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statisticsprobabilitymontecarlopoker

Calculating poker preflop equity efficient

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

Solution

  • 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)

    Plot.png

    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.