Calculating poker preflop equity efficient

Question

I've read many articles about the Monte Carlo algorithm for approximating the preflop equity in NL holdem poker. Unfortunately, it iterates over only a few possible boards to see what happens. The good thing about this is that you can put in exact hand ranges.

Well, I don't need exact ranges. It's good enough to say "Top 20% vs Top 35%". Is there a simple formula to tell (or approximate) the likelihood of winning or losing? We can ignore splits here.

I can imagine that the way to calculate the odds will become much simpler if we just using two (percentile) numbers instead of all possible card combinations.

The thing is, I don't know if for example the case "Top 5% vs Top 10%" is equal to "Top 10% vs Top 20%". Does anyone know of a usable relation or a formula for these inputs?

Thanks

Answer 1

Okay, I've made a bit analytical work and I came up wit the following.

The Formula

eq_a(a, b) := 1/2 - 1/(6*ln(10)) * ln(a/b)

Or if you like:

eq_a(a, b) := 0.5 - 0.072382 * ln(a/b)

Where a is the range in percent (0 to 1) for player a. Same for b. The function outputs the equity for player a. To get the equity for player b just swap the two ranges.

When we plot the function it will look like this: (Where a = x and b = y)

As you can see it's very hard to get an equity greater than 80% preflop (as even AA isn't that good mostly).

How I came up with this

After I've done some analysis I became aware of the fact that the probability of winning is dependent on just the ratio of the two ranges (same for multiway pots). So:

eq_a(a, b) = eq(a * h, b * h)

And yes, Top 5% vs Top 10% has the same equities as Top 50% vs Top 100%.

The way I've got the formula is I've done some regressions on sample data I've calculated with an app and picked the best fit (the logarithmic one). Then I optimised it using special cases like eq_a(0.1, 1)=2/3 and eq_a(a, a)=1/2.

It would be great if someone will do the work for multiway preflop all-ins.