Phantom bonuses
November 22, 2004
1 Introduction
The game is defined by a list of payouts u
1
, u
2
,. . . , u
`
, and a list of probabil-
ities p
1
, p
2
, . . . , p
`
,
`
P
i=1
p
i
= 1. We allow u
i
to be rational numbers, not just
integers, to include games like blackjack, n-play video poker or the banker bet
in baccarat. We assume that the casino has the advantage, so
`
P
i=1
p
i
u
i
< 0.
The player can bet any positive integer k up to his current bankroll or
the maximum bet b, whichever is smaller, and his bankroll increases by u
i
k
with probability p
i
. To cater for blackjack or poker type games, only the
player’s initial bet is limited, and we allow the player to borrow money for
splits, doubles or raises if necessary, but he has to stop if he loses and ends
up with a negative bankroll. Each game is independent of all others.
The player has a phantom bonus m without wagering requirements. At
any point he can decide to cash in, if his bankroll is greater than m, then m
is deducted from his balance in and the bonus is lost, if his bankroll is m or
less, he forfeits his whole balance.
2 The question
If the player’s current bankroll is n, what strategy should he follow to max-
imize the expectation of the real money he can cash in after deducting the
bonus, and what is this expectation a
n
?
n can be a rational number whose denominator is the least common mul-
tiple of the denominators of the u
i
, for example, when dealing with blackjack,
1
n can be a half integer, but the stake is always an integer and let us also
define a
n
= n for n < 0. Define the value of the phantom bonus to be
a
n
− max(0, n − m), this is the amount the player expects to gain by playing
optimally instead of cashing in immediately.
In games involving an element of skill, we assume that the player is play-
ing a fixed strategy, possible adjustments to playing strategy in view of the
changed expected values are not considered, strategy will only mean betting
strategy.
3 The solution
In theory, there is an easy way to calculate to a
n
. Start with a
n,0
= n for
n < 0, a
n,0
= 0 for 0 ≤ n ≤ m, and a
n,0
= n − m for n > m. Then for each
n, calculate
`
P
i=1
p
i
a
n+u
i
k,0
for all integers k, 1 ≤ k ≤ min(n, b), and let a
n,1
be the maximum of these values and a
n,0
. We are calculating the optimal
strategy using a
n,0
as approximation to a
n
. Repeat with a
n,1
, and so on. In
general, let
a
n,j+1
= max
³
a
n,j
,
n
`
X
i=1
p
i
a
n+u
i
k,j
|1 ≤ k ≤ min(n, b)
o´
.
a
n,j
≤ n for all j, since the casino has the advantage, so the expected
value of the player’s cash-in cannot exceed current bankroll, even ignoring
the bonus. a
n,j
≤ a
n,j+1
, and it is a well-known theorem in analysis that
a bounded monotonically increasing sequence has a limit, so a
n
= lim
j→∞
a
n,j
exists. Unfortunately, this procedure as described involves an infinite amount
of calculation.
I conjecture that the phantom bonus is worthless, i.e., a
n
= n − m if
n ≥ N =
m
P
u
i
<0
p
i
u
i
`
P
i=1
p
i
u
i
, (1)
and that if the player plays a game in which all the payouts are integers ≥ −1
and b ≥ N, then the optimal strategy is simply to bet everything if n < N
and to cash in if n ≥ N. Under these conditions N is exactly the point at
2
which the expected value of betting everything is the same as that of cashing
in. If the game only has two possible outcomes, u
1
= −1 with probability
1 − p and u
2
= u, a positive integer, with probability p, then a
n
can be
calculated by finding the integer r such that N/(u + 1)
r
≤ n < N/(u + 1)
r−1
,
and then a
n
= p
r
((u + 1)
r
n − m), so a
n
is a piecewise linear function on
n. I cannot prove all this, but I have quite good numerical evidence. The
idea for the proof I have depends on some kind of convexity of the sequence
a
n
. (Convexity means a
n
≤ (a
n−1
+ a
n+1
)/2, in other words, that the player
should always take a bet with no house edge. I would only need something
weaker, that a
n
/n increases with n, and a
n+k
≤ a
n
+ k, the latter meaning
that value of the phantom bonus decreases as n increases.)
If the conjecture is true, then the calculations can be restricted to n < N ,
which makes each iterative step finite.
Another approach to the problem is that if the optimal strategy is known
for each n, then we can write down the linear equations relating the a
n
and
solve them, this gives exact answers, unfortunately the optimal strategy is
not known in advance.
The method described below is a combination of iterations and linear
equations.
Step 1. Calculate N by (1). Set a
n
= n for n < 0, a
n
= 0 for 0 ≤ n ≤ m,
and a
n
= n − m for m < n ≤ N.
Step 2. Check for each n, 0 < n ≤ N, whether betting the maximum possible
amount improves the current value of a
n
, and if so, update the value of a
n
.
Do this 10, 20 or 50 times, the exact number does not seem to make much
difference in the running time. For some reason I do not understand, going
in decreasing order of n gives faster convergence than in increasing order.
Step 3. Calculate the optimal strategy using the current values for a
n
, and
solve the resulting linear equations. This is equivalent to using the currently
optimal strategy and letting it converge. This is the slowest step because
it involves solving a large, but sparse system of linear equations, usually
consisting of several hundred equations.
Step 4. For each n, calculate the expected value of betting any of possible
bet sizes using the current values of a
n
. If this does not give an improvement
for any n, then the current numbers are the exact values of a
n
, and we can
also deduce the optimal strategy. If improvement is possible, then go back
to Step 2.
The calculations were carried out using Mathematica, which can handle
3
rational numbers with arbitrary numerators and denominators, so it can
give exact answers. These exact answers can be used to justify the use
of the unproven conjecture retrospectively. This method produced results
reasonably quickly for all the games are I looked at.
4 The results
The tables in the Appendix show the results for various games. The size
of the phantom bonus was always taken to be m = 10. The first column
shows the maximum bet, the second column the largest bankroll at which
it is still in the player’s interest to play, rather than to cash in. The re-
maining columns show the value of the phantom bonus, a
n
− max(0, n − m),
from n = 5 to 40 in steps of 5. These numbers were rounded to 3 dec-
imal places. The probabilities for blackjack were taken from a simulation
by Michael Shackleford, the “Wizard of Odds”, available on the web at
http://www.wizardofodds.com/games/blackjack/bjapx4.html.
I believe that the problem scales linearly as long as the payouts are inte-
gers, so choosing m = 10 is not a real restriction.
The tables confirm some of the expected phenomena. The value of the
phantom bonus, and also the size of the bankroll at which the player should
stop playing both increase with b. The value of the phantom bonus is max-
imal at n = m. The house edge is not the primary factor in determining
the value of the phantom bonus, a large probability of losing and a small
probability of winning a large amount is the good for the player. Among all
the roulette bets, which all have the same house edge, betting on a single
number is the best. Similarly, Jacks or Better video poker with doubling is
better than without doubling, while the house edge is the same. The line
bet (6 numbers) in roulette with house edge 2.7% is better for the player
than coin flip with probability of winning 0.495 and house edge 1%. The
numbers for coin flip with probability of winning 0.499 and house edge 0.2%
are similar to those for Jacks or Better video poker with house edge 0.46%.
5 Details of the strategy
Assume that all the payouts are integers ≥ −1. Empirical evidence suggests
and I may be able to prove it rigorously, too, that if player’s bankroll is a
4
multiple of b, he should always bet the maximum, b . In this case the numbers
a
jb
(j = 0, 1, 2, . . .) satisfy the recurrence relation a
jb
=
`
P
i=1
p
i
a
(j+u
i
)b
. Let
u = max{u
i
|1 ≤ i ≤ `}. The solution is of the form a
jb
=
`
P
i=0
c
i
λ
j
i
, where the
λ
i
, i = 0, 1, . . . , u, are the roots of the characteristic equation 1 =
`
P
i=1
p
i
λ
u
i
,
assuming that the roots are distinct.
Let kb be the highest multiple of b at which the player should still play.
The coefficients c
i
, i = 0, 1, . . . , u, can be determined from the equations
a
0
= 0, a
(k+1)b
= (k+1)b−m, a
(k+2)b
= (k+2)b−m,. . . , a
(k+u)b
= (k+u)b−m.
Define α
jb,t
by α
jb,t
=
`
P
i=1
p
i
α
(j+u
i
)b,t
and α
0,t
= 0, α
(t+1)b,t
= (t + 1)b − m,
α
(t+2)b,t
= (t + 2)b − m,. . . , α
(t+u)b,t
= (t + u)b − m. These are the values of
the a
jb
using the assumption that k = t. k can be found as the largest value
of t for which α
tb,t
≥ tb − m, but unfortunately, there is no exact method of
solving for k.
Example 1. Coin flip with the probability of winning p
The characteristic equation is 1 = (1 − p)λ
−1
+ pλ, the roots are λ
0
= 1,
λ
1
= (1 − p)/p, and the solution is
α
jb,t
=
(((1 − p)/p)
j
− 1)(tb − m)
((1 − p)/p)
t+1
− 1
.
Now let m = b = 10, p = 0.495. The table below shows the value of α
tb,t
calculated using the above formula for various values of t.
t 1 2 3 4 5 6 7 8 9 10
tb − m 0 10 20 30 40 50 60 70 80 90
α
tb,t
4.95 13.2 22.3 31.7 41.2 50.9 60.6 70.4 80.2 89.97
For t = 9, α
tb,t
> tb − m, for t = 10, α
tb
< tb − m, so k = 9. This means
that the player should still play with a bankroll of 90, but not with a bankroll
of 100, which agrees with the table in the appendix, which says that largest
bankroll at which the player should still play is 98.
Example 2. Betting on a single number in roulette
The characteristic equation is 1 = 36λ
−1
/37 + λ
35
/37, it cannot be solved
5
exactly, but Mathematica is able to provide numerical solutions. Let m = 10,
b = 50, the results are in the table below.
t 1 2 3 4 5 6 7
tb − m 40 90 140 190 240 290 340
α
tb,t
48.4 96.8 145.3 193.8 242.3 290.9 339.5
For t = 6, α
tb
> tb − m, for t = 7, α
tb,t
< tb − m, so k = 6. This
means that the player should still play with a bankroll of 300, but not with
a bankroll of 350, which agrees with the table in the appendix, which says
that largest bankroll at which the player should still play is 332.
There are some curious phenomena that I do not understand. The table
below shows some values of n and the corresponding optimal bet size when
betting on a single number in roulette with m = b = 10. If the last digit
of n is 7, 8 or 9, then it is never correct to bet 10, but only 7, 8, or 9,
respectively. There are many other numbers for which betting the maximum
is not correct, in all of these cases the correct bet size is the last digit of n.
n 1 2 3 4 5 6 7 8 9 10
bet 1 2 3 4 5 6 7 8 9 10
n 11 12 13 14 15 16 17 18 19 20
bet 10 10 10 10 10 10 7 8 9 10
n 101 102 103 104 105 106 107 108 109 110
bet 10 10 10 10 10 6 7 8 9 10
n 141 142 143 144 145 146 147 148 149 150
bet 10 10 10 10 5 6 7 8 9 10
n 171 172 173 174 175 176 177 178 179 180
bet 10 10 10 4 5 6 7 8 9 10
n 201 202 203 204 205 206 207 208 209 210
bet 10 10 3 4 5 6 7 8 9 10
n 221 222 223 224 225 226 227 228 229 230
bet 10 2 3 4 5 6 7 8 9 10
n 241 242 243 244 245 246 247 248 249 250
bet 1 2 3 4 5 6 7 8 9 10
If we consider the strategy for the split bet with the same m and b, then
this phenomenon does not start until much later, the player should bet the
maximum possible for all n ≤ 157.
6
I can only guess at the reasons. It seems that multiples are b are somehow
preferable, so when the optimal bet is not b, it is the difference between n and
the largest multiple of b less than n, so that if the player loses, his bankroll
will be a multiple of b and from that point on he will always bet b. This
phenomenon also occurs in other games especially near the upper bound.
The reason why 7, 8 and 9 are different in the first example seems to be
that the maximum bankroll at which the player should still play in the first
example is 252, and winning bet with a stake of 7 or more would get him
to this bound or above it. In the second example, the maximum bankroll at
which the player should still play is 198, this cannot be reached by betting 10
or less, this is why the optimal strategy behaves differently. Other examples
also support this, but I have no rigorous explanation for this observation, I
can only speculate.
6 The effects of using the wrong strategy
The traditional wisdom about phantom bonuses is that the player should bet
big, but the previous section shows that betting the maximum is not always
correct. I also calculated the expectations if the player only bet min(n, b) and
the difference from the a
n
was very small, typically only a few thousandths
or even less, so for practical purposes always betting the maximum possible
is a good strategy.
I also considered what happens if the player chooses a different target, for
example he gets a 100% match bonus on his deposit and aims to increase his
bankroll tenfold, so if m = 10 as in the previous calculations, he aims for 200.
The typical situation is that if the optimal strategy suggests that he should
aim higher, say, for 400, he does not gives up much in terms of expectation
by stopping at 200. On the other hand, if the optimal strategy suggests
stopping earlier, say, at 100, then trying to go for 200 can be expensive.
7
7 Appendix
Baccarat (8 decks): player bet
1 30 2.308 4.953 2.984 1.46 0.443 0.01
2 40 2.916 6.046 4.388 2.976 1.807 0.919 0.312 0.025
5 61 3.573 7.245 6.018 4.896 3.881 2.976 2.185 1.51
10 84 3.921 7.95 6.98 6.119 5.263 4.515 3.774 3.143
20 114 4.16 8.436 7.716 7.105 6.396 5.784 5.179 4.684
50 169 4.339 8.797 8.294 7.838 7.475 6.956 6.533 6.17
100 225 4.42 8.962 8.532 8.172 7.873 7.438 7.131 6.846
200 286 4.46 9.044 8.66 8.339 8.071 7.698 7.398 7.185
500 371 4.47 9.065 8.712 8.38 8.071 7.804 7.536 7.269
Baccarat (8 decks): tie bet
1 24 1.99 4.39 2.288 0.796
2 30 2.598 5.534 3.664 2.211 1.035 0.154
5 42 3.33 6.82 5.453 4.216 3.097 2.084 1.168 0.339
10 50 3.411 7.612 6.175 5.452 4.215 3.497 2.447 1.729
20 57 3.496 7.612 6.894 6.177 4.871 4.153 3.435 2.717
50 61 3.577 7.612 6.894 6.177 5.459 4.741 4.023 3.305
100 63 3.577 7.612 6.894 6.177 5.459 4.741 4.023 3.305
Blackjack
1 57.5 3.32 6.884 5.587 4.418 3.377 2.467 1.692 1.057
2 80 3.801 7.736 6.726 5.786 4.911 4.104 3.365 2.696
5 123 4.154 8.423 7.722 7.053 6.414 5.806 5.228 4.679
10 171 4.353 8.811 8.29 7.808 7.315 6.86 6.401 5.973
20 236 4.491 9.077 8.677 8.32 7.936 7.592 7.25 6.944
50 355 4.545 9.308 8.984 8.759 8.489 8.244 7.985 7.784
100 475 4.603 9.422 9.151 8.977 8.754 8.568 8.354 8.201
8
Coin flip, probability of winning 0.45
1 14 0.448 1.669
2 17 1.016 2.717 0.502
5 23 1.93 4.288 2.171 0.694
10 30 2.436 5.413 3.508 2.03 0.929 0.116
20 39 2.734 6.075 4.604 3.5 2.254 1.341 0.532
50 52 2.87 6.379 5.125 4.175 3.225 2.5 2. 1.5
100 54 2.87 6.379 5.125 4.175 3.225 2.5 2. 1.5
Coin flip, probability of winning 0.48
1 19 1.244 3.099 0.868
2 25 1.929 4.301 2.181 0.719 0.02
5 36 2.787 5.806 4.076 2.619 1.457 0.615 0.12
10 48 3.251 6.773 5.32 4.111 2.979 2.06 1.275 0.671
20 64 3.562 7.42 6.311 5.458 4.355 3.564 2.795 2.204
50 91 3.747 7.807 6.995 6.265 5.736 4.989 4.404 3.885
100 115 3.822 7.963 7.165 6.589 6.012 5.344 4.952 4.56
200 129 3.822 7.963 7.209 6.589 6.012 5.436 4.952 4.56
Coin flip, probability of winning 0.49
1 25 1.936 4.302 2.191 0.719 0.03
2 34 2.589 5.459 3.624 2.127 0.996 0.271
5 51 3.323 6.781 5.38 4.127 3.026 2.084 1.308 0.704
10 69 3.713 7.578 6.453 5.465 4.51 3.674 2.895 2.218
20 93 3.983 8.129 7.281 6.589 5.76 5.063 4.385 3.856
50 137 4.17 8.511 7.917 7.369 6.958 6.362 5.882 5.446
100 179 4.25 8.674 8.168 7.703 7.353 6.874 6.483 6.128
200 228 4.291 8.757 8.259 7.871 7.483 7.059 6.765 6.471
500 254 4.291 8.757 8.276 7.871 7.483 7.095 6.765 6.471
9
Coin flip, probability of winning 0.495
1 34 2.594 5.46 3.628 2.129 0.998 0.274
2 47 3.159 6.487 4.979 3.656 2.516 1.58 0.846 0.336
5 71 3.753 7.582 6.489 5.474 4.54 3.687 2.919 2.236
10 98 4.065 8.212 7.362 6.589 5.827 5.136 4.462 3.856
20 134 4.282 8.651 8.022 7.476 6.854 6.308 5.765 5.305
50 203 4.449 8.988 8.554 8.157 7.819 7.382 7.021 6.68
100 270 4.524 9.14 8.791 8.465 8.214 7.861 7.57 7.302
200 354 4.566 9.225 8.92 8.636 8.415 8.122 7.876 7.649
500 502 4.588 9.268 8.975 8.723 8.478 8.233 8.02 7.823
Coin flip, probability of winning 0.499
1 73 3.753 7.582 6.489 5.474 4.54 3.688 2.92 2.237
2 102 4.085 8.213 7.381 6.592 5.844 5.14 4.478 3.86
5 159 4.403 8.823 8.261 7.717 7.191 6.682 6.192 5.72
10 222 4.566 9.15 8.735 8.337 7.94 7.561 7.183 6.822
20 309 4.681 9.381 9.081 8.799 8.5 8.218 7.937 7.674
50 478 4.777 9.574 9.377 9.186 9.013 8.811 8.627 8.449
100 660 4.825 9.669 9.52 9.376 9.251 9.097 8.961 8.83
200 900 4.856 9.731 9.613 9.501 9.405 9.284 9.178 9.079
500 1335 4.879 9.777 9.689 9.594 9.508 9.436 9.349 9.266
European roulette: straight up (single number)
1 105 4.183 8.398 7.646 6.928 6.243 5.592 4.975 4.391
2 140 4.395 8.83 8.26 7.73 7.197 6.701 6.204 5.744
5 201 4.611 9.229 8.852 8.483 8.119 7.764 7.419 7.084
10 252 4.654 9.459 9.123 8.934 8.606 8.422 8.104 7.924
20 295 4.688 9.459 9.324 9.189 8.885 8.671 8.535 8.4
50 332 4.711 9.459 9.324 9.189 9.054 8.919 8.784 8.649
100 347 4.721 9.459 9.324 9.189 9.054 8.919 8.784 8.649
200 355 4.726 9.459 9.324 9.189 9.054 8.919 8.784 8.649
500 359 4.726 9.459 9.324 9.189 9.054 8.919 8.784 8.649
10
European roulette: split bet (2 numbers)
1 76 3.868 7.798 6.792 5.85 4.974 4.165 3.423 2.75
2 103 4.161 8.377 7.606 6.89 6.188 5.54 4.909 4.329
5 152 4.459 8.932 8.42 7.923 7.44 6.972 6.518 6.078
10 198 4.547 9.218 8.781 8.462 8.041 7.734 7.336 7.046
20 249 4.619 9.304 9.054 8.919 8.559 8.261 8.031 7.896
50 302 4.68 9.382 9.102 8.919 8.784 8.649 8.514 8.378
100 327 4.704 9.418 9.14 8.919 8.784 8.649 8.514 8.378
200 341 4.704 9.438 9.159 8.919 8.784 8.649 8.514 8.378
500 349 4.704 9.438 9.171 8.919 8.784 8.649 8.514 8.378
European roulette: street bet (3 numbers)
1 63 3.63 7.352 6.166 5.077 4.085 3.194 2.406 1.724
2 85 3.981 8.032 7.113 6.265 5.45 4.704 3.997 3.356
5 126 4.337 8.695 8.074 7.475 6.898 6.343 5.811 5.302
10 167 4.457 9.054 8.536 8.151 7.658 7.292 6.823 6.475
20 216 4.562 9.192 8.875 8.649 8.246 7.906 7.614 7.407
50 277 4.638 9.311 9.018 8.736 8.514 8.378 8.243 8.108
100 309 4.668 9.358 9.092 8.804 8.537 8.378 8.243 8.108
200 328 4.668 9.401 9.134 8.843 8.576 8.378 8.243 8.108
500 339 4.668 9.401 9.134 8.868 8.601 8.378 8.243 8.108
European roulette: corner bet (4 numbers)
1 54 3.431 6.982 5.655 4.455 3.388 2.456 1.665 1.02
2 74 3.826 7.74 6.701 5.749 4.848 4.034 3.277 2.608
5 110 4.231 8.492 7.781 7.1 6.449 5.826 5.233 4.671
10 147 4.377 8.902 8.314 7.865 7.312 6.891 6.372 5.977
20 192 4.492 9.071 8.72 8.419 7.946 7.557 7.236 6.968
50 256 4.617 9.261 8.932 8.625 8.338 8.108 7.973 7.838
100 293 4.617 9.35 9.011 8.744 8.444 8.178 7.973 7.838
200 315 4.617 9.35 9.083 8.817 8.509 8.242 7.976 7.838
500 329 4.617 9.35 9.083 8.817 8.55 8.283 8.017 7.838
11
European roulette: line bet (6 numbers)
1 44 3.097 6.368 4.822 3.469 2.32 1.386 0.68 0.214
2 60 3.564 7.246 6.011 4.897 3.875 2.976 2.179 1.508
5 90 4.048 8.14 7.276 6.458 5.686 4.96 4.281 3.649
10 121 4.248 8.637 7.935 7.366 6.718 6.193 5.599 5.119
20 160 4.388 8.88 8.429 8.069 7.517 7.058 6.651 6.327
50 221 4.503 9.129 8.752 8.39 8.1 7.769 7.484 7.297
100 263 4.539 9.204 8.937 8.563 8.255 7.988 7.675 7.401
200 292 4.563 9.204 8.937 8.671 8.404 8.137 7.815 7.526
500 309 4.563 9.204 8.937 8.671 8.404 8.137 7.871 7.604
European roulette: 9 numbers
1 35 2.68 5.614 3.826 2.344 1.195 0.411 0.027
2 48 3.226 6.617 5.152 3.862 2.735 1.795 1.037 0.48
5 72 3.805 7.679 6.624 5.641 4.732 3.897 3.139 2.458
10 97 4.068 8.285 7.433 6.722 5.951 5.313 4.623 4.06
20 129 4.275 8.629 8.058 7.574 6.94 6.378 5.888 5.473
50 183 4.389 8.918 8.475 8.042 7.707 7.3 6.976 6.661
100 227 4.462 8.997 8.608 8.342 8.075 7.635 7.254 6.988
200 260 4.462 9.067 8.641 8.342 8.075 7.809 7.542 7.275
500 279 4.462 9.067 8.672 8.342 8.075 7.809 7.542 7.275
European roulette: dozen or column bets (12 numbers)
1 30 2.312 4.96 2.994 1.468 0.448 0.004
2 40 2.913 6.052 4.393 2.984 1.815 0.925 0.317 0.015
5 60 3.574 7.247 6.021 4.899 3.883 2.977 2.184 1.506
10 81 3.873 7.95 6.941 6.118 5.231 4.512 3.75 3.134
20 108 4.108 8.331 7.668 7.103 6.34 5.687 5.145 4.683
50 155 4.288 8.705 8.22 7.685 7.226 6.84 6.453 6.079
100 200 4.333 8.87 8.36 7.942 7.615 7.348 6.979 6.54
200 233 4.37 8.87 8.475 8.081 7.64 7.348 7.082 6.815
500 249 4.37 8.87 8.475 8.081 7.686 7.348 7.082 6.815
12
European roulette: even money bets (18 numbers),
no en prison rule or half of your stake back on 0
1 22 1.635 3.778 1.586 0.265
2 30 2.311 4.966 3.001 1.473 0.457 0.0001
5 44 3.102 6.377 4.833 3.482 2.333 1.398 0.689 0.219
10 59 3.526 7.249 5.989 4.9 3.866 2.977 2.181 1.502
20 80 3.812 7.836 6.89 6.108 5.193 4.442 3.713 3.111
50 114 4.011 8.245 7.535 6.948 6.487 5.767 5.267 4.837
100 146 4.087 8.402 7.784 7.27 6.876 6.278 5.807 5.5
200 189 4.109 8.447 7.883 7.364 6.876 6.481 6.087 5.692
Full pay Jacks or Better video poker
1 108 4.28 8.577 7.892 7.226 6.579 5.953 5.35 4.769
2 140 4.426 8.867 8.318 7.785 7.262 6.756 6.26 5.783
5 201 4.595 9.197 8.805 8.42 8.043 7.672 7.308 6.952
10 264 4.683 9.393 9.084 8.802 8.501 8.227 7.935 7.669
20 344 4.746 9.516 9.293 9.092 8.847 8.626 8.411 8.219
50 480 4.799 9.622 9.456 9.294 9.148 8.988 8.84 8.695
100 605 4.825 9.675 9.534 9.398 9.274 9.149 9.028 8.904
Full pay Jacks or Better video poker with doubling once on every win
1 149 4.45 8.912 8.385 7.871 7.369 6.88 6.404 5.941
2 196 4.576 9.163 8.754 8.356 7.962 7.578 7.2 6.833
5 284 4.71 9.423 9.141 8.861 8.586 8.314 8.046 7.782
10 374 4.774 9.568 9.346 9.144 8.926 8.727 8.513 8.319
20 485 4.819 9.655 9.497 9.35 9.174 9.015 8.861 8.719
50 680 4.859 9.733 9.614 9.499 9.397 9.279 9.175 9.075
100 860 4.878 9.772 9.673 9.576 9.49 9.392 9.303 9.217
13