I've always thought that the "best" games are ones that accomodate a huge range of skill levels. In this regard, Go and football ("soccer" in the U.S.) are my favorites, although I do not play either. Both also have rules that are about as simple as it can get, another reason I admire them. In football there is an Elo-based rating system but I can't figure out what they're saying at [1a]. Obviously the team nature of the game and the fact that individual players get swapped from one team to another makes mathematical modeling more difficult, but I think the original questions (Thane Plambeck and rcs) are equally relevant and answers equally useful. The English football league system [1b] has a pyramid of about 20 levels, the top 14 of which have an exponential distribution: each level has about 1.5 times as many teams as the level above it. The churn rate is about 25 percent per year: out of 22 clubs typically 5 or 6 clubs leave each year (either to move up or move down). I expect that "200 Elo points" for English football would be more than one level of this pyramid, (because 200 point spread means the higher team will win 75% of the time, and hearing the football coverage on the BBC it seems winning is more random than that.) I imagine it's about 100 Elo points per pyramid level. That makes the pyramid about 2000 Elo points high (ignoring the very few leagues at the bottom). This does not include semi-professional and amateur teams, many of which also have leagues but can't ever make it to the Premier League. In Go, the scale is about 3000 points high[2a], and the size of "200 Elo points" varies from about 100 points at the top end to about 300 at the bottom [2b]. I suppose this comes from the way the game changes as you get really good at it, but it also looks suspiciously like a grade inflation problem. - Robert [1a] http://en.wikipedia.org/wiki/World_Football_Elo_Ratings [1b] http://en.wikipedia.org/wiki/English_football_league_system#The_system [2a] http://en.wikipedia.org/wiki/Go_ranks_and_ratings#Elo_Ratings_as_used_in_Go [2b] http://en.wikipedia.org/wiki/File:Estimated_Win_Probabilities_under_EGF_Rati... On Fri, Feb 10, 2012 at 13:19, <rcs@xmission.com> wrote:
I'll add one more vaguely-specified question to Thane's list: He estimated that the spread of tennis skill is at least six steps of 'near certain victory', and maybe much higher.
What's the story for other games?
I think Chess has a smaller spread: IIRC, 400 Elo rating points is one Plambeck step, and the Elo spread is 1200 - 2900, only 4.25 Pbk. I'll guess that Go has a bigger spread, while soliciting more information from real players.
Rich
---------- Quoting Thane Plambeck <tplambeck@gmail.com>:
I'd welcome pointers to statistical models of the following vaguely-specified situation
I play tennis. There are probably plenty of people who also play tennis whom I could defeat, if they were chosen at random amongst all players who know how to play and have a racket.
However, I could easily point to a tennis player X whom I would little chance (essentially zero) of defeating. (S)he in turn could point to a player Y with the same property. And Y could point to player Z.
I'm sure that for a middling tennis player such as me, there must be at least five levels, and perhaps many more, of players with this transitive "I'd have little chance of defeating that person" property
Eventually we'd reach Novak Djokovic at the top of the world tennis rankings. There are probably only ten players who have any reasonable chance of beating him in a match today.
What I'm looking for is a statistical model of such a situation, which I view as somewhat in common in competitive sports. The closest thing I can think of is the ELO chess rating system.
I'm interested in answering questions like this.
Say I have two randomly chosen worldwide tennis players X and Y. Let's say X defeats Y twice in two matches. What is the chance X will defeat Y in a third match, if no other information is provided?
I'm also interested in (again vaguely-defined) "churn" parameter (ie the mixing of player skill levels over time). For example, in chess it seems to be possible for a player to play at specific level for many years. In tennis, most players begin to fade by their thirties if not sooner
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