### A model for explaining home field advantage between sports

It occurred to me that it might be possible to predict, or explain, the difference in home field advantage between different sports, based on their rules and outcomes. In this post, I'll just talk about the "absolute" home field advantage, in terms of goals or points. (The translation to winning percentage is easy after that, but I'll save that for a future post.)

Take a look, and let me know what you think.

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Here's the home field advantage (HFA) for three different sports, in terms of goals or points.

0.453 - Premier League Soccer (2010-11)

4.000 - NBA (estimate)

0.783 - NHL (1980-81 to 1984-85)

As expected, the HFA is highest for basketball, where the most points are scored, and lowest for soccer, where the fewest "points" are scored. Still, they're very different in terms of rates. Here are the percentages by which the home team outscores the visiting team:

38% - Premier League (+0.453 home goals per 1.1737 road goals)

4% -- NBA (roughly, +4 per 100)

27% - NHL (+.783 per 2.937)

These HFAs are all over the place. In basketball, the home team only outscores the visiting team by 4 percent. But, in the NHL, it jumps to 27%, and in soccer, it's almost 40%!

Why the big differences?

It has nothing to do with the length of the game. If the home team scores 4% more points over 48 minutes, you can also expect it to score 4% more points over a minute, a quarter, or a season. (It's the percentage of how many more *wins* the home team gets that depends on game length, but, again, that's not what we're discussing in this post.)

So, what is it then? There are probably many contributing factors, but I think the biggest has to do with the structure of the individual games. That's because it's easy to get a higher or lower HFA just by changing the rules.

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Start by looking at the NBA, where the HFA is about 4 points a game. Let's change the way basketball works, to move that difference away from 4 points.

In fact, let's do that while keeping many aspects of the game constant. We'll stay with a game where each team gets 100 possessions, has to throw a ball through a hoop on a basketball court, and has an average score of 100. We'll just change the "internal" rules.

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Suppose we change the game to consist of only foul shooting. Each possession, the team gets two foul shots. If it sinks them both, it gets two points. Otherwise, it gets zero. We can assume the average player shoots 71%, so that the probability of two straight makes is almost exactly 50%. (If 71% seems a bit low, just imagine that we make the hoop a bit smaller at the same time we change the rules.)

We have empirical data that lets us figure out what HFA would be, thanks to King Yao, who compiled home and road free-throw percentages for a few recent NBA seasons. The numbers were:

75.95% home team

75.72% visiting team

For the chance of making two straight shots, then, we can just square those numbers:

57.68% home team

57.34% visiting team

The difference is 0.34%. Over 100 possessions, that's .34 extra scores, or around 0.7 extra points. That's much smaller than the 4 point HFA in "real" basketball. We've reduced HFA by 80 percent just by changing the rules!

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How can we construct a game where HFA is higher? Again, that's easy -- try "double or nothing" basketball.

In that game, when you score a field goal, you don't get the two points yet. Instead, you immediately get a second possession, and you have to score on that one too. If you do get two in a row, you get 4 points. If you don't get the second one too, you get zero.

A game consists of 100 possessions for each team (so each team will get somewhere between 100 and 200 attempts to make a field goal).

In this game, the HFA will be roughly double. How do we know? Well, the real life HFA is 4 points, and each team gets roughly 100 attempts. So, we can guess that, in normal basketball, the home team might score on around 52% of attempts, while the visiting team will score only on 50%.

But, now, each team has to make two in a row. The home team will do that around 27% of the time (52% squared), while the visiting team will be at 25% (50% squared). That's still two extra scores per game, but now each score is worth four points. So, instead of winning 104-100, the home team will win 108-100.

The change in the rules has increased the HFA from 4 points to 8 points -- from 4% to 8%.

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So: three different hoops games, three different HFAs. That shows that you have to examine the rules in order to understand where HFA is coming from. It can't be just crowd influence, or referee bias, or familiarity with the home court, or anything like that. Those are things that could *cause home field advantage to exist*. But they aren't things that could, on their own, cause the level of home field advantage to *vary between sports*. For that, you need to examine the rules.

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So, is there a factor that explains how the HFAs change for the three games?

It seems to me that the answer is: the level of "compounding" of events, the number and difficulty of the things that all have to go right for you to score.

For the "two in a row" game, you need to score twice in a row, not just once. If you have a 4% advantage on each one, you'll have an 8% advantage on two of them compounded. (1.04 squared is about 1.08.)

For the foul shooting game: In "real" basketball, there's more than just shooting. To score a field goal, you have to do a whole bunch of things right. For instance, you have to (a) pass the ball around accurately; (b) deke out a defender enough to get a good shot; (c) have the other members of your team distract the other defenders so they can't block; and then (d), take an accurate shot.

That's four things that might all have to go right. From the discussion above, we know the HFA for two consecutive foul shots is 0.34%. Suppose each of those four of those things, from (a) to (d), have that same 0.34% advantage. Then the home team gets an advantage of roughly four times 0.34%, or roughly 1.4 percent.

That hypothetical only gets us to 1.4 percent, not to 4 percent. That suggests that more than four compoundings are necessary -- maybe as many as 10. That's certainly reasonable. Foul shooting seems to be something that's simpler than most other basketball skills. We assumed, for convenience, that "taking an accurate shot" was exactly as complex as foul shooting ... but it might be twice as complex. You have to shoot accurately, but first you also have to judge the shot. If the other three things are also twice as complex, which doesn't seem implausible, then we have eight compoundings.

Anyway, the point is not to get this particular example to work out perfectly, but to show that it's at least a decent approximation.

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The "compounding" explanation also seems to work if you compare hockey to basketball.

In basketball, the net is unguarded. In hockey, there's a goalie trying to stop the puck. The goalie has his own HFA, while the hoop presumably does not. So, that's at least one extra compounding in favor of hockey.

In basketball, it's difficult to force a turnover; it's a fairly rare event. In hockey, it's easy, since physical play is allowed -- you just run into the puck carrier, if you can, and dislodge him.

So, in hockey, part of the goal scoring process is avoiding checks from the opposition. If four players touch the puck before a shot, and each one has the same chance of losing the puck as the entire team has in a basketball possession ... then you have three extra compoundings, since there's an additional HFA for each of the four players.

In soccer, it's even more extreme: it isn't unusual to take 10 or 15 passes before you get a decent chance at a shot. So, ten things have to go right. If each pass in soccer has the same chance of being intercepted as a pass in basketball, but soccer requires three times as many passes before a goal ... well, now you have three more compoundings.

This is all theoretical, of course, but we can check the numbers to see if they're reasonable.

In soccer, the absolute HFA was 38 percent more goals. In basketball, it was only 4 percent. To get from 4 percent to 38 percent, you need about eight times as many compoundings (since 1.04 to the eighth power equals approximately 1.38).

Is eight compoundings reasonable? I think it is, because we can get a similar result another way.

In the NBA, about 50 percent of possessions result in a score. In soccer, it's probably, what, around 2 percent?

If soccer had two compoundings to every one basketball compounding, the scoring rate would still be 25 percent (you'd have to do something with a 50 percent success rate, twice). If it had three compoundings, it would be 12.5 percent. Four compoundings, about 6 percent. Five compoundings, 3 percent. Six compoundings, 1.6 percent, and we're there.

So, a naive estimate is that it to score a goal in soccer, you have to be skilled enough to do what it takes to score a goal in basketball, six consecutive times.

We were expecting 8 compoundings from the "compare HFA" argument. The "probability of scoring" method suggests 6 compoundings.

Not bad! Those two estimates, 8 and 6, are pretty close. Why aren't they closer?

Well, it could be that some of my estimates were off, like the one where I guessed that 2 percent of soccer possessions score.

Or, it could be that there's a large difference in competitive balance between the two leagues. The more lopsided the talent, the lower the HFA (when the better team is so good that it always wins, the HFA is obviously zero).

But, most importantly, it could be that there are factors outside of "compoundings". For instance, referee bias, which need not be anywhere near the same order of magnitude as player HFA. Actually, that could very well be it: in soccer, the referee can have a very large impact on the game. In the NBA, a blown call is worth a couple of points out of 100. But, in soccer, a blown penalty call could be one goal out of two.

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Anyway, if you buy the idea that these two estimates of compounding should be the same, that suggests a way to get a rough estimate of what HFA should be in other sports, at least for sports that are similar to basketball/hockey/soccer, in the respects we used in our arguments. Specifically:

-- you can divide the game into possessions

-- each team gets roughly an equal number of possessions

-- you can only score once per possession

-- you can estimate a probability of an average team scoring on each possession

-- what keeps you from scoring at will is a defense that's similar to defenses in basketball/hockey/soccer and also subject to HFA (which excludes, say, foul shooting or skills competitions)

-- referee bias is roughly the same order of magnitude as for basketball/hockey/soccer.

To get an estimate, you start with a known sport and a known HFA, and you adjust it by the differences. Let's use the NBA as our reference point. It has a 50% percent chance of scoring on each possession (0.5), and an absolute HFA of 4% (1.04).

We now adjust for the number of compoundings based on the difference in probability of scoring on a possession. That leads to this formula:

Let p be the probability of scoring on a single possession in your particular sport. Then:**HFA = 1.04 ^ [log(p)/log(0.5)]**

(Checking that it works for the "double or nothing" basketball variation: p=0.25 gives HFA=1.08, which is 8% more points, which is correct.)

That's your rough estimate of HFA. I emphasize it's *rough*. You'd still need to adjust it (slightly) up if your league has more competitive balance than the NBA, or down if it has less. And you'd have to adjust it up (perhaps substantially) if you think the effect of referee bias on HFA is higher, or down if you think referee bias is lower.

And, of course, there might be other factors I haven't thought of.

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Does someone want to try this for other leagues or other sports, and see how close it comes? I'd be curious to see the NLL, which has, maybe, 12 goals per team per game. The problem is estimating possessions, which is hard for lacrosse but easier for, say, the WNBA.

Labels: basketball, hockey, home field advantage, NBA, NHL, soccer

## 3 Comments:

My initial impression on the discrepincy of the 3 sports' HFA you cited is in the difference in point totals for each. The difference in competition is probably consistently close in all 3, but the winning team in a 1-0 soccer game, (not an unusual result) has scored all of the game's goals/points. A professional basketball game would never finish with a score of, say, 100-0. I would further speculate that if you adjusted adjusted basketball scoring to mimic soccer scores, you may see a more similar HFA. For instance, if you gave each team a "goal" for scoring 50 points in basketball (making a score of 102-98 change to a score of 2-1), you may get similar HFA. I would agree with most of what you've written here, but there is a lot of oversimplifications. You raise many interesting points that would be too lengthy to fully respond to.

Better: determine the relative frequency of games in which soccer teams score no goals, say 18%. Determine the 18th percentile basketball score, let's say 75 points. Then any team that scores 0-75 points is given 0 "goals". Then determine the relative frequency of soccer teams that score 1 goal, say 40%. Determine the 58th percentile score in basketball, let's say 93 points, and any team that scores 76-93 is given 1 goal. And so on and so on. I'll bet that if basketball scores were converted to soccer scores in this fashion, you'd get the same HFA.

If that were true, the "free throw game," the "double or nothing" game, and regular basketball, would all have the same HFA. Wouldn't they?

Because, under all three games, the chance of scoring is 50% per possession, and the average score is 100.

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