A Note on Variance

[Orlando Salazar]

People always talk about running bad, getting coolered then going busto on these forums. In short, I tested the veracity of these claims. The following chart show about 31k hands where a 30 Heros have committed exactly 1/20th of their bankroll in pots they are either a favorite or an underdog. The average P (hero is a favorite in 1 hand) is 50.02%. This equates to a “2% edge” or 2bb/100. However, since Heros cannot always be exactly a 50.02% favorite in each individual hand, I’ve randomized Hero equity for each hand into 10 common equity buckets. When Hero’s are ahead, they are: 80%, 75%, 60%, 55%, or 66% favorites. When behind, they have equity inverse to that above. Heros are more likely to be slim favorites than huge ones (ie, you’ll have 1 over card vs. villain more frequently than an over pair). The pool of random equity must therefore adjust for commonness of situations AND Hero edge. To do this, I used excel to select random equity from a column of over 2.2k rows. The frequency (as % of possible equity situations) of being a favorite 2%, 20%, 50%, 1.25% and 30% when Hero has equity of 80%, 75%, 60%, 55%, or 66%, respectively. In other words, 50% of the time hero has positive equity, he will be a 60/40 favorite. I’ve upward adjusted the positive equity frequency to account for a 2% edge. So the average equity of the sample is 50.02%. The graph is the cumulative log returns on an original bankroll.

As you can see, 2 of 30 will be superstar robustos (up more than 3000%) in practically no time and 5 will go busto. Since in practice Heros almost always have less than 1/20th of their roll at risk when they play a pot (usually pots are 15bbs not 100), the variance and results of this chart could take 8x-10x as many hands to be realized. Yet, I believe it is a reflection of pokers true variance. Comments and critiques are welcome. Trolling is not. GL at the tables.