Friday, March 25, 2011

Value of a top player

I got a comment on the previous thread and posted a response but wanted to make a new post about it:
I would like to hear your statistically informed opinion on the following thought experiment: assume that there are thirteen players of roughly average (on the scale of all ultimate players) and equal ability (compared to each other). The fourteenth is a player of outstanding ability--someone widely thought to be one of the best players in the game.

They play a pickup game in which everyone is trying their best to win. What is the probability that the team with the elite player wins?

Hey, good question. I did some simulations about 15 years ago for a UPA Newsletter article. I will use the chart in there to make estimates.

(First, what is an "average" ultimate player? What is the average income between a homeless guy, Joe the Plumber, and Bill Gates? When you have such a range between high and low, "average" becomes a funny concept. I'll assume "average" is someone who would fit in nicely on a low-level regionals team.)

Two teams that score at equal rates will of course win an equal amount of the time (with a slight advantage to the team that receives in the first half, but we'll ignore that). A team that has a 5 percentage point advantage (e.g., 40% vs 35% of the time they touch the disc, they score) will win 65-75% of the time (with the bigger advantage when the percentages are at the lower end). A 10 point advantage goes from 76-87.

With the average groups, I'll assume that teams score about 30% of the time. Top Open teams playing against top Open teams in moderate wind might be around 50%. What effect does this awesome player have?

First, I think the effect on defense will be less than on offense. He will get some poach blocks but since there is no star on the other team he won't be able to thwart their offense. Let's assume he gets 3 additional blocks but otherwise has no effect on their offensive efficiency (such a player at the elite Open level would be possibly the best player in history). Previously they were 15/50 in a game to 15, change that to 15/53, that's a drop to only 28.3%. To lower their % to 25%, he'd need to get 10 blocks a game.

Let's pause for a minute and consider what a superstar team would do against this team. I'd guess 15-1 or 15-2 is a fairly typical score for a game like this, though there is a question of whether they are trying their best to win, if for no other reason than they have 4 games that day (but so does the other team, and I'll guess they aren't in as good shape so would be further from peak efficiency). If they had 5 turnovers, that'd only be 75%. So, adding 7 elite players to an average team would take you from 30% up to 75%. I suspect that most of the benefits come from the first one or two, and almost nothing from 5-7. (Dennis suggested 20 years ago that the highest marginal value is provided by the second player, because that gives the first player someone to throw to). So, to get those 45 percentage points, I'll say it's 14, 14, 9, 4, 2, 1, 1 for each added player.

That puts the O efficiency at 44%, D efficiency at 28%. That means the O will score 15/34 times instead of 15/50. The other team will score 28.3% of 33 times or 9.3 goals. Set the point spread at 5.5.

Using a Pythagorean exponent of somewhere between 4 and 6, which my earlier research has suggested, that gives an expected winning percentage of 87-95%. Interpolating my table would give an estimate of about 93%.

Also, IIRC, a 40 point difference in RRI translated to a 1 point difference in expected score.


#28 said...

As a thought, what if, instead of playing man D, where each defender has approximately the same effect on the disc (i.e. near or away from the play the play), the superstar's team plays zone or junk, allowing him to disproportionately be near the disc and make plays. Do you think this would increase his defensive activity?

Anonymous said...

Are you suggesting that the difference between a top-level player and a player on a low-finishing regionals team is comparable to the difference in wealth between Bill Gates and some guy who owns a Kia dealership out on I-95? I don't see it quite that way. My intuitive feeling is that the winning percentage would depend a great deal on the type of elite player in this hypothetical pickup game. In particular, an experienced coach and student of the game would be the most likely to influence the outcome, especially if coupled with a patient and calm manner.

parinella said...

I was thinking that the difference in zone would be less than in man because zones require a lot more teamwork, but I hadn't thought of the possibilites of focusing the defense on the player.

parinella said...

I was just picking an extreme example to show that "average" is hard to define when there is a large range in the data set. If there was a Bill Gates in ultimate, his team would win every game 15-0 unless he was playing Warren Buffett (who, by the way, is so convinced that really rich people should be taxed a lot more, so why doesn't he just pay more himself?).

Anonymous said...

As far as your parenthetical goes, Brecht says, "Was ist Einbruch in eine Bank gegen die Grüdung einer Bank?" (Think of taxation as breaking into the bank. . .)

But my other point was that the players considered the game's best (look at the Huddle's fantasy draft, for example) vary quite a bit in their relevant characteristics for this scenario. A player from the Florida system would probably do the best with it, since the set-up would most closely approximate their normal style. But the morale issue might worry me a bit.

Anonymous said...

If one person pays more in taxes, that accomplishes nothing.

If all rich people are taxed more, you can tax the middle/lower classes much less and/or reduce the deficit.

parinella said...

Warren Buffett could pay enough extra in taxes to save any number of "worthwhile" projects (say, NPR with its $90M). Yet he thinks that his money is better managed by letting Bill Gates' foundation solve problems.

Anonymous said...

Certain types of zone D can certainly cause a "superstar" to have a huge effect of defensive efficiency

My league team when I first started playing had a superstar level (at least compared to the skill level of the rest of our team) player - we played a 5 man cup with him playing the deep-deep man behind it.

I guess the question is really what's average - the entire defense basically assumed that nobody could throw a decent hammer over the cup

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luke said...

paga what?