[eastbay]

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We see this statement so often on this forum, and I don't understand why so many people think it's a meaningful statement (or question).

The phrase "true ROI" presupposes two things, both of which are false.

First, it assumes that your game is static.

Second, it assumes that the skill level of your average opponent is static.

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I think this old "debate" is boring, and I think your objections are mostly academic and miss the point of what most "true ROI" discussions are getting at.

ROI has an expectation value. This is generally what people are referring to as "true ROI." This has a well-defined meaning.

How well you can "measure" the expectation value is open to some speculation. Maybe you will improve, maybe the player pool will improve. Then again, maybe neither will happen fast enough that an active player can get a sample over which the game conditions are effectively stationary. Do you really think ZeeJustin's game is going to change that much over the next 1k SnGs he's going to play this week, or do you really think the player pool is going to evolve in that week's time? No way. Those are infinitesimal deltas over that time frame. The stationarity assumption is a good one in that case, so sample size estimates based on a stationarity assumption are useful.

Now if you're talking about someone who started playing three months ago and they play 5 SnGs a week, yes, there is major non-stationarity in what he is trying to measure and him trying to measure his "true ROI" is silly. There's no way he can do it before it will move on him significantly.

Now, returning to what I think is the main problem with your objection is that estimates of sample sizes needed for knowing "true ROI" are generally used to support the position that a sample isn't high enough to draw a conclusion. The stationarity assumption that goes into those required sample sizes only makes them more optimistic than they would be for a non-stationary system. So the point remains. Non-stationarity only lengthens the sample needed to get an idea of expected ROI.

Bottom line: You do have a "true ROI" and anybody who thinks differently is wrong. However, it may not be possible to measure it, depending on how active you are, and how dynamic both your game and the player pool are. But we knew that was really hard to do in any case.

eastbay