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# Gaming Guru

### Know When to Fold'em

3 July 2004

In my May article on dealer qualification, I indicated that I would address the topic of folding at a later date. Folding can be quite complicated. In regular Poker, for example, your position at the table, the actions of the other players, as well as the strength of your hand all contribute to the decision. On the other hand, in some casino games the folding criterion can be quite simple and only depends upon the cards in your hand. I am going to illustrate this by explaining how one determines the folding criterion for the Ante/Play wager in Three Card Poker. There is another wager in Three Card Poker called Pair Plus but that will not concern us here; it is essentially a side bet.

The Ante/Play wager begins with the player making a unit bet called the Ante. The player then receives his three-card hand. After examining his hand the player can fold, and thereby forfeit the Ante, or he can elect to Play. The latter is accomplished by matching the Ante bet with an additional unit wager, which places two units at risk. If the decision is to play, the dealer then receives his three cards. If the dealer's hand does not contain a queen high hand or a hand of greater rank, then the dealer is said to have not qualified and the player is paid even money on his Ante, and the Play wager is returned. On the other hand, if the dealer qualifies then the dealer's and player's hands are compared and the higher-ranked hand wins; ties are pushes. If the player wins he is paid even money on both the Ante and Play wagers. If the player loses the dealer collects the Ante and Play wagers.

A word about the ranking of the hands is appropriate here. In Three Card Poker, the rankings are:

• Straight Flush (Suited Ace, King, Queen is highest)
• Three of a Kind
• Straight
• Flush
• Pair
• High Card

These rankings are determined by the frequency of each type of hand when dealt from a 52-card deck. It turns out that of the 22,100 possible hands, there are only 52 Three of a Kinds, whereas there are 720 Straights and 1096 Flushes.

There is an additional payoff to the player called Ante Bonus. If you are dealt a Straight or better you are automatically awarded a payoff on your Ante wager whether or not the dealer qualifies and whether or not your hand beats the dealer's. The payoff is 5 to 1 on a Straight Flush, 4 to 1 on Three of a Kind, and even money on a Straight.

So when should a player fold? I first addressed this several years ago when I was consulting with the British Columbia Lottery Commission. They were considering a version of the game called Tre Card Poker for adoption in BC casinos. As I recall, that version did not have an Ante Bonus but paid even money on the Ante and a bonus on the Play wager that was determined by a pay table. Thus the dealer had to qualify and the player had to win in order to collect the bonus. Anyway, I dug out the software I wrote to analyze Tre Card and fortunately it just needed minor modifications to help me in writing this article. Let me describe how the modified program works and I think you'll see how the folding criterion is obtained.

The program cycles through each of the 22,100 three-card hands that are possible to obtain from a 52-card deck. For each of these hands, the cards are removed from the deck, and then three cards are dealt to the dealer from the 49 remaining cards in the deck. There are 18,424 such hands and the program cycles through each of them. Before this cycle starts the program sets a variable called "stake" to zero. As each dealer hand is dealt, it is checked for qualification and if the dealer hand does not qualify the stake is increased by 1. If the dealer does qualify the program checks the rank of the player's hand versus the dealers hand and settles the bet according to the rules--that is, the stake is increased by 2, decreased by 2, or remains the same. After all 18,424 dealer hands have been checked the variable stake is divided by 18,424 thereby obtaining the average return the player would expect for the particular player hand under consideration; I called the variable exp for expectation.

Here is the important point. If exp < -1 then the player would have been better off folding his hand. When this happens the program sets exp equal to -1, since the player would have folded, and records that particular player hand as a hand that should be folded. If that hand is greater in rank than a previously recorded fold hand, then the old fold hand is replaced with the current fold hand. That way the program retains the largest ranking hand that should be folded. The program also adds exp to a running variable called totexp that totals the players expected return for all of the 22,100 hands played. The overall expected return for the game is then totexp divided by 22,100.

I should mention that the program also adjusts the player's stake accordingly when the player's hand merits an Ante Bonus. This feature could be omitted if one were only looking for the folding criterion since a Straight or better would never be folded. However, I wanted to get the house edge at the same time so I included it.

Well, here are the results. The house edge per game was 3.3729809%. The average bet was 1.6742081 units, which means that the house edge per wager is 2.0146724%. The last recorded fold hand? That turned out to be Queen-Six-Three unsuited. In other words the player should fold all hands lower in rank than Queen-Six-Three and should play Queen-Six Four and higher.

The overall lesson here is that given some initial information in a game one should fold if the expected return in this situation is less than -1. Sometimes, as in the case of Three Card Poker, it is easy to come up with a folding criterion (although I wouldn't want to check it by hand). For other games the folding criterion might be quite complicated and depend on things that are hard to quantify, but the question is still the same: Do I expect to lose more than my unit bet by staying in the game?

I don't want to leave this topic without mentioning a related one, namely, Surrender in Blackjack. Since in Blackjack you can surrender your hand for half of your wager, the criterion is simply to see if your expected return in a particular situation is less than -1/2. For example, in a multiple-deck Blackjack game the expected return when holding a 16 versus the dealer's 10 is around -0.54, so strategy tables tell you to surrender this hand.

If you want to learn more about Three Card Poker, I recommend the booklet Mastering the Game of Three Card Poker by Stanley Ko. This booklet not only reports the above figures but also gives the player additional strategy in case he sees other player's cards or sees one of the dealer's cards. This is available at gambling bookstores or at Gambology, P.O. Box 82225, Las Vegas, NV 89180.

Folding or surrendering is tough to do since it represents a sure loss. Sometimes, however, not doing it is worse! See you next month.

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Best of Donald Catlin
Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers