackpot that high the overall return of the machine becomes a whopping 102.70% (compared to 100.76% with no progressive) although I’m sure there must be other strategy adjustments you need to make to take full advantage of the monster jackpot.

In fact, Rick did tell me that when the meter gets to $1350, that’s when you throw away the wild card if you’re dealt a royal flush with one wild deuce.  That’s correct, although he’s probably not accounting for the fact that any jackpot of $1200 or more will trigger tax paperwork.

A $2000 progressive would happen only very rarely the Palms because the meter rises very slowly.  When it does, you can be sure that the bank of machines will be perma-occupied by pros until it hits, and then they’ll all go home.

And it was thinking about that kind of mass exodus – which you really do see after a brief moment of fake, bitter congratulations when a substantial progressive is hit – that got me thinking.  Rick’s bad advice not only cost me a little EV, but it also cost himself – and every other player sitting at that bank of machines!

When I make the correct play and hold the three aces, I will hit a royal flush on the draw exactly 0% of the time.  However, when I hold the three card royal draw, it is a possibility.  It’s a very slim possibility (the odds are 1080-1 against) but nevertheless it still happens more often than never.

The key thing is that when I do get lucky, not only do I win the jackpot but the progressive meter resets to $1000.

Let’s pretend I did hit it.  A few seconds ago, Rick was playing a game with a $1122 jackpot which had an expected return of 100.97%.  Now it suddenly became a standard 100.76% game.

A couple of tenths of one percent doesn’t sound like much, but even with that modest progressive his hourly win rate (before comps) was 28% bigger before the jackpot got hit.

If the progressive had crept up to about $1400, that would have been a 101.5% game until my jackpot just went and ruined it by cutting the player’s edge in half!

It’s easy to think that it’s completely irrelevant how well or how poorly the guy next to you plays at video poker because it’s a solo game.  In fact, when a progressive jackpot is involved, your edge can be affected directly by the decisions that those around you make.

Therefore a sharp video poker player would have absolutely encouraged me to hold the three-of-a-kind.  It makes no difference to him whether I’m making the optimal play on this hand, but it would make a difference if I made a donkey play that reset a jackpot which he still had a shot at winning.

And so, in anticipation of my inevitable 2+2 book deal, I proudly present the Fundamental Theorem of Video Poker.  You may call it the FTOVP if you wish.

"Whenever a player incorrectly holds a royal flush draw, they lose and everyone else playing that game also loses.  However, whenever a player incorrectly fails to hold a royal flush draw, they lose but everybody else gains".

OK, I admit it should really be called "an occasionally relevant theorem of multi-player progressive jackpot video poker", but it’s not quite as catchy is it?