The "Kelly" number for a given game provides the optimal bet size to the fastest possible expected growth of the (geometric mean) bankroll. There is no trivial way to get the Kelly number for a given game -- I use interpolation to find the value f that maximizes the sum of p_i*ln(1+f*g_i), where i indexes the possible outcomes from a unit bet, and outcome i has probability p_i and gain g_i. The Bankroll is then just 1/f. You can generate this table yourself if you have a Perl interpreter -- just use my script. For video poker these number are rather high. The numbers can reduced substantially by cash back. |
In video poker, we don't usually have a choice of bet size (or, at least, only have the choice of a small number of widely separated bets, typically $1.25, $5, $25, etc.). However, we can turn the Kelly value on its head and determine the optimal bankroll size in for a given bet. Kelly tells us that if our bankroll is less than half the "full" Kelly value shown in the table, then taking the bet will actually reduce the expected growth of our bankroll (the rate of increase drops from its maximum to break even as the bankroll decreases). Essentially what this means is that we should not take the bet unless our bankroll is at least as half as large as the Kelly value. In reality, we probably want a bankroll that is comfortably larger than this minimum to allow for errors and early bad runs. |
The following table gives the full Kelly bankrolls for a number of games that I have analyzed (see my video poker page for strategy cards for these games). The table emphasizes the importance of cash back in choosing a game. Many of the games are not playable without cash back. I have taken the cash back numbers out to 2%, which is quite high but might be available occasionally during promotions, etc. Note that all values are given in terms of units bet, i.e., a unit on a $1 machine is $5 since we always bet max coins. For example, to play 9/6 Jacks at $1 with 1% cash back, the full Kelly bankroll is 2917 units or $14,585. |
Cash Back | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Game | Base Exp | 0.0% | 0.2% | 0.4% | 0.6% | 0.8% | 1.0% | 1.2% | 1.4% | 1.6% | 1.8% | 2.0% |
9/6 DB/DJ | 100.09% | 95430 | 29987 | 17355 | 11990 | 9024 | 7145 | 5848 | 4901 | 4180 | 3614 | 3158 |
Pick'em | 99.9536% | -- | 9393 | 3895 | 2386 | 1687 | 1287 | 1030 | 851 | 721 | 621 | 543 |
8/6 JP2400 | 103.21% | 3450 | 3114 | 2817 | 2551 | 2312 | 2097 | 1902 | 1725 | 1564 | 1416 | 1282 |
8/6 JP2040 | 102.08% | 4411 | 3855 | 3390 | 2995 | 2656 | 2361 | 2104 | 1878 | 1677 | 1498 | 1338 |
8/6 JP1700 | 101.04% | 7235 | 5806 | 4776 | 3999 | 3393 | 2907 | 2509 | 2179 | 1899 | 1661 | 1456 |
All American | 100.72% | 3369 | 2542 | 2016 | 1652 | 1386 | 1185 | 1027 | 900 | 797 | 712 | 640 |
10/7 DblB80 | 100.52% | 5213 | 3661 | 2787 | 2229 | 1843 | 1560 | 1345 | 1176 | 1040 | 929 | 836 |
10/7 DblBon | 100.17% | 16511 | 7346 | 4630 | 3331 | 2572 | 2075 | 1726 | 1468 | 1269 | 1113 | 986 |
Flush Att50 | 100.04% | 64612 | 11702 | 6274 | 4213 | 3131 | 2466 | 2017 | 1694 | 1452 | 1263 | 1112 |
8/6 JP1360 | 99.99% | -- | 28637 | 13675 | 8695 | 6208 | 4791 | 3728 | 3022 | 2495 | 2087 | 1763 |
9/6 Jacks | 99.55% | -- | -- | -- | 11792 | 4875 | 2917 | 1998 | 1468 | 1128 | 893 | 723 |
8/3 Fl Attack | 99.24% | -- | -- | -- | -- | 61755 | 11356 | 6088 | 4083 | 3029 | 2382 | 1944 |
7/4 Fl Attack | 99.19% | -- | -- | -- | -- | -- | 14221 | 6845 | 4417 | 3213 | 2495 | 2020 |
9/7 Dbl Bonus | 99.10% | -- | -- | -- | -- | -- | 28154 | 9188 | 5367 | 3731 | 2824 | 2250 |
9/6 DB/DJ | 9/6 Double Bonus with Double Jackpot. This game is appearing in Atlantic City. It is slightly positive even without cash back, but beware the variance! |
Pick'em | This is a half-stud/half-draw game. You must keep the first two cards dealt, but get to keep exactly one of the next two. You then get two more cards to complete the hand. Lowest variance I've seen yet in a VP game. |
8/6 JPxxxx | 8/6 Jacks or Better with a Jackpot. The "xxxx" indicates the jackpot size in terms of number of bets. On a quarter machine the values are 2400 units = $3000, 2040 = $2550, 1700 = $2125, and 1360 = $1700. |
All American | Payoff schedule: 1,1,3,8,8,8,40,200,800. Hard to find, but worth it! |
10/7 DblB80 | Payoff schedule: 1,1,3,5,7,10,[80,50,160],80,800. The [80,50,160] indicates 80 units for Quad 2-4, 50 units for 5-K, and 160 units for Aces. |
10/7 DblBon | Payoff schedule: 1,1,3,5,7,10,[80,50,160],50,800. Same as DlbB80 except Straight Flush pays 50. |
Flush Attack | Three versions as shown. Flush Attack 50 is the Las Vegas version, with a payoff schedule of: 1,1,3,4,25,8,[80,50,160],50,800. We assume continuous play and that it takes 4 off flushes to turn it on. The other two version are from Atlantic City. It only takes 3 flushes to turn the machine on, but the payoff schedules are reduced to 1,1,3,3,25,8,[80,50,160],50,800 (8/3) and 1,1,3,4,25,7,[80,50,160],50,800 (7/4). |
9/6 Jacks | The well-know "full pay" version of Jacks or Better. Payoff schedule: 1,2,3,4,6,9,25,50,800. |
The Kelly value can help us to choose which game to play for a given bankroll. Given two choices that both meet our criteria, we still need to consider the risk of ruin. That is, the Kelly number alone is not sufficient to derive the chances that we will bust out, although it is a good indicator. For example, notice that 9/6 Jacks and 10/7 Double Bonus have a "cross-over" at 1.4% cash back -- that is, they have the same Kelly value. Since 1,000,000 hands represents approximately one year's play for a serious player, I asked the question: "What is my expected low point for 1,000,000 hands for each of these games with 1.4% cash back?". |
I ran 49 separate simulations of one million hands each for each game, assuming perfect play (using my best basic strategies). I found the expected low point was -684 units for 9/6 Jacks, with a standard deviation of 650, while 10/7 Double Bonus had a low point of -928 units and a standard deviation of 791. The large standard deviations mean we can't trust these median values too much, but it seems clear that 10/7 DB is more risky. You have to weigh that against the higher expected earnings (0.5% over 1,000,000 hands at $5 a hand is a $25,000 difference). |