•TWO-TIER BETTING
Introduction
Two-tier betting involves making higher variance (usually large) bets until reaching a target gain, then switching to normal variance bets to complete the remaining wagering requirement. This strategy increases EV over a typical smaller bet size, with both cashable and sticky bonuses. The EV increase is large for sticky bonuses and cashable bonuses where you are expected to lose a large portion of the bonus during wagering, but the increase may be negligible for cashable bonuses where you are expected to lose little of the bonus during wagering.
Cashable Bonuses
The EV of a cashable post-wager bonus or other cashable bonus with ~0% chance of busting is Bonus - Wagering Requirement * House Edge (per unit wagered). However, when there is a chance of busting before completing wagering, the formula for EV changes to:
Bonus - Average Wagering * House Edge (per unit wagered)
As you increase bet size, the chance of busting before completing wagering increases; so the average wagering decreases, and the average gain increases. A summary is below for a $100 bonus with 60xB wagering on French Roulette (standard Microgaming Clearplay bonus). The change in average gain is negligible for $1-2 bets since you are expected to complete most of the wagering during the times that you bust. However, the change is quite large for $20+ bets when you have a realistic chance of busting early on.
Average
Gain & Win/Bust Chance with Fixed Bet Size
for $100 Bonus with 60xB Wagering on French Roulette
| Bet Size | Chance
of Bust |
Chance
of Win |
Average
Gain |
| Post-Wager Bonus | 0% |
Varies |
$19 |
| $1 Flat Bets | 9% |
59% |
$20 |
| $2 Flat Bets | 21% |
56% |
$23 |
| $5 Flat Bets | 39% |
50% |
$32 |
| $10 Flat Bets | 53% |
43% |
$41 |
| $20 Flat Bets | 64% |
35% |
$51 |
| $50 Flat Bets | 75% |
24% |
$64 |
| $100 Flat Bets | 81% |
18% |
$73 |
| $200 Flat Bets | 87% |
13% |
$79 |
| All Full-bankroll Bets | 97% |
3% |
$87 |
Average gain can be increased to a similar degree with a smaller bust risk by making large bets to quickly meet a target gain or bust, then switching to smaller bets after reaching the a target gain. If you initially make large bets of the values listed in the tables above, then you can often nearly match the listed average gain for that fixed bet size, even though bet size is much smaller for the majority of the wagering. EV is high because of the chance of ending wagering early on, and bust risk is lower than using all fixed bets of that size because of the reduced chance of busting later on.
A summary of average gain with various two-tier betting strategies on the $100 60xB French Roulette bonus described earlier is in the table below. These results were obtained using the two-tier bet size simulator. You can use this program to simulate other bonuses. With a target of 2x(B+D), the average gain is slightly under $60. The ~$40 expected loss with a 2x target and large bets is roughly half the ~$80 expected loss with small flat bets since average wagering is roughly cut in half. With a 4x target and $200 bets, the expected loss is roughly 1/4 the loss with small flat bets. However, with $100 or smaller bets and a 4x target, the average gain instead approaches the fixed bet size numbers in the table above. For example, the table above lists an average gain of $63 with $50 bets. Regardless of how high the target is, the average gain will not exceed $63 with $50 bets. Similarly the average gain will not exceed $74 with $100 bets. The average gain will usually approach either Bonus - (house edge * wagering requirement) / target ratio, or the numbers in the table above for a flat fixed bet size, whichever is smaller.
At a given target, full-bankroll bets results in the highest average gain. However, with a 2x target you may prefer smaller bets than full-bankroll since there is little difference in average gain, there is less chance of a casino labeling you as advantage player, and you typically get to play more hands. Deciding on the target size is a matter of risk tolerance. Would you rather have an average gain of $57 and busting half the time, an average gain of $78 and busting 3/4 of the time, or an average gain of $84 and busting 7/8 of the time? Different players will give different answers.
Average
Gain & Win/Bust Chance with Two-Tier Betting
for $100 Bonus with 60xB Wagering on French Roulette
| Initial Bet Size (with $1 final bet size) |
Target |
Chance
of Bust |
Chance
of Win |
Average
Gain |
| $25 | $400 |
55% |
45% |
$54 |
| $50 | $400 |
53% |
47% |
$55 |
| $100 | $400 |
52% |
48% |
$56 |
| $200 | $400 |
51% |
49% |
$57 |
| $50 | $800 |
74% |
24% |
$63 |
| $100 | $800 |
77% |
23% |
$72 |
| $200 | $800 |
76% |
24% |
$78 |
| Repeating Full-bankroll | $800 |
76% |
24% |
$78 |
| $100 | $1600 |
81% |
18% |
$74 |
| $200 | $1600 |
86% |
14% |
$82 |
| Repeating Full-bankroll | $1600 |
88% |
12% |
$84 |
Blackjack Bonuses
With blackjack bonuses, the average gain is more complex. The expected loss over a wagering requirement is often small with blackjack bonuses, so the increase in EV with larger bets may be negligible. Full bankroll bets have a high house edge due to not being able to double or split. If you lose enough hands such that your bet size is a full bankroll bet, then average gain decreases. A summary of average gains with a fixed bet size is below, for a $100 bonus with 40xB wagering on blackjack. House edge of the blackjack game is assumed to be 0.48% and house edge per unit wagered 0.48%/1.13 = 0.42%. The average gain levels off around $89 for $20 to $100 bets, and is much lower for $200 bets (full-bankroll initial bet). By far the highest average gain occurs with all half-bankroll bets. Using half-bankroll permits large bets and avoids the full-bankroll house edge increase on losses.
Average Gain
& Win/Bust Chance with Fixed Bet Size
for $100 Bonus with 40xB Wagering on Blackjack
| Bet Size | Chance
of Bust |
Chance
of Win |
Average
Gain |
| Post-Wager Bonus | 0% |
Varies |
$83 |
| $1 Flat Bets | 1% |
89% |
$83 |
| $2 Flat Bets | 6% |
80% |
$83 |
| $5 Flat Bets | 22% |
68% |
$85 |
| $10 Flat Bets | 37% |
57% |
$87 |
| $20 Flat Bets | 44% |
53% |
$88 |
| $50 Flat Bets | 69% |
30% |
$89 |
| $100 Flat Bets | 78% |
21% |
$89 |
| All Half-bankroll Bets | 86% |
13% |
$96 |
| $200 Flat Bets | 85% |
15% |
$82 |
| All Full-bankroll Bets | 97% |
3% |
$63 |
The table below shows average gain with two-tier betting on the blackjack bonus described above, instead of a fixed bet size. Again, these results were obtained using the two-tier bet size simulator. The highest average gain occurs with repeating half-bankroll bets until reaching the target gain. This strategy may not be practical due to max bet limits. The next highest gains occurs with flat bets of $100 when possible and betting half of your bankroll when not possible. A sequence of bets might go $100 (lose, balance=$100), $50 (lose, balance =$50), $25 (double and win $50, balance = $100), $50 (win, balance = $150), $75 (win, balance = $225), $100 (bet $100 instead of half bankroll since balance is >$200). You could further improve the strategy by betting half-bankroll when under the max table limit and betting the max table limit when over. To keep things simple and speed up wagering, you may prefer simply using a flat bet until reaching the target gain. The difference in EV between flat bets until reaching the target and varying half-bankroll bets is small for a 2x(B+D) target. In this example, a higher target increases EV when half-bankroll bets are made on losses instead of maintaining a fixed bet size. However, with a fixed bet size, EV is similar for all of the listed target sizes, approaching the fixed bet size average gains in the table above.
Average Gain &
Win/Bust Chance with Two-Tier Betting
for $100 Bonus with 40xB Wagering on Blackjack
| Initial Bet Size (with $10 final bet size) |
Half-bankroll
on loss? |
Target |
Chance
of Bust |
Chance
of Win |
Average
Gain |
| $25 | No |
$400 |
52% |
46% |
$87 |
| $50 | No |
$400 |
55% |
44% |
$89 |
| $100 | No |
$400 |
56% |
43% |
$89 |
| $50 | No |
$800 |
69% |
30% |
$89 |
| $100 | No |
$800 |
76% |
24% |
$89 |
| $100 | No |
$1200 |
78% |
21% |
$89 |
| $25 | Yes |
$400 |
51% |
46% |
$88 |
| $50 | Yes |
$400 |
54% |
44% |
$90 |
| $100 | Yes |
$400 |
54% |
44% |
$91 |
| Repeating Half-bankroll | Yes |
$400 |
60% |
39% |
$93 |
| $50 | Yes |
$800 |
66% |
31% |
$93 |
| $100 | Yes |
$800 |
71% |
26% |
$94 |
| Repeating Half-bankroll | Yes |
$800 |
80% |
20% |
$96 |
| $100 | Yes |
$1200 |
72% |
24% |
$94 |
| Repeating Half-bankroll | Yes |
$1200 |
84% |
15% |
$96 |
Game Choice
The optimal game choice for initial bets to reach a target game is often not obvious. The tables below show average gain with various game choices for the $100 40xB blackjack bonus described in the blackjack section above. $10 flat bets on blackjack are used to complete the remaining wagering after reaching the target gain. The first table is for a 2x(B+D) target, and the second is for a 4x(B+D) target. In the tables, "half-bankroll" and "full-bankroll" indicate repeating half/full-bankroll bets until reaching the target gain. Note that the average gain with $10 flat bets on this bonus is $87. For a strategy to increase EV on this bonus, the average gain must be above $87.
In this example, the highest average gain occurs with betting the full $200 bonus on a single hand of Jacks or Better (or other low house edge video poker game). Unfortunately this is usually not a practical option. A full-bankroll bet is likely to exceed max bet limits, and the listed average gain includes extremely rare outcomes, like hitting a royal flush on the full-bankroll bet. If you do not play enough bonuses to hit rarer outcomes on a single hand, then the average gain is significantly lower.
For typical bonuses, I recommend choosing blackjack. If bet size is reduced on losses such that large full-bankroll bets are not made, then blackjack has a high average gain for a given bust risk. Reducing bet size with each loss can lead to small bets and time consuming wagering. To avoid this problem, you may wish to set a minimum bet limit, such as 1/8 of bankroll. If you reach the minimum bet limit, then make a full-bankroll bet or choose a different game. The lower the limit, the less effect a full-bankroll bet will have on average gain.
Baccarat and French roulette are also good choices. They have a high average gain and have a more simple betting strategy and faster play than blackjack (with reducing bet size on losses). For bonuses with a high expected loss during wagering, the difference in EV between baccarat / French Roulette and blackjack is far smaller. At a given bust level, the difference in EV may become negligible. It is counterintuitive that player bets on baccarat have a higher average gain than banker bets, since banker bets have a lower house edge. The slight increased bust risk with player bets results in a greater decrease in average wagering, and a greater average gain according the formula listed earlier. Video poker can also be a good choice, if the double feature is used to reach high gains.
Average Gain with 2x(B+D) target for $100 Bonus with 40xB Wagering on Blackjack
| Game |
Initial Bet Size | Half-bankroll
on loss? |
Target |
Chance
of Bust |
Chance
of Win |
Average
Gain |
| Jacks or Better | $200 | -- |
$400 |
70% |
29% |
$94 |
| Blackjack | Half-bankroll | Yes |
$400 |
60% |
39% |
$93 |
| Blackjack | $100 | Yes |
$400 |
54% |
44% |
$91 |
| Jacks or Better | $10 | -- |
$400 |
62% |
37% |
$90 |
| Blackjack | $100 | No |
$400 |
56% |
43% |
$89 |
| Baccarat: Player | $200 | -- |
$400 |
55% |
44% |
$89 |
| French Roulette | $200 | -- |
$400 |
55% |
44% |
$89 |
| Baccarat: Banker | $200 | -- |
$390 |
54% |
44% |
$88 |
| Single Zero Roulette: 1:1 bet |
$200 | -- |
$400 |
55% |
43% |
$86 |
| 3-Card Poker: PP | $200 | -- |
$400 |
76% |
23% |
$85 |
| Pai Gow Poker | $200 | -- |
$400 |
55% |
43% |
$81 |
Average Gain with 4x(B+D) target for $100 Bonus with 40xB Wagering on Blackjack
| Game |
Initial Bet Size | Half-bankroll
on loss? |
Target |
Chance
of Bust |
Chance
of Win |
Average
Gain |
| Blackjack | Half-bankroll | Yes |
$800 |
80% |
20% |
$96 |
| Blackjack | $100 | Yes |
$800 |
71% |
26% |
$94 |
| Baccarat: Player | Full-bankroll | -- |
$800 |
76% |
24% |
$91 |
| French Roulette | Full-bankroll | -- |
$800 |
76% |
24% |
$91 |
| Baccarat: Banker | Full-bankroll | -- |
$760 |
76% |
24% |
$91 |
| Single Zero Roulette: 3:1 bet |
Full-bankroll | -- |
$800 |
76% |
24% |
$90 |
| Blackjack | $100 | No |
$800 |
76% |
24% |
$89 |
| Single Zero Roulette: 1:1 bet |
Full-bankroll | -- |
$800 |
76% |
24% |
$86 |
| Pai Gow Poker | Full-bankroll | -- |
$760 |
76% |
24% |
$78 |
Sticky Bonuses
The EV of a sticky bonus is Chance of Bust * Bonus - Average Wagering * House Edge (per unit wagered). The only difference between this sticky EV formula and the cashable EV formula above is the "chance of bust" multiplier. If you choose a strategy such that chance of bust is very low, then the EV of a sticky bonus will be low. If you choose a strategy such that the chance of bust is very high, then the sticky bonus EV will approach the EV of a cashable bonus with the same wagering.
The table below shows average gain with average gain with a fixed bet size for a $100 sticky bonus with 24xB wagering on blackjack (typical IOG sticky). As bet size increases, average gain increases. This increase in average gain relates to the increased chance of bust and reduced average wagering with a larger bet size. $200 bets (full-bankroll initial bet) have a much larger difference between chance of bust and average gain than other bet sizes because of the increased house edge with not being able to double or split. The average gain values are well below the numbers for the 40xB cashable bonus in the blackjack section. The difference in average gain is roughly bonus * (1- chance of bust), as expected from the formula above. For example, the average gain with $100 flat bets on the sticky bonus is $64, and the average gain with $100 flat bets on the cashable bonus is $89. The difference between these numbers is $25, which is roughly (1- chance of bust) * $100 = $27. The values are not exactly the same because the two bonuses have different wagering requirements.
Average Gain
& Win/Bust Chance with Fixed Bet Size
for $100 Sticky Bonus with 24xB Wagering on Blackjack
| Bet Size | Chance
of Bust |
Chance
of Win |
Average
Gain |
| $1 Flat Bets | 1% |
43% |
-$10 |
| $2 Flat Bets | 5% |
44% |
-$7 |
| $5 Flat Bets | 15% |
46% |
$5 |
| $10 Flat Bets | 28% |
46% |
$19 |
| $20 Flat Bets | 44% |
43% |
$35 |
| $50 Flat Bets | 62% |
33% |
$52 |
| $100 Flat Bets | 73% |
25% |
$64 |
| All Half-bankroll Bets | 81% |
18% |
$78 |
| $200 Flat Bets | 81% |
18% |
$65 |
| All Full-bankroll Bets | 94% |
6% |
$60 |
Average gain is often negative with small bets, so using large or high variance bets at some point during the wagering is critical. Two-tier betting can be useful with sticky bonuses. At a given chance of bust, average gain is higher with two-tier betting than with flat bets since average wagering is lower. This difference is in average gain between two-tier betting and flat bets is often small. When using a bet size such that there is a 50% chance of bust, EV is only approximately ~$2 higher with two-tier betting (on blackjack) than with a fixed bet size. This small difference relates to the small expected loss during the 24xB wagering. With a greater expected loss during wagering, the difference in average gain between two-tier betting and a fixed bet size increases.
The tables below show the average gain with two-tier betting for a $100 sticky bonus with 24xB wagering on blackjack (typical IOG sticky). The first table shows a target of 2x(B+D). The second shows a target of 4x(B+D). Again, these results were obtained using the two-tier bet size simulator. Note that the order of games differs from the tables in the game choice section above. 3-card poker has the highest average gain among as listed in the 2x target below, but 3-card poker is listed near the bottom of the corresponding cashable bonus table. The higher average gain with 3-card poker for sticky bonuses relates to the higher chance of bust. However, I do not recommend choosing 3-card poker since the average gain is tied to rare events on a single hand, and the difference between average gain and chance of bust is greater for 3-card poker than for other games with a comparable bust level due to the higher house edge.
Instead I recommend large or half-bankroll bets on blackjack until reaching a target gain (again). At a given bust level, average gain is higher with half-bankroll bets on blackjack than with other strategies. Select a target corresponding to your risk tolerance. If blackjack is not available or you wins to choose another game for other purposes, some good options are large bets on baccarat, french roulette, or video poker. With video poker, use the double feature to reach a target gain.
Average Gain with 2x(B+D) target for $100 Sticky Bonus with 24xB Wagering on Blackjack
| Game |
Initial Bet Size | Half-bankroll
on loss? |
Target |
Chance
of Bust |
Chance
of Win |
Average
Gain |
| 3-Card Poker: PP | $200 | -- |
$400 |
75% |
23% |
$69 |
| Jacks or Better | $200 | -- |
$400 |
70% |
28% |
$66 |
| Blackjack | Half-bankroll | Yes |
$400 |
59% |
39% |
$55 |
| Jacks or Better | $10 | -- |
$400 |
63% |
36% |
$54 |
| Blackjack | $100 | Yes |
$400 |
54% |
42% |
$47 |
| Blackjack | $100 | No |
$400 |
54% |
42% |
$46 |
| Baccarat: Player | $200 | -- |
$400 |
52% |
43% |
$44 |
| French Roulette | $200 | -- |
$400 |
52% |
43% |
$44 |
| Baccarat: Banker | $200 | -- |
$390 |
51% |
43% |
$43 |
| Single Zero Roulette: 1:1 bet |
$200 | -- |
$400 |
52% |
42% |
$43 |
| Pai Gow Poker | $200 | -- |
$390 |
52% |
42% |
$39 |
Average Gain with 4x(B+D) target for $100 Sticky Bonus with 24xB Wagering on Blackjack
| Game |
Initial Bet Size | Half-bankroll
on loss? |
Target |
Chance
of Bust |
Chance
of Win |
Average
Gain |
| Jacks or Better | $200 | -- |
$800 |
84% |
14% |
$81 |
| 3-Card Poker: PP | $200 | -- |
$800 |
87% |
13% |
$79 |
| Blackjack | Half-bankroll | Yes |
$800 |
81% |
19% |
$78 |
| Jacks or Better | $50 | -- |
$800 |
78% |
20% |
$73 |
| Baccarat: Player | Full-bankroll | -- |
$800 |
76% |
24% |
$68 |
| French Roulette | Full-bankroll | -- |
$800 |
76% |
24% |
$68 |
| Single Zero Roulette: 3:1 bet |
Full-bankroll | -- |
$800 |
76% |
24% |
$67 |
| Baccarat: Banker | Full-bankroll | -- |
$760 |
74% |
26% |
$67 |
| Single Zero Roulette: 1:1 bet |
Full-bankroll | -- |
$800 |
76% |
24% |
$65 |
| Blackjack | $100 | No |
$800 |
72% |
26% |
$64 |
| Blackjack | $100 | Yes |
$800 |
67% |
27% |
$61 |
| Pai Gow Poker | Full-bankroll | -- |
$760 |
76% |
24% |
$56 |
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