Making the NHL draft lottery more fun

The NHL draft lottery is to be held today in Toronto. Indeed, it might have already happened; I can't find any reference to what time the lottery actually takes place. The NHL will be announcing the results just before 8:00 pm tonight.

(Post-lottery update: someone on Twitter said the lottery took place immediately before the announcement.  Which makes sense; the shorter wait time minimizes the chance of a leak.  Also: the Oilers won, but you probably know that by now.)

The way it works is actually kind of boring, and I was thinking about ways to make it more interesting ... so you could televise it, and it would hold viewers' interest.

(Post-lottery update: OK, they did televise a medium-sized big deal for the reveal.  But, since the winner was already known, it was more frustrating than suspenseful. This post is about a way to make it legitimately exciting while it's still in progress, before the final result is known.)

Let me start by describing how the lottery works now. You can skip this part if you're already familiar with it.

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The lottery is for the first draft choice only. The 14 teams that missed the playoffs are eligible. Whoever wins jumps to the number one pick, and the remaining 13 teams keep their relative ordering. 

The lower a team's position in the standings, the higher the probability it wins the lottery. The NHL set the probabilities like this:

 1. Buffalo Sabres       20.0%
 2. Arizona Coyotes      13.5%
 3. Edmonton Oilers      11.5%
 4. Toronto Maple Leafs   9.5%
 5. Carolina Hurricanes   8.5%
 6. New Jersey Devils     7.5%
 7. Philadelphia Flyers   6.5%
 8. Columbus Blue Jackets 6.0%
 9. San Jose Sharks       5.0%
10. Colorado Avalanche  3.5%
11. Florida Panthers      3.0%
12. Dallas Stars          2.5%
13. Los Angeles Kings     2.0%
14. Boston Bruins         1.0%

It's kind of interesting how they manage to get those probabilities in practice.

The NHL will put fourteen balls in a hopper, numbered 1 to 14. It will then randomly draw four of those balls.

There are exactly 1,001 combinations of 4 balls out of 14 -- that is, 1,001 distinct "lottery tickets". The 1001st ticket -- the combination "11, 12, 13, 14"  -- is designated a "draw again." The other 1,000 tickets are assigned to the teams in numbers corresponding to their probabilities. So, the Sabres get 200 tickets, the Hurricanes get 85 tickets, and so on. (The tickets are assigned randomly -- here's the NHL's .pdf file listing all 1,001 combinations, and which go to which team.)

It's just coincidence that the number of teams is the same as the number of balls. The choice of 14 balls is to get a number of tickets that's really close to a round number, to make the tickets divide easily.

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So this works like a standard lottery, like Powerball or whatever: there's just one drawing, and you have the winner. That's kind of boring ... but it works for the NHL, which isn't interested in televising the lottery live.

But it occurred to me ... if you DID want to make it interesting, how would you do it?

Well, I figured ... you could structure it like a race. Put a bunch of balls in the machine, each with a single team's logo. Draw one ball, and that team gets a point. Put the ball back, and repeat. The first team to get to 10 points, wins.

You can't have the same number of balls for every team, because you want them all to have different odds of winning. So you need fourteen different quantities. The smallest sum of fourteen different positive integers is 105 (1 + 2 + 3 ... + 14). That particular case won't work: you want the Bruins to still have a 1 percent chance of winning, but, with only 1 ball to Buffalo's 14, it'll be much, much less than that.

What combinations work? I experimented a bit, and wrote a simulation, and I came up with a set of 746 balls that seems to give the desired result. The fourteen quantities go from 70 balls (Buffalo), down to 39 (Boston). 

In 500,000 runs of my simulation, Buffalo won 20.4 percent of the time, and Boston 1.1 percent. Here are the full numbers. First, the number of balls; second, the percentage of lotteries won; and, third, the percentage the NHL wants.

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                                    result  target
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 1. Buffalo Sabres        70 balls   20.4%   20.0%
 2. Arizona Coyotes       63 balls   13.5%   13.5%
 3. Edmonton Oilers       61 balls   11.5%   11.5%
 4. Toronto Maple Leafs   58 balls    9.3%    9.5%
 5. Carolina Hurricanes   57 balls    8.5%    8.5%
 6. New Jersey Devils     56 balls    7.3%    7.5%
 7. Philadelphia Flyers   54 balls    6.5%    6.5%
 8. Columbus Blue Jackets 53 balls    5.9%    6.0%
 9. San Jose Sharks       51 balls    4.7%    5.0%
10. Colorado Avalanche    49 balls    3.7%    3.5%
11. Florida Panthers      47 balls    3.2%    3.0%
12. Dallas Stars          45 balls    2.5%    2.5%
13. Los Angeles Kings     43 balls    1.9%    2.0%
14. Boston Bruins         39 balls    1.1%    1.0%
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The probabilities are all pretty close. They're not perfect, but they're probably good enough. In other words, the NHL could probably live with awarding the Bruins a 1.1% chance instead of a 1% chance.

If you did the lottery this way, would it be more fun? I think it would. You'd be watching a race to 10 points. It would have a plot, and you could see who's winning, and the odds would change every ball. 

Every team would have something to cheer about, because they'd probably all have at least a few balls drawn. The ball ratio between first and last is only around 1.8 (70/39), so for every 9 points the Sabres got, the Bruins would get 5. 

The average number of simulated balls it took to find a winner was 72.4. If you draw one ball every 30 seconds ... along with filler and commercials, that's a one-hour show. Of course, it could go long, or short. The minimum is 10; the maximum is 127 (after a fourteen-way tie for 9). But I suspect the distribution is tight enough around 72 that it would be reasonably manageable.

Another thing too, is ... every team would have a reasonable chance of being close, and an underdog will almost always challenge. Here's how often each team would finish with 7 or more points (including times they won):

 1. Buffalo Sabres         53.3 percent
 2. Arizona Coyotes        43.4
 3. Edmonton Oilers        40.2
 4. Toronto Maple Leafs    35.7
 5. Carolina Hurricanes    33.9
 6. New Jersey Devils      32.4
 7. Philadelphia Flyers    29.5 
 8. Columbus Blue Jackets  27.7  
 9. San Jose Sharks        25.2 
10. Colorado Avalanche     22.1 
11. Florida Panthers       19.5 
12. Dallas Stars           16.7 
13. Los Angeles Kings      14.5 
14. Boston Bruins          10.6 


And here's the average score, by final position after it's over:

10.0  Winner
 8.0  Second
 7.3  Third
 6.6   ...
 6.1
 5.6
 5.2
 4.8
 4.3
 3.9
 3.5
 3.0
 2.4
 1.6

That's actually closer than it looks, because you don't know which team the bottom one will be. Also, just before the winning ball is drawn the first-place team would have been at 9.0, which means, at that time, the second-place team would, on average, have been only one point back. 

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The problem is ... it still takes 746 balls to make the odds work out that closely. That's a lot of balls to have to put in the machine. Of course, that's just what I found by trial and error; you might be able to do better. Or, you could use a smaller number of balls, and accept a probability distribution that's different from the NHL's, but still reasonable.

Or, you could add a twist. You could give every ball a different number of points. Maybe the Sabres' balls are worth from 5 points down to 1, and the Bruins' balls are only 3 down to 1, and the first team to 20 points wins. I don't think it would be that hard to find some combination that works. 

That's kind of a math nerd thing. I'd bet you can find a system that comes as close as I got with less than 100 balls, and I bet you'd be able to get to it pretty quick by trial and error. 

At least, the NHL could, if it wanted to.