Reconciling results with previous work by others

Much excellent work has done by on identifying advantage play risks on Baccarat by prominent casino mathematicians. We took a slightly different approach influenced by Asian gaming and addressing the risk of Bot play and computer-aided rings. The following is reconciliation between our results and those reported elsewhere.

Reconciling Baccarat Advantage-Play Results

Published baccarat advantage-play studies often report their results using different wager sizes, time periods, pay tables, cut-card positions and betting methods.

These differences can make broadly consistent analyses appear contradictory. A useful comparison begins by converting each result to the same unit basis. The remaining difference can then be separated into betting frequency, average edge, penetration, pay-table and strategy effects.

This review compares our composition-dependent results with figures published by Wizard of Odds, Advanced Advantage Play and Eliot Jacobson’s articles for 888casino.

Common measurement basis

Our table reports profit per shoe using five-unit wagers. Each result was converted to expected units won per 100 hands using a one-unit wager.

Our simulations averaged approximately 79.38 hands per shoe:

Normalized units per 100 hands = (profit per shoe ÷ 5) × (100 ÷ 79.38)

The same result can also be checked from betting frequency and average edge:

Units won per 100 hands = betting frequency × average edge × 100

For wagers offered separately on Player and Banker, the two results are combined only when the published figure also assumes that both sides are played.

Normalized comparison

Wager	Our normalized units/100 hands	Published units/100 hands	Reconciliation
Big Tiger, 50-to-1	0.343	0.256	Explained by our lower betting frequency but higher average edge
Dragon 7, 40-to-1	1.086	0.734	Explained by both higher betting frequency and higher edge in our method
Panda 8, 25-to-1	0.456	0.293	Similar betting frequency; our selected wagers have a higher average edge
Pairs, 11-to-1, both sides	0.401	0.405	Reconciles closely after combining both sides and normalizing wager size
Lucky 7	0.078	0.146	Different pay tables and penetration materially change betting frequency
Small Tiger	0.383	0.518	Different pay tables and penetration materially change betting frequency
Lucky Six	0.234	0.194–0.289	Our result falls between the published Level 1 and Level 2 counting results
Tie, 8-to-1	0.005	0.0101	Same small order of magnitude; penetration and rounding can explain much of the gap
Tie, 9-to-1	0.096	Approximately 0.096	Reconciles closely
Player base bet	Approximately 0.0025	Approximately 0.0010	Both are extremely small; our rounded summary is too coarse for a precise comparison
Banker base bet	Approximately 0.0040	Approximately 0.0008	Not fully reconciled from the rounded summary; requires direct hand-level review
High Tie 7	1.184	2.65	Different pay table, penetration and betting frequency explain the direction of the gap

Back-of-the-envelope reconciliation

Big Tiger

Our table shows approximately 2.48 bets per shoe. With 79.38 hands per shoe:

Our betting frequency = 2.48 ÷ 79.38 = 3.12%

Our average optimal edge is 10.95%:

3.12% × 10.95% × 100 = 0.342 units per 100 hands

The published method reports a 3.536% frequency and 7.227% edge:

3.536% × 7.227% × 100 = 0.256 units per 100 hands

The arithmetic reconciles. Our method bets slightly less frequently but reports a higher average edge on the selected wagers. The remaining difference is attributable to strategy selection and penetration.

Dragon 7

Our table shows approximately 8.51 bets per shoe and a 10.13% edge:

(8.51 ÷ 79.38) × 10.13% × 100 = 1.086 units

The published method reports a 9.16% frequency and 8.03% edge:

9.16% × 8.03% × 100 = 0.736 units

The published 0.734 result is reproduced by the published frequency and edge. Our higher result comes from a higher betting frequency and a higher average edge. Exact-composition selection is a reasonable explanation, although a same-shoe comparison would be needed to allocate the difference precisely.

Panda 8

Our table shows approximately 3.52 bets per shoe and a 10.27% edge:

(3.52 ÷ 79.38) × 10.27% × 100 = 0.456 units

The published method reports a 4.609% frequency and 6.329% edge:

4.609% × 6.329% × 100 = 0.292 units

The betting frequencies are similar. Most of the difference comes from our higher average edge on the hands selected for wagering.

Pairs

Our table reports Banker Pair and Player Pair separately:

Banker Pair profit: 0.80 units per shoe using five-unit wagers
Player Pair profit: 0.79 units per shoe using five-unit wagers

(0.80 + 0.79) ÷ 5 × (100 ÷ 79.38) = 0.401 units

The published method reports a 1.74% betting frequency per side and an 11.63% edge:

2 sides × 1.74% × 11.63% × 100 = 0.405 units

The results reconcile closely. Our method bets somewhat less frequently at a higher average edge, while the published method bets more frequently at a lower edge.

Lucky 7

Our table shows approximately 0.52 bets per shoe and an 11.88% edge:

(0.52 ÷ 79.38) × 11.88% × 100 = 0.078 units

The published 7/15 version reports a 2.52% frequency and 5.79% edge:

2.52% × 5.79% × 100 = 0.146 units

The arithmetic reconciles, but the games are not like-for-like. Our two-card payoff is 6-to-1 rather than 7-to-1. The higher published payoff makes many more shoe states profitable, increasing betting frequency from about 0.66% to 2.52%.

Small Tiger

Our 22-to-1 version shows approximately 2.39 bets per shoe and a 12.70% edge:

(2.39 ÷ 79.38) × 12.70% × 100 = 0.383 units

The published 23-to-1 version reports a 5.96% frequency and 8.70% edge:

5.96% × 8.70% × 100 = 0.519 units

The difference is primarily betting frequency. The additional payoff unit and deeper 14-card penetration create more positive-expectation opportunities. Earlier like-for-like testing produced about 0.556 units per 100 hands, reasonably close to the published 0.518.

Lucky Six

Our table shows approximately 1.39 bets per shoe and a 13.42% edge:

(1.39 ÷ 79.38) × 13.42% × 100 = 0.235 units

Wizard of Odds reports the following for its 12/23 pay table:

Level 1: 3.09% × 6.27% × 100 = 0.194 units

Level 2: 4.13% × 7.00% × 100 = 0.289 units

Our normalized result falls between the two published counting systems. This is a reasonable reconciliation, although our three-card payoff is 20-to-1 rather than 23-to-1.

Tie paying 8-to-1

Our rounded table reports approximately 0.03 bets per shoe with a 14.07% edge:

(0.03 ÷ 79.38) × 14.07% × 100 ≈ 0.005 units

The published computer-perfect result is 0.0101 units per 100 hands at a 14-card cut.

Both results show a very small opportunity. The difference is approximately half of one-hundredth of a unit per 100 hands. Deeper penetration and rounding of our very small shoe-level values can explain much of this gap.

Tie paying 9-to-1

Our table shows approximately 1.62 bets per shoe and a 4.71% edge:

(1.62 ÷ 79.38) × 4.71% × 100 = 0.096 units

The refined published computer-perfect estimate is also approximately 0.096 units per 100 hands.

This result reconciles closely and provides useful support for the composition-dependent calculation.

Player base bet

Our rounded table shows approximately 0.06 bets per shoe and a 3.30% edge:

(0.06 ÷ 79.38) × 3.30% × 100 ≈ 0.0025 units

The published computer-perfect estimate is approximately 0.001 unit per 100 hands.

Both estimates are extremely small. Because our table rounds bets per shoe and profit per shoe, the displayed values do not contain enough precision for a firm reconciliation. The underlying hand-level totals should be used.

Banker base bet

Our rounded table shows approximately 0.06 bets per shoe and a 5.32% edge:

(0.06 ÷ 79.38) × 5.32% × 100 ≈ 0.0040 units

The published computer-perfect estimate is approximately 0.0008 units per 100 hands.

The results agree that the opportunity is very small, but the difference is not fully explained by the rounded summary. The comparison should be repeated using unrounded, directly summed hand-level results and identical final-round rules.

High Tie 7

Our 40-to-1 version shows approximately 5.25 bets per shoe and a 17.88% edge:

(5.25 ÷ 79.38) × 17.88% × 100 = 1.183 units

The published UR Way Egalite version reports a 22.4% frequency and an 11.8% edge:

22.4% × 11.8% × 100 = 2.643 units

The published 2.65 result is reproduced. The published wager pays 45-to-1 rather than our 40-to-1 and uses a 14-card cut. Those differences substantially increase betting frequency. The direction and approximate source of the gap are explainable, but the two results are not directly comparable.

How the approaches relate

The published studies use several methods:

simple level-one counts;
more detailed level-two counts;
specialized trigger-and-weight systems;
separate counts for individual tie totals; and
computer-perfect play based on the exact remaining-card composition.

Our method is closest to computer-perfect play. It calculates expected value from the detailed remaining-card composition.

A traditional count compresses the shoe information into one number. This makes the method more practical, but different shoe compositions can receive the same count even when their exact expected values differ.

The methods can be related in the following way:

Exact-composition analysis estimates the opportunity available from full shoe information.
A counting system approximates that opportunity using fewer inputs.
Betting frequency measures how often each method identifies an opportunity.
Average edge measures the quality of those selected opportunities.
Units won per 100 hands is the product of frequency and edge.

Summary of the differences

Most of the reported differences can be explained:

Pairs: reconciles after combining both sides and normalizing wager size.
Tie paying 9-to-1: reconciles closely with the computer-perfect result.
Lucky Six: falls between the published Level 1 and Level 2 systems.
Big Tiger, Dragon 7 and Panda 8: differences are explained by the balance between betting frequency and average edge, together with method and penetration.
Lucky 7 and Small Tiger: differences are primarily explained by different pay tables and penetration.
High Tie 7: the direction of the difference is explained by the higher published payoff and greater betting frequency.
Tie paying 8-to-1: both methods find a very small opportunity; penetration and rounding can explain much of the small absolute difference.

The Player and Banker base-bet comparisons are not fully reconciled from the displayed summary. Their expected profits are so small that rounding materially affects the comparison. These two results should be recalculated from unrounded hand-level totals using identical burn, cut-card and final-round rules.

Overall, the comparisons are internally understandable. The largest apparent contradictions usually disappear after separating wager size, betting frequency, average edge, pay table and penetration.