Loss rebates

December 27, 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.

Several casinos have promotions where they give the player 10% of his

losses each month or on certain days, and VIP Casino also gives a 1% bonus

on the winnings. In this paper we investigate the value of these bonuses and

the optimal strategy for them. We assume that the bonus is withdrawable

without further wagering requirements.

2 The questions

If the player’s initial bankroll is m, his current bankroll is n, and the casino

oﬀers a 10% rebate on losses, what strategy should the player follow to maxi-

mize his expectation and what is this expectation a

n

? What if the casino also

oﬀers a 1% bonus on winnings?

1

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,

n can be a half integer, but the stake is always an integer and let us also

deﬁne a

n

= n + 0.1m for n < 0.

Deﬁne the value of the promotion to be 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

These problem can be solved by the same iterative method as the phantom

bonus problem, so the details are not repeated here. For the simple 10%

rebate the starting values need to be set to a

n,0

= 0.9n+0 .1 m for 0 ≤ n ≤ m,

and a

n,0

= n for n > m. For the VIP Casino problem, set a

n,0

= 1.01n −

0.01m for n > m. There are a priori bounds on how high the player should

aim, he should stop if his bankroll exceeds

³

m

X

u

i

<0

p

i

u

i

´.³

10

`

X

i=1

p

i

u

i

´

in the

case of the loss rebate, or

³

0.11m

X

u

i

<0

p

i

u

i

´.³

1.01

`

X

i=1

p

i

u

i

´

in the case of the

VIP Casino promotion.

4 The results

The tables in the Appendix show the results for various games. The player’s

initial bankroll was always taken to be m = 100, this way the maximum

amount of the loss rebate is 10 and the ﬁgures can be compared to those

obtained for the phantom bonus. 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.

The ﬁrst table for each game is for the 10% loss rebate, the second table

for the VIP Casino promotion. In each table the ﬁrst column shows the max-

imum bet, the second and third columns the smallest and largest bankrolls,

2

respectively, at which it is still in the player’s interest to play, rather than to

cash in. If the number in the second column is 1, it means that the player

should play until his bankroll exceeds the number in the third column or

go bust. The fourth column shows the value of the promotion at the initial

bankroll.

These calculations required much more computer time than for the phan-

tom bonus, the most diﬃcult case was Cryptologic Double Bonus video poker

with 1% bonus on winnings with b = 1, this took about 16 hours on a com-

puter with a 2.2 GHz Intel Pentium processor. This is probably due to the

fractions involved. The calculations had to be done using exact arithmetic,

the standard 16 digits numerical precision was not enough, it sometimes

caused the iterations to diverge. Some calculations were done using smaller

numbers (e.g., m = b = 10 instead of m = b = 100) and then scaled up. As

a consequence, the numbers in the second and third columns are not always

exact, but they are always accurate within one maximum bet.

There are some new phenomena with the loss rebate which do not occur

with the phantom bonus. It is clearly never in the player’s interest to stop

if his bankroll does not exceed the phantom bonus. With the loss rebate, if

the house edge is too high, the player should simply not play. Such games

are the tie bet in baccarat and the coin ﬂip with the probability of winning

0.45, these games are not included in the tables. If it is not in the player’s

interest to play at the initial bankroll, he should not play at all. Even for

other games, the strategy is not always “aim for a high target or go bust”,

but “aim for a high target or stop at a stop loss limit greater than 0”. For

example, when playing blackjack with the 10% loss rebate, the player should

stop when his losses exceed 7.5 maximum bets or when his winnings exceed

7 maximum bets, with the additional 1% bonus on proﬁts, the limits are 8

maximum bets either way. This assumes that the initial bankroll is more

than 7.5 or 8 maximum bets. The additional 1% bonus on winnings calls for

a slightly more aggressive strategy in general.

It is interesting to note that when the stop loss limit it 0, the target

bankroll for the 10% loss rebate on 100 units is the same as for the 10 unit

phantom bonus, the calculations in the next section will explain this. This

also shows that the loss rebate requires a much less aggressive strategy.

Many observations are the same as for the phantom bonus. The value of

the promotions, and also the upper bound on the bankroll at which the player

should stop playing both increase with b, while the lower bound decreases

with b. The value of these promotions is maximal at the initial bankroll.

3

The house edge is not the primary factor in determining the value of the

promotions, 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%. The best game is

Cryptologic Double Bonus video poker, which combines both a low house

edge (0.06%) and the possibility of big wins. The value of the VIP Casino

promotion for this game with large enough bets may exceed 10, of course,

this comes at the expense of extremely large variance.

The value of the loss rebate is usually less than that of the phantom bonus,

it is possible to realize about 90% of the nominal value of the phantom bonus

on most games with high enough maximum bet, while only video pokers, the

single number bet in roulette and the coin ﬂip with the probability of winning

0.499 achieve more than 70% with the loss rebate. The extra 1% bonus on

winnings increases the value of the promotion by about 15%.

5 Details of the strategy

Let kb be the largest and k

0

b the smallest multiple of the maximum bet

at which the player should still play. They can be calculated by a similar

method as in the case of phantom bonuses.

Assume that all the payouts are integers ≥ −1. Empirical evidence

suggests that if player’s bankroll is a multiple of b, he should always bet

the maximum, b. Let u = max{u

i

|1 ≤ i ≤ `}. The numbers a

jb

(k

0

≤

j ≤ k) satisfy the recurrence relation a

jb

=

`

P

i=1

p

i

a

(j+u

i

)b

together with the

boundary conditions a

(k

0

−1)b

= 0.9(k

0

− 1)b + 0.1m and a

(k+1)b

= (k + 1)b,

a

(k+2)b

= (k + 2)b,. . . , a

(k+u)b

= (k + u)b in the case of the 10% loss re-

bate. In the case of the VIP Casino promotion, the equations have to be

changed to a

(k+1)b

= 1.01(k + 1)b − 0.01m, a

(k+2)b

= 1.01(k + 2)b − 0.01m,. . . ,

a

(k+u)b

= 1.01(k + u)b − 0.01m.

Given a pair of positive integers t

0

≤ m/b and t ≥ m/b, we can deﬁne

4

α

jb,t

0

,t

by substituting it for a

jb

in the above equations. The pair is suitable

if a

t

0

b,t

0

,t

≥ 0.9t

0

b + 0.1n and a

tb,t

0

,t

≥ tb in the case of the 10% loss rebate,

while in the case of the VIP Casino promotion, the second inequality has to

be replaced by a

tb,t

0

,t

≥ 1.01tb − 0.01m.

k

0

is the smallest possible value of t

0

and k is the largest possible value

of t in a suitable pair, they can be found by searching, but there does not

appear to be any method of ﬁnding them directly.

If k

0

= 1, meaning that the player should play until he either reaches his

goal or goes bust, the 10% loss rebate with initial bankroll 100 units and the

phantom bonus of 10 units give the same equations for k, this explains the

observation in the previous section.

If the maximum bet is large enough or there is no maximum bet and all

the payouts are integers greater than or equal to −1, then the player should

bet his whole bankroll until he either reaches the goal given in Section 3 or

goes bust.

It might be expected that the player should always bet the table limit

or his whole bankroll, and just like in the case of the phantom bonus, this

is usually correct, but not always. With the phantom bonus, the correct

strategy was always to bet either the maximum or diﬀerence between the

current bankroll and the largest multiple of the maximum bet not exceeding

it, so that if the player lost, his bankroll would be an integer multiple of

the maximum bet. There is a tendency to favor multiples of the maximum

bet with the loss rebate, too, but the bet size may be such that the player’s

bankroll will be equal to or just slightly greater than an integer multiple

of the maximum bet if he wins. For even money bets, 1/2, 1/4 or 1/8 of

the maximum bet may also be a desirable goal. The tables below show the

bankroll and optimal bet size for various games to illustrate these principles.

Roulette, even money bets, b = 100, 10% loss rebate

n 52 53 54 55 . . . 70 71 72 73 74 75

bet 48 47 46 45 . . . 30 29 28 27 26 75

For 52 ≤ n ≤ 74, the player aims to get to 100 if he wins.

Roulette, column or dozen bets, payout 2:1, b = 100, 10% loss rebate

n 21 22 23 24 . . . 33 34 35 36 37 38 39 40 41 42

bet 6 5 5 24 . . . 33 33 33 32 32 31 31 30 30 29

For 34 ≤ n ≤ 42, the player aims to get to 100 or 101 if he wins.

5

Coin ﬂip, probability of winning 0.49, b = 100, 10% loss rebate

n 34 35 36 37 38 39 40

bet 16 15 14 13 12 39 40

Coin ﬂip, probability of winning 0.49, b = 100, VIP Casino promotion

n 27 28 29 30 31 32 33 34

bet 23 22 21 20 19 18 17 34

In the above two cases, for 34 ≤ n ≤ 38 and for 27 ≤ n ≤ 33, respectively,

the player aims to get 50 = b/2 if he wins.

Baccarat, player bet, b = 100, VIP Casino promotion

n 9 10 11 12 13 14 15 16 17

bet 4 10 11 12 12 11 10 9 17

For 13 ≤ n ≤ 16, the player aims to get 25 = b/4 if he wins. There is a

similar example with b = 200, the optimal bet with a bankroll of 13 is 12.

The last two tables show cases where the optimal strategy cannot be

explained easily.

Roulette, split bet (2 numbers), payout 18:1, b = 50, VIP Casino promotion

n 50 51 52 53 . . . 60 61 62 63 64 65 66 67 68 69

bet 50 1 50 50 . . . 50 50 1 50 50 50 50 17 18 19

Roulette, corner bet (4 numbers), payout 9:1, b = 100, VIP Casino promotion

n 101 102 103 104 105 . . . 109 110 111 112 113

bet 100 2 3 4 100 . . . 100 2 3 100 100

6 The eﬀects of using the wrong strategy

As the previous section shows, the optimal betting strategy is quite far from

obvious. Fortunately, the simple strategy of betting the whole bankroll or the

maximum allowed, whichever is smaller, is quite good, and in the few cases

I looked at it never caused a loss of more than 0.5% of the initial bankroll

compared to the optimal strategy.

The other element of the strategy is the correct target and stop loss limit.

Too timid a strategy will fail to realize the value of the loss rebate, while too

agressive a strategy will also cause the player to lose because of the house

edge. The correct target is often around three times the initial bankroll, if

the player choose to double or to quadraple his bankroll instead, he may give

up 0.5% of his initial bankroll, but this can mean 10–15% of the value of

the loss rebate. Too agressive a strategy is more dangerous, playing on after

6

reaching the correct target instead of cashing in can cost 5% or more of the

initial bankroll and even wipe out any expected proﬁt from the loss rebate.

7 Appendix

Baccarat (8 decks): player bet

1 99 101 0.048141

2 98 102 0.0962819

5 93 107 0.240705

10 86 114 0.48141

20 71 128 0.962819

50 28 170 2.40705

100 12 225 4.0399

200 8 286 4.85648

500 7 371 4.85648

1 99 101 0.057868

2 97 103 0.115736

5 92 107 0.28934

10 84 115 0.57868

20 67 131 1.15736

50 22 178 2.8934

100 9 236 4.68845

200 6 303 5.58615

500 5 404 5.60089

Blackjack

1 92.5 107 0.201784

2 85 114 0.403568

5 62.5 135 1.00892

10 25 170 2.01595

20 1 235 3.54223

50 1 350 5.35131

100 1 460 6.35234

1 92 108 0.242936

2 84 116 0.485872

5 60 140 1.21468

10 20 180 2.42494

20 1 245 4.14159

50 1 370 6.11084

100 1 500 7.25468

Coin ﬂip, probability of winning 0.48

1 100 100 0.012

2 100 100 0.024

5 99 101 0.06

10 98 102 0.12

20 96 104 0.24

50 88 111 0.6

100 81 123 1.2

1 100 100 0.0168

2 100 100 0.0336

5 99 101 0.084

10 97 102 0.168

20 94 105 0.336

50 85 114 0.84

100 76 129 1.68

7

Coin ﬂip, probability of winning 0.49

1 100 100 0.031

2 99 101 0.062

5 96 104 0.155

10 92 108 0.31

20 83 116 0.62

50 56 141 1.55

100 34 179 3.1

1 100 100 0.0359

2 99 101 0.0718

5 96 104 0.1795

10 91 109 0.359

20 81 118 0.718

50 51 146 1.795

100 27 190 3.59

Coin ﬂip, probability of winning 0.495

1 98 102 0.0645228

2 96 104 0.129046

5 90 110 0.322614

10 79 121 0.645228

20 57 142 1.29046

50 11 203 3.13008

100 4 270 4.73984

1 98 102 0.0790729

2 96 104 0.158146

5 88 111 0.395365

10 76 123 0.790729

20 51 146 1.58146

50 7 212 3.71214

100 3 285 5.39322

Coin ﬂip, probability of winning 0.499

1 88 112 0.328972

2 75 124 0.657944

5 36 162 1.64486

10 1 222 3.17062

20 1 309 4.75243

50 1 478 6.4383

100 1 660 7.39252

1 86 113 0.396093

2 72 127 0.792186

5 30 168 1.98047

10 1 232 3.7247

20 1 323 5.42949

50 1 501 7.22636

100 1 692 8.23941

European roulette: straight up (single number)

1 67 121 0.823681

2 33 143 1.64736

5 1 201 3.69786

10 1 252 5.2068

20 1 295 6.15924

50 1 332 6.80058

100 1 347 7.02703

1 64 124 0.991716

2 27 149 1.98343

5 1 212 4.28873

10 1 268 5.94317

20 1 317 7.00561

50 1 358 7.7206

100 1 376 7.97297

8

European roulette: split bet (2 numbers)

1 84 110 0.400049

2 68 121 0.800097

5 20 152 2.00024

10 3 198 3.63066

20 1 249 5.14822

50 1 302 6.31848

100 1 327 6.75676

1 83 112 0.481785

2 65 124 0.96357

5 13 160 2.40893

10 1 209 4.21609

20 1 265 5.88137

50 1 324 7.187

100 1 352 7.67568

European roulette: street bet (3 numbers)

1 90 106 0.258869

2 80 113 0.517739

5 50 133 1.29435

10 9 167 2.5842

20 2 216 4.25364

50 1 277 5.85099

100 1 309 6.48649

1 89 107 0.311842

2 78 115 0.623684

5 44 138 1.55921

10 7 176 3.0839

20 1 228 4.88587

50 1 296 6.66947

100 1 332 7.37838

European roulette: corner bet (4 numbers)

1 93 104 0.188205

2 86 109 0.37641

5 64 124 0.941024

10 28 149 1.88205

20 6 192 3.491

50 2 256 5.3981

100 1 293 6.21622

1 92 105 0.226707

2 84 111 0.453414

5 60 127 1.13354

10 20 155 2.26707

20 3 203 4.06435

50 1 272 6.16801

100 1 314 7.08108

European roulette: line bet (6 numbers)

1 96 103 0.11745

2 92 106 0.234901

5 79 115 0.587252

10 57 130 1.1745

20 18 160 2.34901

50 5 221 4.53616

100 3 263 5.67568

1 96 103 0.141873

2 91 106 0.283746

5 76 117 0.709364

10 52 134 1.41873

20 15 168 2.83746

50 3 234 5.21329

100 2 282 6.48649

9

European roulette: 9 numbers

1 98 101 0.0707495

2 96 103 0.141499

5 88 108 0.353748

10 76 117 0.707495

20 52 134 1.41499

50 14 183 3.35281

100 9 227 4.86486

1 98 102 0.084644

2 95 104 0.169288

5 87 110 0.42322

10 73 120 0.84644

20 46 140 1.69288

50 12 194 3.90175

100 6 240 5.59459

European roulette: dozen or column bets (12 numbers)

1 99 101 0.0470651

2 98 102 0.0941302

5 93 105 0.235325

10 86 111 0.470651

20 71 122 0.941302

50 36 156 2.35325

100 21 200 4.05405

1 99 101 0.0571817

2 97 102 0.114363

5 92 106 0.285908

10 84 112 0.571817

20 67 125 1.14363

50 30 164 2.85908

100 16 210 4.7027

European roulette: even money bets (18 numbers),

no en prison rule or half of your stake back on 0

1 100 100 0.0243243

2 100 100 0.0486486

5 98 102 0.121622

10 96 104 0.243243

20 91 109 0.486486

50 76 123 1.21622

100 52 147 2.43243

1 100 100 0.0291892

2 99 101 0.0583784

5 97 102 0.145946

10 94 105 0.291892

20 88 111 0.583784

50 70 128 1.45946

100 48 154 2.91892

Full pay Jacks or Better video poker

1 67 121 0.725652

2 34 143 1.4513

5 1 201 3.27731

10 1 264 4.6667

20 1 344 5.81142

50 1 480 6.98917

100 1 605 7.67061

1 62 124 0.908339

2 24 149 1.81668

5 1 212 3.88772

10 1 275 5.377

20 1 360 6.60106

50 1 505 7.83973

100 1 640 8.56562

10

Full pay Jacks or Better video poker with doubling once on every win

1 26 150 1.63572

2 1 196 3.06156

5 1 284 4.90873

10 1 374 6.03806

20 1 486 6.93285

50 1 680 7.84278

100 1 860 8.33571

1 14 157 2.04428

2 1 207 3.65335

5 1 299 5.63043

10 1 390 6.83604

20 1 510 7.78084

50 1 720 8.74025

100 1 910 9.2598

Cryptologic Double Bonus Video Poker

1 1 545 7.1477

2 2 718 7.80724

5 1 1025 8.46746

10 1 1335 8.8471

20 1 1710 9.13194

50 1 2280 9.39536

100 1 2740 9.52055

1 1 574 8.00639

2 2 756 8.70235

5 1 1080 9.39366

10 1 1410 9.79079

20 1 1810 10.0896

50 1 2420 10.3664

100 1 2920 10.5039

11