How should your fantasy team use bench spots?

Ranking the top 200 players is only going to get you so far in fantasy hockey. If you want to rob your competition (read: co-workers) of trash talking opportunities you need to get the most out of your roster. You’re not Mike Babcock, so that might mean looking at value above replacement, or thinking through how valuable each category is. You can also get a leg up by making the most of your bench spots.

What matters when we look at bench spots?

A bench player is useful if enough players at the same position aren’t playing the same day. That means the schedule matters, injuries matter, and positions matter.

There are probably gains you could make by going deep into which teams have complementary schedules, but that granular information is easier to process when evaluating a trade, or week-to-week add/drops. You don’t have the mental space at the draft to worry about those combinations. You’ll weigh that information badly and your plans will crumble as the perfect piece for your strategy gets snatched up.

I’m also ignoring goalies, since they’re very different.

Value varies across positions

Depending on your league’s fantasy rules dynamics might be slightly different, but in most leagues centres put up the best numbers, but have fewer line-up spots than other positions. Your best centre on the bench probably puts up better raw numbers than one of the wingers in your line-up, and would be a big improvement over even a second-tier defenceman. How much better does a player who isn’t going to play as much need to be to provide more value to your team?

NHL Scheduling

How often do teams play the same day? We exclude weird weeks, because which actual day is busy is less important than the mix of busy and slow days. Mon: 25% of teams play; Tue: 66%; Wed: 25%; Thu: 62%; Fri: 34%; Sat: 72%; Sun: 45%

A typical week in the NHL has three busy days, three slow days, and an in-between. A bench player is rarely going to get into the line-up on a busy day, but they’re also not going to play many games on slow days.

Injuries

Elite players don't miss many games. If you look at top point producers and check how many games they played the next season, over half played at least 80 games. As a whole, these players played 94% of the year. Cumulative Density: 82 Games: 29%; 80: 50% (median); 77: 67% (mean); 65: 94% (95%ile)

Bench spots serve as insurance for injuries, but elite players really don’t spend that much time injured; You can expect about five games missed per player. Still, the effect is big enough that we’d be negligent to not include it.

The effect of injuries is small, but not insignificant. Injuries pump up the value of depth players, but how much you'll get out of them depends on the size of your line-up.

(The above chart is based on running the analysis below with and without injuries factored in)

The Math

For any given day of the week, line-up size, and depth spot, we can turn to some probability and combinatorics:

P(n+1)=1-x^n; P(n+2)=1-x^(n+1)-(n+1)(1-x)x^n; P(n+3)=1-x^(n+2)-(n+2)(1-x)x^(n+1)-(n+2)!(1-x)^2*x^n/n!2!

where n is the number of line-up spots and x is the probability of any player having a game that day. So, for the sixth winger on a team with four wing spots on a Thursday, take the P(n+2) equation, use n = 4 and x = 0.94 * 0.62 = 0.58 (the probability a player is not injured and has a game on any given Thursday).

(Note that this assumes schedules are independent, which isn’t true, but accounting for that gets into the unhelpful territory of checking every player’s compatibility instead of building a “satisficing” model that you can apply to every player in the league)

To expand over a whole season, you take the value for each day of the week, and multiply it by the proportion of games that are played each day. To continue the previous example, 19% of games are played on Thursdays (excluding weird weeks, because again, we don’t care about the day of the week, we care about busy and slow days). Add up those numbers from each day of the week and you have the proportion of games you expect that player to get into the line-up.

Results

Bench value is a function of line-up size. Balancing your roster is vital. All else being equal, a 3rd player in a 2 spot position provides about the same nmber of starts as a 6th fighting for one of 4 spots, or a 9th trying to break into the top 6.

Line-up spot,1,2,3,4,5,6,7,8,9,10,11,12
n+1,0.491008,0.714896,0.830904,0.896319,0.935141,0.958898,0.973714,0.983073,0.989039,0.992869,0.995342,0.996946
n+2,0.267120,0.482881,0.634660,0.741029,0.816361,0.869999,0.908201,0.935345,0.954570,0.968142,0.977695,0.984403
n+3,0.159239,0.331103,0.475106,0.590364,0.682265,0.755394,0.813197,0.858446,0.893497,0.920376,0.940800,0.956196

An example

Let’s posit an unlikely situation. We’re in a league where only points from last year count. The league’s line-up is 3C / 6W / 4D and a giant bench. We’ve drafted the following nine players for the bench. How much do we expect from them? (We’re going to ignore the fact that some of these players missed games)

C – Derek Stepan (55 points)
C – Anze Kopitar (52)
C – Tyler Johnson (45)
W – Marian Hossa (45)
W – Patrick Maroon (42)
W – Mikko Rantanen (38)
D – Andrei Markov (36)
D – Damon Severson (31)
D – Morgan Reilly (27)

Easy! Just multiply those values by our factors! We get:

C – Derek Stepan: 55 * 0.83 = ~46 points
W – Marian Hossa: 45 * 0.96 = ~43
W – Patrick Maroon: 42 * 0.87 = ~37
C – Anze Kopitar: 52 * 0.63 = ~33
D – Andrei Markov: 36 * 0.90 = ~32
W – Mikko Rantanen: 38 * 0.76 = ~29
D – Damon Severson: 31 * 0.74 = ~23
C – Tyler Johnson: 45 * 0.48 = ~21
D – Morgan Reilly: 27 * 0.59 = ~16

Tyler Johnson’s value takes a beating, and the difference between Kopitar and Markov is negligible. Note that even if Markov was going to be your 4th defenceman (i.e. not a bench player) Stepan, Hossa, and Maroon on the bench still all provide more value, but that might not be true if you’re looking at replacement value (note that replacement is irrelevant for bench players). Let’s look at another worked example:

C – Bo Horvat (52 points) ~43 expected points
W – Evander Kane (43) ~41
W – Rick Nash (38) ~33
D – Jacob Trouba (33) ~30
C – Artem Anisimov (45) ~29
W – Andrei Burkovsky (35) ~26
D – Marc-Edouard Vlasic (28) ~21
C – Mike Fisher (42) ~20
D – Mattias Ekholm (23) ~14

In this case Jacob Trouba is more valuable than Anisimov, despite putting up 12 fewer points. Mike Fisher tumbles even further. In these cases, if you compare “best available” drafting with these results you see the difference it makes. If you have Horvat and Kane and then pick Nash and Trouba (71 real points) instead of Anisimov and Fisher (87 real points) you can expect 14 more points (63 vs 49) to show up in your match-ups.

“I don’t wanna do timesing, just gimme the gist”

Okay that’s fine. I did examples across a variety of scenarios (league sizes, line-ups, &c) and here’s the quick and dirty (and intuitive) guideline that will probably do alright for you if you follow it until your bench is full

1. For your first two bench spot take a winger and a centre (1C 1W)
2. Next, take another winger. (1C 2W)
3. Take a defenceman or another centre. (1C 2W 1D or 2C 2W)
4. Make sure you have a defenceman, and don’t go over two centres (1C 1D 3W or 2C 1D 2W)
5. Bring yourself up to 2C 1D 3W

Your mileage may vary, but it won’t steer you horribly wrong. If your league differentiates between left and right wing, centres become a lot more valuable and I’d strongly recommend playing through a few scenarios using the factors.

What about dual eligibility?

You’re golden. Look at the table: If you’re in a 2C 4W league, even if your C/W is your ninth best forward, they’re going to play over 70% of their games. They end up playing even more than the math says, so you might not want to even bother discounting their value. These guys just don’t end up sitting out often. That’s why they’re useful.

“But I drafted this guy last year and he always got to play!”

That’s lucky! There’s a ton of variability in this. Sometimes your fifth winger will play less than your third centre. It’s rare but even with the assumptions I made above, it’ll happen with about 1% of skater / winger pairs (I did the math). I just don’t have faith in your ability to figure that out in advance and make a decision based on it. We need to make our best guess, and it’s a bad bet to make.

Break a leg

You can be aware that the bench devalues player contributions, but if you can’t measure it, you can’t manage it. In the Moneyball era the A’s didn’t bother paying for defensive contributions. It wasn’t because defence doesn’t matter, but if you can’t measure it (they didn’t know how to at the time), how can you value it properly? Hopefully with this tool you can properly value the contributions from your bench.

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