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
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.
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
9/6 Jacks||The well-know "full pay" version of Jacks or Better.
Payoff schedule: 1,2,3,4,6,9,25,50,800.