[DICE] Rules of Caribbean Stud Poker

From the rec.gambling Frequently-Asked-Questions (by Steve Brecher):

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The player antes, and is then dealt a five-card hand; the dealer is also dealt five cards of which only one is exposed. The player now either folds, losing his ante, or bets an additional amount equal to exactly twice the ante. The dealer then reveals his remaining four cards. If the dealer's hand is not Ace-King or better, the player is paid even money on the ante and nothing (i.e., a push) on the bet. If the dealer's hand is Ace-King or better it is said to "qualify" (for play against the player). In that case if the dealer's hand is better than the player's, the player's ante and bet are collected by the house. If the dealer's qualifying hand is worse than the player's hand, the player is paid even money on the ante and an amount on the bet according to the player's hand as follows:

        AK or pair                1:1
        two pair                  2:1
        three of a kind           3:1
        straight                  4:1
        flush                     5:1
        full house                7:1
        four of a kind           20:1
        straight flush           50:1
        royal flush             100:1

There is an optional independent side bet of $1.00 available for which the player is paid for being dealt premium hands (flush or better); the payoff of this side bet is based on a progressive jackpot for straight flushes (10% of jackpot) and royal flushes (100%), although some places cap the straight flush payoff (e.g., $5000 max). The jackpot bet is extremely unfavorable except for the case of a very large jackpot. If the jackpot payoff is $50/75/100 for flush/full house/quads and there is no straight flush cap, then the expected return per $1 jackpot bet is approximately $0.23 plus 2.924 cents for each $10,000 in the jackpot; if the flush/ full house/quads payoff is $100/250/500, the expected return is approximately $0.68 plus 2.924 cents for each $10,000 in the jackpot. Examples:

        Jackpot        Expectation per $1 bet
        -------        50/75/100  100/250/500  --flush/full/quads payoffs
                       ---------  -----------
        $10,000          0.26        0.71
         20,000          0.29        0.74
         50,000          0.38        0.82
         75,000          0.45        0.90
        100,000          0.52        0.97
        110,542          0.55        1.00
        150,000          0.67        1.12
        200,000          0.82        1.26
        250,000          0.96        1.41
        263,228          1.00        1.45
        400,000          1.40        1.85
        500,000          1.69        2.14

If the jackpot payoffs are different, you can calculate the expectation from the following formula:

     0.0019654*flush$ + 0.0014406*fullHouse$ + 0.00024010*quads$ +
     f(0.00000013852*straightFlush%*JP, straightFlushCap$) + 0.0000015391*JP

--where * denotes multiplication, JP is the size of the jackpot, and f(x,y) is equal to the smaller of x and y if there is a cap on the straight flush payout or equal to x if there is no cap.

My analysis of the basic game:

When the dealer doesn't qualify the player's bet wins the ante and the dealer's payoff on the ante. In other words, if the dealer doesn't qualify the player is paid even money on the bet. However, in the long run the dealer will qualify 56.3% of the time. A bluff is always an unfavorable bet. Even the best possible bluff--where the player holds an Ace or King, another card which matches the dealer's upcard, and a four-flush of the same suit as the dealer's upcard--is unfavorable. This means that a player who always folds hands worse than Ace-King will lose less in the long run than one who sometimes bluffs.

A pair or better should always be bet. A bet on even the worst possible pair--deuces, with no Ace nor King, no card matching the dealer's upcard, and no card of the same suit as the dealer's upcard--yields an expected profit. This means that a player who always bets a pair of deuces or better will lose less in the long run than one who sometimes folds such hands.

The dealer will fail to qualify 43.7% of the time, and will qualify with an Ace-King (no pair) 6.4% of the time. The player who holds an Ace-King and bets will win even money more than 43.7% of the time (because the player's holding Ace-King reduces the chance of the dealer qualifying), and will be paid two to one (1:1 bet payoff plus 0.5:1 ante plus 0.5:1 ante payoff) when the player's Ace-King beats the dealer's. Therefore, there are some player Ace-King hands which should be bet, depending on what other cards the player holds. For example, if the player holds a card having the same value as the dealer's upcard, the chance of the dealer having a pair is reduced.

The optimum strategy is to bet when the player holds:

    (1)   AKQJ or better (including any pair or better)
  or
    (2a)  AKQxx   with any card in player's hand matching dealer's upcard; or
    (2b)          with both x cards having higher value than dealer's
                  upcard; or
    (2c)          with a four flush of the same suit as dealer's upcard and:
                           at least one of the x cards being either:
                                  8 or better (i.e., 8, 9, or 10)
                          or
                                  of higher value than dealer's upcard.
  or
    (3)   AKJ with any card in player's hand matching dealer's upcard
  or
    (4)   AKxxx with any x card matching dealer's upcard

The results of this strategy and two simpler strategies are shown below, each based on computer simulation of 200 million deals. "Expected loss per ante amount per hand" is the average amount that the player will lose per hand in the long run as a percentage of the ante amount. "Payback per $1 risked" is the average long run total payback on each dollar wagered--on antes plus bets.

                                  Expected loss per
  Strategy        Bet frequency   ante amount per hand   Payback per $1 risked

  Optimum                 52.0%           5.23%              $0.9743
  Bet any pair or better  49.9%           5.48%              $0.9726
  Bet Ace-King or better  56.3%           5.75%              $0.9729

For the casual player, "Bet any pair or better" is the recommended strategy. The expected difference in total loss versus the optimum strategy over a couple of hundred hands is about half of one ante. "Bet Ace-King or better" provides more betting action at the cost of another half an ante per couple of hundred hands.

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