High Country Poker

   At the World Gaming Congress and Expo that was held in Las Vegas in September of 1999 there was, as usual, a plethora of new gambling machines as well as several new table games on display.  Three of the latter were of special interest to me since I had done the mathematical analysis of these games for their developers and I was interested to compare them to other games being offered.  The three I mention are High Country Poker^TM developed by Glen Garrod of Nevada City, California, Hit and Win^TM and JackBlack^TM both developed by Derek Webb of Prime Table Games of Las Vegas, Nevada.  Derek, by the way, is the man who developed and marketed the popular game Three Card Poker^TM which was recently sold to Shuffle Master.  All three of these games require some skill on the part of the player and that is why I am going to devote this and the next two columns to explaining how to derive the optimum strategy to play these games.  I'll also explain why I think these games will be winners.

    This month I will discuss High Country Poker.  It is easy to predict that this game will be a winner because it has already been approved for play by the Nevada State Gaming Commission and is playing very successfully.  The game has been playing for several months at Cache Creek Indian Bingo and Casino in northern California.  It had a short trial run at the Reno Hilton pending approval by the Nevada Gaming Commission.  The Commission approved the game for general play in Nevada in late 1999.  The game opened at Harvey's Resort Hotel/Casino in South Lake Tahoe on January 14th of this year, at John Ascuaga's Nugget in Sparks on January 28th, at the Atlantis in Reno on February 14th, and at the famous Cal Neva Lodge in Crystal Bay on February 18th.  I expect that by the time this article appears the game will be in several other Nevada casinos as well.

    The game is banked by the casino and is dealt by a dealer who plays a hand in competition with the players.  The game is played on a standard 21 shaped table.  The dealer's hand always contains the Ace of Spades and the remaining hands are dealt from the Ace depleted 51 card deck.  The region in front of each player displays three betting spaces marked "Ante 1 Unit", "Bet Two Units", and "Optional Raise 1 - 3" respectively.  Before the deal each player is required to ante one unit in the 'Ante Space' and twice that amount in the 'Bet Space'.  When all of the bets are in place, each player is dealt two cards face down and the dealer is dealt one card face down, that is, the dealer's initial hand consists of the Ace of Spades and an unknown card.  The player can look at his own hand but not any of the other hands at the table (this may change by the time this article appears).  The dealer's card remains hidden.

    At this point the player has the option of folding any two card hand that contains a Deuce; all other hands remain in play.  The rationale for this rule will be clear later in the article.  If the player does not fold and decides to continue, he or she has the option of playing for the three units already wagered (the combined Ante and Bet) or making a raise by placing up to three additional units in the 'Raise Space'.  This done, the dealer deals out five common cards face up on the table.  The five common cards, referred to as the flop, are to be used by all players who have not folded, including the dealer.  Each player remaining in the game must now make the best five card Poker hand he or she can from the seven available cards, that is, the five common table cards (the flop) and the two cards in the player's hand.  The dealer also uses these same five cards in the flop together with his face down card and the Ace of Spades to make the dealer's best five card Poker hand.  A player beats the dealer if the player's hand is higher in rank than the dealer's hand, the ranking being the usual five card Poker ranking, or if the player's hand ties the dealer's hand; otherwise the player loses.  As you can see the game is essentially a form of Texas Hold-Em.

    In addition to the basic Poker game described above, High Country Poker also has some bonus payoffs.  Here is the schedule:
 
 

Now you can see the rationale for folding Deuce hands.  When such a hand is folded the player's Ante and Bet are both returned and the player is paid even money based on the Ante for a single Deuce hand and 5 to 1 based on the Ante for a pair of Deuces.  The other bonuses in Figure 1 are called High Bonus hands and are paid whether or not the player wins the hand.

    The initial analysis of this game was started by Lenny Frome in March of 1998.  In early March Lenny contacted me and asked if I would like to help with the analysis.  I accepted.  There are C(51,2) or 1275 two card starting hands for the player (see my August 1999 article Oh, New York, Bring Back Those Big Dippers for a discussion of this function). Lenny had decided to partition these into suited and unsuited pairs.  This resulted in 169 different types of two card hands.  There was some discussion about considering whether or not the hand contained a Spade, but the results did not seem sensitive to this issue.  So, our initial job was to determine the probability that the player's final hand beats the dealer's final hand starting with each of the 169 types.  This would have been a big job were it not for a very nice Poker simulation program called Poker Probe^TM that is marketed by Mike Caro.  The program has a Hold-Em option.  Both Lenny and I owned this software so we were able to split up the work and obtain the necessary 169 probabilities by simulation.  We had our numbers; the analysis lay ahead.

    Here fate stepped in.  On Friday March 13th I received a fax from Lenny about the game.  I answered it on Sunday the 15th and on Tuesday the 17th one of Lenny's sons called to tell me that Lenny had died over the weekend from a heart attack.  The gaming world in general, and me in particular, was shocked.  We had been working together for about two and a half years at that point and had become good friends.  I miss Lenny.

    With Lenny's death the analysis came to a standstill.  The week following Lenny's death I received a call from Glen Garrod asking if I would continue on alone with the analysis and, of course, I was happy to do so.

    The next step in analyzing this, or any game involving decisions by the player, is to determine the optimum strategy for the player since the house edge against such a player will be the smallest for the house.  Should the player always fold Deuce hands?  When should the player raise?  What if the player could fold non-Deuce hands for the cost of the Ante; would this be to the player's advantage?  I mention this last item because in an earlier version of the game this option was available to the player and there is currently some talk of making this option available again (it is available at Cache Creek).  Let me address this last issue first.

    Assuming that the player does not fold, let's assume that his total wager, Bet plus Ante, is w.  Furthermore, assume that the player's probability of beating the dealer given the player's specific two card hand is p.  In this case the player's expected return is the following:

Since the player's expected return by folding is -1 it is clear that the player should continue whenever the expected return in (1) is greater than -1.  In other words the condition for continuing is:

which is equivalent to

The term (w - 1)/2w on the right of (3) is called the folding number.  From condition (3) it is clear that whatever value w has, the folding number is a number less than 1/2.  This means, ostensibly, that there will be hands that are not fold candidates for which the probability of beating the dealer is less than 1/2.  From (1) we see that whenever p < 1/2, the expected return by not folding is (2p - 1)w, a negative number, and the expected loss is smallest if the Bet plus Ante is set at the minimum of w = 3.  Setting w in (3) equal to 3 we obtain a folding number of 1/3.  On the other hand, if p > 1/2 then (2p - 1)w is positive and w should be set to the maximum of 6 (Ante + 2 Bet + 3 Raise).

    To illustrate what happens in this game let me choose the very worst non-Deuce starting hand that the player can have; it is the unsuited 8/3.  The probability of this hand leading to a final winning hand is 0.317.  According to the criterion above, this hand should be folded.  There are 12 such hands among the 1275 starting hands so, according to (1) above, playing these 12 hands would lead to an expected return for the player of 12 x (2 x 0.317 - 1) x 3 or - 13.176.  Folding the hand only loses 12 units.  But what about the effect of High Bonuses?  These 12 hands have to be subdivided into 3 types of hands to calculate the High Bonuses: 6 no Spade hands, 3 Eight of Spades and no-Spade 3, and 3 no-Spade Eight and Spade Three.  Counting these is tricky and you can give it a try if you wish.  The results, using specific examples, are as follows
 

The last column in the table is calculated as follows; I'll use the first line:

There are C(49, 5) or 1,906,844 ways to complete each two card starting hand.  This means that the number for the total in Figure 2 represents the total amount one would collect from High Bonus hands if each of the 12 unsuited 8/3 hands were completed each of the 1,906,844 ways possible.  Hence, if we divide this total of 3,068,010 by 1,906,844 we obtain the average gain for these 12 hands that we could expect by always playing them  The result is approximately 1.609 units meaning that if we played each of these 12 hands once we would expect to gain 1.609 units.  Adding this to the - 13.176 figure we obtained above give us an expected return by playing this hand of - 11.567.  The return from folding is - 12.000 so the hand should be played.

    Every non-Deuce hand behaves the same way; all of them should be played.  A calculation similar to the one just made can be made for the Deuce hands but the results aren't even close.  Playing an unsuited Q/2, for example, results in an expected return of  - 11.465 from playing the 12 such hands versus a sure return of  +12.000 by folding them.  All of the Deuce hands should be folded.   The optimum strategy for Playing High Country Poker is simplicity itself:
 

    To calculate the house edge for High Country Poker one would have to first calculate the expected return for the 1275 starting hands with no High Bonuses but with folded Deuces.  This is somewhat tedious but reasonably easy to do.  The result is -269.231 and the total amount at risk is 4401 units.  Then one would, ostensibly, have to calculate the return from High Bonuses for each of the non-Deuce hands, add these numbers, and add the total to the above figure.  Whew!  Fortunately there is an easier way.  Since I already have the High Bonuses for the 194 Deuce hands, I can add these to obtain a result of 54,068,040.  This figure represents the amount we would collect in High Bonuses if we played all 194 Deuce hands (which, of course, we don't).  If we played every one of the 1275 hands it is reasonably easy to calculate the total return from all the High Bonuses; it is 397,873,350 (write to me if you are interested in the details).  The difference between these two numbers is 343,873,350 and this represents the return from High Bonuses for the hands actually played.  Dividing this by 1,906,844, the number of flops per hand, we get the expected gain to the player from High Bonuses for the 1275 starting hands.  This number is 180.297.  Adding this to -269.231 we obtain -88.934, the expected return from playing each of the 1275 starting hands once.  Dividing this by the amount at risk, 4401 units, we obtain the house edge for the game.

    As new games go, a house edge of 2.02% is quite reasonable.  The game seems to be popular with the players in the Reno/Tahoe area and I expect it to eventually take a permanent place in the array of popular casino games.  If you try this game, make sure you follow the strategy in Figure 3.  A player only raising the maximum of 3 units on non-Deuce pairs, for example, would raise the house edge in this game to around 4.1%.  See you next month with Hit and Win.