In last month’s article the case was made that we should play a tight aggressive strategy before the first draw and either three-bet or fold against an opening raise. The big question to answer this month is what range of hands do we three-bet and what do we fold?
Estimation of Equity versus a Range
We know that we should tighten up our range against a raise but the key question is by how much? You must ask yourself two questions; what is my opponents range and how does my hand fare against it? Answer these questions to estimate your equity.
If you can create dead money you do not require 50% equity in order for your re-raise to be profitable because if you successfully isolate a raiser from the button you are only putting in 40% of the money. Thus your estimated equity versus a range can be lower than 50% and still be profitable. However, it is probably ideal to target somewhat close to 50% equity to account for the times your isolation attempt does not work and/or you are re-raised by a big holding.
Let’s walk through an example where we attempt to estimate our equity. A solid tight player open raises from the cutoff. You are on the button with a three card 456. This holding is not a powerhouse but it is certainly a hand you would have raised yourself if the action was folded to you.
Assume the cut-off is opening with the following range:
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What is the estimated equity of 456 against this range? For other games software is available that can calculate an equity % based upon an assumed range. To my knowledge that does not exist for Badugi but we can use an equity calculator for various individual matchups and then take a weighted average of results. We need to break this into pieces so let’s first examine how our holding fares given that we are up against a badugi.
| Calculation of | |||||
| Cumulative | Estimated | Weighted Average | |||
| # Hands | % of Badugis | % of Badugis | Equity | Equity | |
| 4 High | 1 | 0% | 0% | 0% | 0% |
| 5 High | 4 | 1% | 1% | 0% | 0% |
| 6 High | 10 | 1% | 2% | 0% | 0% |
| 7 High | 20 | 3% | 5% | 17% | 0% |
| 8 High | 35 | 5% | 10% | 23% | 1% |
| 9 High | 56 | 8% | 18% | 30% | 2% |
| 10 High | 84 | 12% | 29% | 35% | 4% |
| J High | 120 | 17% | 46% | 41% | 7% |
| Q High | 165 | 23% | 69% | 43% | 10% |
| K High | 220 | 31% | 100% | 48% | 15% |
| 715 | 100% | 40% |
The calculations above assume that we were up against the median badugi in each range and that we would keep a queen or better since badugis are hard to make and a good queen will be beat around 50% of pat dealt badugis.
In the assumed range the cut-off is raising all badugis so as the chart indicates your opponent will hold a king badugi approximately 31% of the time. The estimated equity of 456 vs the median king badugi is approximately 48%. So you multiply 31% by 48% to get 15% which is the contribution of the estimated equity to account for the times you are up against a king badugi. You repeat that process for a queen badugi through to the nut A234 badugi and sum up the results to arrive at the total estimated equity of 40% for 456 versus all badugis.
In a similar fashion we can compute 456’s estimated equity against eight or better three card badugis:
| 456 VS Opening of Tri 8s | Calculation of | ||||
| Weighted Average | |||||
| # Hands | # Hands Beat or Tie | 456 Tri Equity | % of Tris | Equity | |
| A23 | 39 | 1554 | 38% | 2.5% | 0.010 |
| A24 | 36 | 1515 | 38% | 2.3% | 0.009 |
| A34 | 36 | 1479 | 38% | 2.3% | 0.009 |
| 234 | 36 | 1443 | 39% | 2.3% | 0.009 |
| A25 | 33 | 1407 | 39% | 2.1% | 0.008 |
| A35 | 33 | 1374 | 39% | 2.1% | 0.008 |
| 235 | 33 | 1341 | 39% | 2.1% | 0.008 |
| A45 | 33 | 1308 | 40% | 2.1% | 0.008 |
| 245 | 33 | 1275 | 41% | 2.1% | 0.009 |
| 345 | 33 | 1242 | 42% | 2.1% | 0.009 |
| A26 | 30 | 1209 | 41% | 1.9% | 0.008 |
| A36 | 30 | 1179 | 41% | 1.9% | 0.008 |
| 236 | 30 | 1149 | 42% | 1.9% | 0.008 |
| A46 | 30 | 1119 | 43% | 1.9% | 0.008 |
| 246 | 30 | 1089 | 45% | 1.9% | 0.009 |
| 346 | 30 | 1059 | 46% | 1.9% | 0.009 |
| A56 | 30 | 1029 | 45% | 1.9% | 0.009 |
| 256 | 30 | 999 | 46% | 1.9% | 0.009 |
| 356 | 30 | 969 | 47% | 1.9% | 0.009 |
| 456 | 30 | 939 | 50% | 1.9% | 0.010 |
| A27 | 27 | 909 | 57% | 1.7% | 0.010 |
| A37 | 27 | 882 | 57% | 1.7% | 0.010 |
| 237 | 27 | 855 | 57% | 1.7% | 0.010 |
| A47 | 27 | 828 | 57% | 1.7% | 0.010 |
| 247 | 27 | 801 | 57% | 1.7% | 0.010 |
| 347 | 27 | 774 | 57% | 1.7% | 0.010 |
| A57 | 27 | 747 | 57% | 1.7% | 0.010 |
| 257 | 27 | 720 | 57% | 1.7% | 0.010 |
| 357 | 27 | 693 | 57% | 1.7% | 0.010 |
| 457 | 27 | 666 | 57% | 1.7% | 0.010 |
| A67 | 27 | 639 | 57% | 1.7% | 0.010 |
| 267 | 27 | 612 | 57% | 1.7% | 0.010 |
| 367 | 27 | 585 | 57% | 1.7% | 0.010 |
| 467 | 27 | 558 | 57% | 1.7% | 0.010 |
| 567 | 27 | 531 | 58% | 1.7% | 0.010 |
| A28 | 24 | 504 | 58% | 1.5% | 0.009 |
| A38 | 24 | 480 | 58% | 1.5% | 0.009 |
| 238 | 24 | 456 | 58% | 1.5% | 0.009 |
| A48 | 24 | 432 | 58% | 1.5% | 0.009 |
| 248 | 24 | 408 | 58% | 1.5% | 0.009 |
| 348 | 24 | 384 | 58% | 1.5% | 0.009 |
| A58 | 24 | 360 | 58% | 1.5% | 0.009 |
| 258 | 24 | 336 | 58% | 1.5% | 0.009 |
| 358 | 24 | 312 | 58% | 1.5% | 0.009 |
| 458 | 24 | 288 | 58% | 1.5% | 0.009 |
| A68 | 24 | 264 | 58% | 1.5% | 0.009 |
| 268 | 24 | 240 | 58% | 1.5% | 0.009 |
| 368 | 24 | 216 | 58% | 1.5% | 0.009 |
| 468 | 24 | 192 | 59% | 1.5% | 0.009 |
| 568 | 24 | 168 | 59% | 1.5% | 0.009 |
| A78 | 24 | 144 | 63% | 1.5% | 0.010 |
| 278 | 24 | 120 | 63% | 1.5% | 0.010 |
| 378 | 24 | 96 | 63% | 1.5% | 0.010 |
| 478 | 24 | 72 | 63% | 1.5% | 0.010 |
| 578 | 24 | 48 | 63% | 1.5% | 0.010 |
| 678 | 24 | 24 | 63% | 1.5% | 0.010 |
| 51.5% | |||||
These calculations show that a 456 tri has approximately 51.5% equity against a range of eight or better three card hands. It should be noted that card removal effects were not accounted for but if they were your equity would probably be slightly higher. Incorporating care removal would mean you are up against another three card fours, fives, and sixes less often and you are an underdog to all of these hands.
Against A2 and A3, the 456 is around a 61% favorite.
Now we can take all of the pieces and calculate the estimated equity against the total assumed range:
| Calculation of | ||||
| % of Opening | Weighted Average | |||
| Range | Equity | Equity | ||
| Badugis | 6.3% | 22% | 40% | 8.9% |
| 8 Better Tri | 18.8% | 66% | 51.5% | 34.2% |
| A2 | 1.7% | 6% | 61% | 3.7% |
| A3 | 1.5% | 5% | 61% | 3.2% |
| 28.3% | 50.0% |
That is quite an interesting result considering it was the very hand that I ever attempted to do this analysis upon.
What if you think your opponent is opening with a tighter range? It turns out that you would have an estimated equity of around 44.6% against someone playing J+ badugis, 7+ three card badugis, and no two card draws.
| Calculation of | ||||
| % of Opening | Weighted Average | |||
| Range | Equity | Equity | ||
| J+ Badugis | 2.9% | 19% | 31% | 5.8% |
| 7 Better Tri | 12.7% | 81% | 48% | 38.8% |
| A2 | 0% | 0.0% | ||
| A3 | 0% | 0.0% | ||
| 15.6% | 44.6% |
This is a borderline result so you should tighten up a little more if you are up against an early position raiser or a tight player who plays snug from any position.
Three Betting Ranges
We can take what we learned from the data points above and try to extrapolate from it some reasonable three betting ranges for different opponents and situations. This is very situational and player dependent so trying to put together a chart of all possibilities is not practical but we can try and set a few guidelines. Against even the tightest of ranges, re-raising with a J+ badugi and A36+ appears to be reasonable. In late position vs the cut-off or loose early position raisers one could probably re-raise with any Q+ badugi and A37+.
Your play from the small blind depends greatly on the situation. On one hand you could be up against a wider range but you are also out of position. A three-betting range against a very loose button opener could be any badugi, A37+, A2, and A3. A loose button opener raises a lot of two card draws and even the worst badugis are big favorites versus these hands. In order to play A2 and A3 profitably, the original raiser would have to be opening greater than 45% of their hands on the button. This helps ensure that their range is heavy enough in weak tris and two card draws that you are not too great of an underdog to their overall range.
It is important to note that you should often turn the weakest of your three card badugis into snows by the end of the hand. And if you aren’t snowing you should often bet the turn and river if you believe you have any fold equity.
There is a gap in between the hands that you would play had the pot been unopened versus those you would play versus a raise. But when compared to other games the gap in Badugi could be a lot smaller. In Hold’em, the gap is wider because there are many spots where you could be an 80%/20% or 70%/30% underdog. You are never really a large underdog in Badugi unless you are up against a very strong dealt pat badugi but those hands are very rare.
This concludes our discussion of strategy before the first draw. Admittedly we only used a few data points in coming up with some three-betting guidelines but now that you are armed with the thinking process it is possible to dive deeper yourself and draw your own conclusions. Next month we will begin to examine middle round strategy which will help guide our play after the first and second draws.