Classifying holdem starting hands

It is well known that some hands "like to be heads up" while other hands, most notably suited connectors "like multiway action" and "thrive on implied odds." How can we objectively classify each of the 169 holdem starting hands into one of these categories? Can we make sense of the shades of grey in between the two extremes?

A full answer would require specifying your opponents' strategies - what hands they are willing to play against you and how agressively they would play them. But as a starting point, we can see how each starting hand fares against a field of randomly dealt hands. This experiment has already been done by a number of people: see the results of such a simulation at the GoCee Poker Center for instance.

But merely listing the percentages, or "sorting the hands from best to worst," is not the full story. A key piece of the puzzle is seeing whether a hand makes more or less money as the number of opponents changes.

Methodology

Playing against more opponents reduces any hand's chance of winning, but increases the size of the potential pot to be won. What we want to know for each hand is: does the increased pot size compensate for the reduced chance of winning?

Consider the first line of Steve Brecher's table:

This means that if AA goes all-in before the flop against one random hand, it expects to win 85.3% of the time. Put another way, for every dollar put into the pot before the flop, the player with AA expects to get a bit more than $1.70 back, 85.3% of two people's money; against two opponents, he expects to get $2.20 back, 73.4% of the money three people contributed to the pot, and so on. For this study, I have used Brecher's data, but instead of plotting chance of winning vs. number of opponents, I have used expected rate of return on investment (chance of winning x number of players contributing money) vs number of players (number of opponents + 1), thus:

Brecher states his table is based on "one million simulated hands"; it is not clear whether he means he played each matchup a million times, or ran a million random matchups, in which each hand was involved on the order of 10,000 times. If the former, each of his reported numbers has an uncertainty on the order of 0.1%; if the latter, on the order of 1%. Multiplying by the number of players magnifies this uncertainty; that is, there is not a statistically significant difference btween the 8-, 9- and 10-handed rates of return for AA.

I examined these rates of return for each starting hand, and classified them based on the shape of the curve outlined by the rate of return plotted against number of players. This divided the 169 starting hands into 5 classes. Each of these can be subdivided according to whether the rate of return is above or below 100%.

Let me emphasize that when I say a hand is "profitable," I mean the word in a very specific sense: profitable if all-in preflop vs. a set of random hands all of whom contributed money to the pot. We will take up the real-world implications of these results later; for now, suffice it to say that any hand deemed "unprofitable" here will be a loser in the real world, but not all hands labelled profitable will be winners in the real world.

The Five Classes

The simplest shape of curve is one that either rises steadily all the way from 2-handed to 10-handed, or falls steadily. A large number of poker hands fall into each of these.

Somewhat more complicated is a curve that rises to a maximum and then falls again, or falls to a minimum and then rises. A smaller number of poker hands are of these two types.

The next notch up the scale of complexity is a curve with two extrema. Only a handful of poker hands improve from 2- to 3-handed, are worse with 5 or 6 opponents, and better again with 10. None at all have a minimum at 3-handed and a maximum at 6-handed.

We call the five classes by their obvious names:

1. Increasing (52 hands, 16.4% frequency) - always more profitable with more opponents
2. Decreasing (47 hands, 42.5% frequency) - always more profitable with fewer opponents
3. Convex (20 hands, 16.3% frequency) - most profitable in a medium-sized field
4. Concave (42 hands, 19.7% frequency) - least profitable in a medium-sized field
5. Compound (8 hands, 5.0% frequency) - better 3-way or multiway than heads up or medium-sized field

The classification of each hand is shown in the table below, along with whether that hand is profitable against all (green) some (yellow) or no (red) field sizes. More detailed coverage of each class continues below the table.