Skill gradient across different levels of the ecosystem

Question

Most online poker sites offer the same game structure (up to slight differences in rake) at multiple stakes. Take, for example, the 6-max hyper SNGs on Pokerstars, which are offered at 11 different buy-ins ranging from a low of $1.50 to a high of $1000.

Now, comparing the level of play at the top end (as showcased on tonkaaaa's Twitch stream, for example) to that at the lowest levels, the quality of play is clearly better at the upper end. But differences in the skill levels of the player pools at adjacent "rungs on the ladder" are much harder to spot; indeed, there is frequently some degree of overlap between pools at one level and its neighbours.

Does the mean skill level of the associated player pool increase at a uniform rate as one moves up the ladder, or is the pattern more complicated? (e.g., does the skill level "saturate" somewhere below the highest stakes, with bankroll considerations, rather than skill edge, determining who plays at the very top?)

While I use the 6-max hypers as an example case, I would be interested in observations from other users on skill gradients across other tournament structures or cash games as well.

Supplemental

A commenter has raised a legitimate question -- how does one measure skill? For the sake of argument, let's use ROI as a proxy measure.

Consider the following thought experiment. Imagine a poker player with a short term memory problem -- the player immediately forgets the hand he/she has just played. (This way, there's no potential learning effect to worry about.)

Let's say we very carefully measure this player's ROI at the $1.50 level by having the player play 1,000,000 tournaments against randomly selected opponents. We repeat the experiment at the $3.50 level, the $7.00 level, and so on.

Our player's ROI is now effectively functioning as our "skill-o-meter" for the average player at each of the rungs on the ladder between the top and bottom of the ecosystem. Naively, we would expect this to show some form of decreasing trend (although the corresponding decrease in rake at the higher levels will likely contaminate our signal in practice).

What I'm wondering about is the form of this decrease. Is the pattern going to be something straightforward like, say, a 2% decrease in ROI with each step up the ladder? Is there a "wall" at some level, at which the ROI starts to drop off far more rapidly? Or will we see the ROI remain the same from, say $100 to $1000, suggesting that the play is (on average) the same at each of these levels, with the distribution of players determined more by bankroll than by pure issues of skill?

Answer 1

I'd say that skill level is going to increase exponentially, not linearly, with the increase in buy-in. Many more people can afford to play microstakes in a casual manner (i.e. being a losing player) than the high stakes. The higher buy-in, the fewer people you're going to encounter with that much disposable income, and the more people you'll find trying to play professionally. Every time you make a step up from one buy-in level to the next, you're losing a portion of that casual crowd.

What will also affect your ROI is the size of the player pool. One reason why players are able to make it at higher stakes are smaller fields. If you're using a HUD then you're able to consistently track your opponents and collect more information on them. With the bigger fields in microstakes, you've got a ton of people that are coming and going, so you have less information on them, which leads to higher variance.

There are natural walls, which most poker sites separate as the "micro", "mid" and "high" stakes levels. Depending on the websites, some populations tend to be softer at the "mid" levels than at the "micros". This just has to do with population tendencies. If the poker website offers sports betting and casino games, it can attract a lot of casual players wanting to throw $20-50 into a mid-stakes tournament. Micros tend to not appeal to these players, because the buy-ins and payouts aren't "worth it".

Something else to consider for cash games is the prevalence of bots/regs at 10nl/25nl. These tend to be players from second world countries who are grinding a living out. Players moving up in stakes sometimes hit this wall and have a hard time getting past it.

Answer 2

You're asking a good question, unfortunately I think data exactly as you ask for (in terms of ROI) hasn't been published yet. (No doubt the game sites have it though, even if just implicitly.) What I could find published is in terms of absolute winrate (and adjusted for rake and expressed in BBs) and as a derived measure from that the CRF, which is [in theory] the number of hours needed to reach the 50%/50% chance/skill breakpoint for a player:

(The small number next to the CRF is the confidence level.) Note that for the bad players, whose winrate is actually negative (although that's unfortunately masked in the table above), the CRF is obviously much lower than for the good players. Alas it makes much longer for a really good player to establish himself among good players than it takes to distinguish a bad player form good ones.

There is some ROI data for two broad groups of players in the World Series: "Those players identified a priori as being highly skilled achieved an average return on investment of over 30 percent, compared to a -15 percent for all other players." But that probably doesn't help you much. However, one thing to note in the latter paper (fig 1 there) is that tournament payoffs are highly convex. And probably for the good reason observed in the former paper: if tournament payoffs were linear it would be really boring and/or it would take a long time for the cash difference to become significant.