Don’t bet on it: Which bets most benefit from a counting system
Contrasting counts…
One judge of a bet’s countability is evaluating how well a counting system provides consistent and accurate information to an advantage player (AP). We call this concept countability. As data scientists, we evaluate a model’s accuracy on the basis of precision and recall where
Precision or when the model predicted an outcome how often was the model correct
Recall or of the outcomes how many were caught by the model
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EV or Expected value is the likely $ value of a bet based on the probability of an outcome. As the payoffs for a winning outcome is fixed by the game rules, the only dynamic in a game is the probability of a winning outcome. For any game dealt out of shoe the probability of a winning outcome will fluctuate based on the remaining composition of the shoe. Advantage players look for shoes with a high concentration cards which favor a winning outcome.
See our assumed game rules and payoffs here or see our reconciliation between our analysis and other Baccarat published work
In essence a computer combinatorics count system is just a probabilistic model but as we can’t train the model to improve recall, we are only interested in precision. Let’s consider a couple worked examples for Monkey – No Monkey, Super Lucky Seven, and Banker.
Exhibit One: precision calculation examples
| Bet | |||||
| Calc | M / NM | SL7 | Banker | ||
| Expected Hit rate | a | 35.0% | 2.0% | 45.9% | Expected frequency bet will win |
| Predicted positive EV Hands | b | 1.9% | 4.2% | 0.1% | Frequency of hands identified as positive player EV |
| Positive EV + Hit | c | 1.0% | 0.1% | 0.0% | The # of hands identified as positive EV which won |
| Hit rate when positive EV | d = c / b | 51.0% | 2.4% | 46.1% | Precision: of the hands identified as positive EV what % won |
| Precision over expected hit freq | e = d - a | 16.0 %pts | 0.4 %pts | 0.2 %pts | Lift from using counting system vs randomly guessing |
| Precision lift over expected | f = e / a | 45.7% | 19.7% | 0.4% | Benefit of using counting system |
M / NM = Monkey / No Monkey
SL7 = Super Lucky Seven
Banker = Banker base bet
In the above example, the Monkey / No Monkey system provided substantial lift (~46% over random guessing) but the frequency of positive EV situations are rare (~2% of hands) in a absolute sense. A counting system can improve an AP’s odds of identifying a winning hands by 20% but in an absolute sense this is only .1% of hands have positive EV. As expected, a computer aided counting system has little practical benefit but this is now the case for all bets.
For reference: we (naively) hoped to improve on combinatorics using machine learning attempting to address several blind spots in combinatorics but we failed and are humbled. (Somewhere Edward Thorpe is laughing.) All of the analysis is based on combinatorics
Graphic one: Countability ranking of Baccarat Side-Bets (ranked by improvement from computer-aided counting system / precision lift)
The most countable bets are Banker 3 card 7, Monkey / No Monkey and the High Tie family. A counter targeting Tiger Tie gains a material benefit from the Tiger Tie count but the game is hardly weak due the payout rules / high house advantage. Monkey / No Monkey stands out as the most countable major bet: an AP who tracks it can beat random selection by roughly 45%, and unlike most edges on this list, the count is simple enough to run in your head. No computer required. The card-by-card influence analysis below shows why.
Exhibit two: card importance for Monkey / No Monkey (SHAP values)
| Rank | Card | Importance | Index | Effect When More Remain |
|---|---|---|---|---|
| 1 | K | 16.56% | 100.0 | Lowers No Monkey |
| 2 | J | 15.98% | 96.4 | Lowers No Monkey |
| 3 | Q | 15.84% | 95.6 | Lowers No Monkey |
| 4 | 8 | 6.76% | 40.8 | Raises No Monkey |
| 5 | 10 | 6.21% | 37.5 | Raises No Monkey |
| 6 | A | 5.98% | 36.1 | Raises No Monkey |
| 7 | 6 | 5.52% | 33.3 | Raises No Monkey |
| 8 | 9 | 5.06% | 30.5 | Raises No Monkey |
| 9 | 5 | 4.96% | 29.9 | Raises No Monkey |
| 10 | 7 | 4.56% | 27.5 | Raises No Monkey |
| 11 | 4 | 4.37% | 26.4 | Raises No Monkey |
| 12 | 2 | 4.37% | 26.4 | Raises No Monkey |
| 13 | 3 | 3.85% | 23.2 | Raises No Monkey |
As expected, the High Tie family of bets also top the league table. The base bets are clearly not susceptible to counting, at least on their own. A online client with an aggressive, turnover based marketing program is targeted by bots seeking to minimize the edge on base bets.
Consistency versus accuracy
Knowing a bet is positive and benefiting from that situation consistently are two different things. Because of the insidious impacts of volatility, even a bet with a strong counting system may not be a candidate for AP. We graded games by the percent of times a computer aided AP leveraging optimal betting would be in the black at the end of one shoe. Games like Monkey / No Monkey are less an obvious risk than accuracy alone would indicate. By contrast, some of the High Tie bets like 0/3/9/8 are easily countable, and yield consistent meaningful returns. Surveillance operators should also be vigilant around Dragon Seven and Four Card Tie.
Chart two: Consistency versus profit: X-Axis is a single shoe will be profitable and Y-Axis is the profit per shoe (Units)
Other bets are risky in conjunction with other bets, an especially difficult and harmful situation called portfolio counting where an AP targets any positive EV bet. Portfolio Counting is the subject of our next blog.
