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