Hole-Carding Three Card Poker: Random Walks & ROR

Hole-carding Three Card Poker has been one of the mainstays of APs for over a decade. I discuss the details in this post. However, most APs have struggled to show the profit that the 3.48% edge should yield for them. There have been a number of theories put forward to explain these struggles.  Most notably, some APs argue that the shufflers are biased in how the cards in the packets of three cards are put together, with the lowest rank card biased towards the hole-card position.  Others have lost bankrolls as large as 250 units, or have gone a year of full-time play showing no profit. What’s going on?

 

The base game has a player edge of 3.48% if the AP has full view of the hole-card and never makes a mistake. Superstar opportunities like this were common through about 2003, when casinos started catching on. But even in the golden days, many APs would still play marginal games. In these situations there were times that the AP couldn’t distinguish which face card he was seeing. There were other times when the AP could not clearly distinguish an Ace from a Two or Three. There are almost no small mistakes when hole-carding. Any mistake may cost the AP a full unit, about the equivalent of a full hour’s expected earnings.

 

I hole-carded Three Card Poker during the period 2002 to 2005. I am in the group who showed no profit. I recall many times when I guessed which face card I was seeing. I recall many times playing on a team when the hole-card spotter signaled the wrong hole-card.  I remember botching the Ace/non-Ace call. I recall mistaking a King for a Jack and playing a busted hand. The 3.48% edge is sitting there for the AP like a ripe apple. But it can be a poisoned apple.

 

The AP should not target Three Card Poker; he should not go looking for Three Card Poker. If the AP happens to observe a superstar flashing dealer at Three Card Poker and there is no better game, that’s the only time he should play. Even with perfect information, it is a tough ride. In the rest of this blog post, I am going to discuss some of the challenges.

 

The following table gives the combinatorial analysis for the AP who sees one hole-card at Three Card Poker:

 

 

In particular, we note:

 

• The AP’s edge with perfect play is 3.4829%.

• The standard deviation for perfect hole-card play is 1.7417.

• The most common outcome is to lose 2 units. This occurs 34.03% of the time.

 

I had a good AP friend who struggled to beat this game in the summer of 2003. He assembled a $50,000 bankroll from investors and played with a unit of $200 per hand. His bankroll was therefore 250 units. The games he played against were exceptionally good. He kept impeccable records, including an honest assessment of the quality of games he was playing and the number of mistakes he made. At 40 hands per hour, his expected hourly profit was $278.63. He lost the entire bankroll in 3 months.

 

To investigate what happened to my friend and more generally to make accurate statements about the risk-of-ruin (ROR) for Three Card Poker with perfect information, I will fix certain parameters of play.

 

Assume:

 

• The player only plays games with perfect hole-card information.

• The player plays 4 hours per day.

• The game pace is 40 hands per hour.

 

Then:

 

• The APs edge on every hand is 3.4829%.

• A day consists of 160 hands of play.

• The expected earnings per day for the AP is 5.5726 units.

 

I wrote a program to model the experience of my friend. The graph below plots 20 simulated samples of 100 days of perfect play. Each of the lines represents a player like my friend who started with 0 units in winnings and went on from there. The horizontal axis counts the number of days. The vertical axis is the number of units won or lost. The jagged path of these lines indicates just how challenging Three Card Poker can be.

 

 

Here are the same 20 random walks, this time with the assumption that the AP loses 1 unit per day due to mistakes:

 

 

If the AP makes errors that costs on average one unit per day, then the player's edge is reduced from 3.48% to 2.86%. The player is giving up about 17.94% of his profit to errors.

 

Of course, it is very unlikely that the AP will get a perfect read on the dealer every hand every day. There may be hands where the face card cannot be distinguished and the AP has to play the QJx strategy. There may be hands where the hole-card is completely missed. There may be times the AP has to play through a break waiting for the superstar to return.

 

Here are the same 20 random walks, this time with the assumption that the AP loses 1 unit per day due to mistakes, and loses an additional 1 unit per day due to playing in poor game conditions:

 

 

Twenty lines are not enough to make firm conclusions, but they do offer some intuitive insight.  Three Card Poker is a tough game.

 

The following graph gives the ROR for the AP based on the number of starting units in the AP’s bankroll. That is, this is the probability that the AP will lose his entire bankroll - that he goes bust.

 

 

The following table gives specific values for the ROR for some starting bankrolls:

 

 

In particular, with the 250 unit starting bankroll my friend had, there is a 0.32% probability that he would lose his entire bankroll. That is, about 1-in-310 times an AP with a 250 unit starting bankroll will go bust. That’s with perfect information and no errors.

 

One memorable day in 2005, I lost close to $4000 in one short session of play with a unit of $75. After losing pairs to higher pairs, straights to higher straights, and having the dealer qualify almost every time he flashed a low card, my pocket was empty. I didn’t have the cash to continue. That day I lost about 53 units. The probability of losing 53 or more units in one day of play (160 hands) is about 1.68%, or about one time per 60 days.  It turns out that it can get much worse.

 

The following table gives a distribution of results from one day of play. Again, I assume that a day consists of 160 rounds of play (4 hours at 40 hands per hour). The horizontal axis represents the number of units won or lost for the day’s play.  The vertical axis represents the probability of the given outcome.

 

 

Many in the AP community believe that Three Card Poker is their salvation. However, Three Card Poker is a siren that has called many APs to be crushed on the jagged rocks of its variance.

 

I am not suggesting that casinos should stop worrying about protecting their hole-cards at Three Card Poker. If dealers are flashing their hole-cards, it does not matter the game. Hole-card protection issues on one game show a lack of adequate supervision and training that may bleed over to other games.

 

Any time a security issue is observed on one game, this issue needs to be addressed from the top down. Casinos should not leave important issues like hole-card protection to the staff on the floor. The Director of Table Games should find ways to implement fixes and audits that will cover the particular issue in all its possible incarnations further down the food chain.