Questions about win-rate distribution among all players

HarroginHarrogin Red Chipper Posts: 5 ✭✭
edited February 27 in Off Topic Chat
Suppose there is a 3 player S&G tournament format. Consider 3 players who all play the exact same strategy. If they played this tournament against each other an infinite number of times, each player would win 33.33% of tournaments. This is the mean win-rate of this tournament.

Now suppose that every current poker player in the world, each with a different strategy, played this S&G format an infinite number of times, each time against two new opponents. We'll define a "winner" as a player who wins more than 33.33% of tournaments, and a "loser" as a player who wins less than 33.33% of tournaments.

1.) Can we assume that the win-rate among all players is a normal distribution with a mean of 33.33%? Basically, are there an equal number of winners and losers, or are there more losers than winners?

2.) What does the probability density/standard deviation look like?
- What do the win-rates of the best players look like? (top 10%/ top 1%)
- Worst players? (10%/ 1%)

3.) How does this distribution change in a HU S&G format? 4 player S&G format?

4.) Would the distribution change if each player faced the same two opponents every tournament?

It's been a while since I've studied statistics, so I apologize if this isn't totally clear. I'd just like to get an idea of how many really good and really bad players there are in relation to average players in the poker ecosystem. What does the distribution actually look like in practice, and what's theoretically possible?

Statistics Stuff:
https://en.wikipedia.org/wiki/Normal_distribution
https://en.wikipedia.org/wiki/Probability_density_function

Comments

  • TheGameKatTheGameKat Posts: 3,645 -
    edited February 28
    Harrogin wrote: »

    1.) Can we assume that the win-rate among all players is a normal distribution with a mean of 33.33%? Basically, are there an equal number of winners and losers, or are there more losers than winners?

    A normal distribution would be a natural assumption, but we can't know it is without data mining. My guess is it might exhibit skewness resulting from very bad losing players playing fewer games. Whether this impacts the kurtosis I don't know.
    Harrogin wrote: »
    2.) What does the probability density/standard deviation look like?
    - What do the win-rates of the best players look like? (top 10%/ top 1%)
    - Worst players? (10%/ 1%)

    When SNGs were effectively solved in the late 00s, max win-rates capped at a fairly low ROI. Interpreting this is plagued with selection effects, however. For example, serious players are attempting to maximize their hourly rate rather than ROI. Thus they tend to drift up in buy-ins to whatever level maxes this out, but they're also typically massively multi-tabling which inevitably reduces their win-rate as measured by the number of times they win.

    As to the standard deviation, it's much lower than in MTTs, but as mentioned in 1) above, you'd need to data-mine to find out what it is. And if the distribution deviates from normal then you'd also need to use a different metric than the standard deviation anyway.
    Harrogin wrote: »
    3.) How does this distribution change in a HU S&G format? 4 player S&G format?

    Probably depends on how skilled "top" players are at bum hunting.
    Harrogin wrote: »
    4.) Would the distribution change if each player faced the same two opponents every tournament?

    Yes, insofar as it would likely reduce systematic effects and produce more well-behaved statistical distributions.
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