Guillotine Leagues: It Helps to Reason Backwards

Guillotine leagues have something in common with the 21 Flags game from Survivor: reasoning backwards from the last move to the present is an important part of winning. In 21 Flags, each team has an optimal strategy that can be discovered by picturing the winning move at the end of the game, and working backwards to figure out how to get there.

For those who are interested in 21 Flags, here are two videos about it. The first one shows the game as it was played on Survivor. The second one is from a Youtube math channel, showing the optimal strategy for winning the game.

CONTENT WARNING: THIS ARTICLE CONTAINS MATH

Players in guillotine leagues should take the same approach of picturing the winning move – meaning, constructing a winning Week 16 lineup – and working backwards. And as the Survivor video shows, many people don’t look ahead this way, even when the logic of the situation demands it.

Based on playing in a guillotine league this year, and looking at Twitter talk about guillotine league bids and decision-making, it seems clear that many teams are spending their free agent money in amounts that would make sense in conventional leagues, but not guillotine leagues. In guillotine leagues, I believe the best strategy is Bid Low, Bid Late, and looking backwards from Week 16 helps to show why.

Why Reasoning Backwards is Critical in Guillotine Leagues

Week 16 match-ups are more important than the preceding ones, but that’s true of fantasy football leagues in general, not just guillotine leagues. Guillotine leagues require reasoning backwards in a way that most leagues don’t, because by far the most important player acquisitions you make are towards the end of the season.

In conventional leagues, a substantial number of the players who play on Week 16 were picked up in the draft (or auction) by the team playing them. Many other good players are picked up early in the season once their value becomes clear. Team-building is ongoing, but the bulk of the talent of the winning team comes from the draft, early season waivers, or trades. Late season waiver and free agent pickups can be important, but they are not the dominant source of the winning team’s top players.

Guillotine leagues are completely different, because most of the championship team’s starting players will be added very late in the season. For example, if the league’s rules call for each team to start 9 players, then the Week 16 game features a total of 18 players. There are 17 teams in a guillotine league. That means that on average, each team starts the season with one player who starts in Week 16. The other eight must be acquired in free agency.

And for all the reasons discussed in the Bid Low, Buy Late article, Week 16 talent gets more concentrated as the season wears on.

That’s why we need to reason backwards from Week 16. If we don’t, we will be tempted to overbid on players who will be useful now, but will not be useful in Week 16. And we will do so at the cost of bidding power later in the season, when more and more Week 16-useful players will be available.

Some More Guillotine League Math

If your FAAB is $1000, and you drafted the average one Week 16 starter in the draft, you need to acquire eight Week 16 starters off waivers. (I am going to keep assuming that your league starts 9). If that is your team, you would have $125 per player to bid on those top players. And that’s if you don’t spend any FAAB on any other players, which you probably should.

Suppose you are a brilliant and very lucky drafter, and you managed to draft 4 players who are Week 16 starters. This is an impressive and rare feat. Assuming you make it to the finals, you would still need to acquire 5 starters before then, at an average of $200 each. It puts those $700 bids to get Tyreek Hill in perspective, doesn’t it?

When do those eight other Week 16 starters become available? Late, very late. The average team does not have two Week 16 starters until Week 9, when 9 teams are left, since 18 ÷ 9 = 2. The number is up to three per team when six teams are left in Week 12. Four per team is surpassed when four teams are left in Week 14, and six per team is reached in the “semifinals” week, Week 15, when three teams are left.

Stretching the FAAB Dollars

Because the number of top players hitting the waivers grows dramatically late in the season, FAAB money spent late makes a much greater impact than FAAB money spent early.

Consider how you could use $300 of FAAB money after Week 1, Week 9, Week 14 and Week 15.

The team that got guillotined after Week 1 might have one player who could be started in Week 16. The top prospects will be vastly overbid on, because (1) everyone is at their maximum FAAB; (2) everyone is engaged and excited to play, and therefore spending-happy; (3) some teams will be unhappy about their draft and worried about early elimination; (4) information about who will be a good Week 16 starter will be unreliable compared to later in the season.

A spending budget of $300 might get you a player who might be a Week 16 starter. In my league, both Le’Veon Bell and Mike Evans went for over $400 in the post-Week 1 waiver auction. A $300 bid would not have gotten you Stefon Diggs, Keenan Allen or Sony Michel in the Week 7 auction, when team’s budgets were so depleted that only 4 teams out of 11 even had $300 to spend.

Now suppose you wait until Week 9 to spend $300. The number of rostered players will be half of the number at Week 1, meaning double the talent is available. Also, there will only be half as many teams bidding for those players. That means your budget goes 4 times farther after Week 9 than it did after Week 1. And that is before considering the fact that some of the teams that made it through more than half the season probably overspent on top players to get there, meaning you are bidding against teams with depleted budgets.

After Week 14, one of the final four teams gets chopped. If you were lucky enough to make it to the Week 15 “semifinals,” you are competing against two other teams for 4 or 5 starters. That is 1/8 the competition for 4X or 5X the talent, compared to week 1, or 32 – 40 times the spending power.

If you make it to the finals, you are competing against one other team for perhaps 6 starters. That is 1/16 the competition for 6X the talent, which is 96 times the spending power.

Now imagine if heading into waivers after Week 15, you had a $300 budget left but your opponent had only $99. You could get 5 of the 6 starters, guaranteed, by identifying the six players you view as starters and bidding $50 on each. Your opponent can only match your bid on one player, because her second-highest bid cannot be more than $49. In general, you can lock in 2n-1 of the top players if you have n times as much FAAB budget, plus 2n. So 3 times your opponent’s budget plus 6 guarantees you 5 out of 6 players every time. And of course, if you have n times your opponent’s budget plus n, you can bid your opponent’s budget plus one on n players, locking up the top n available players.

What a Week 16 Starting Lineup Might Look Like

To have a decent chance of winning, your Week 16 lineup should have a top-2 QB, a top-2 TE, and a half of the top-10 WRs and RBs (assuming you are playing with 2 WRs, 2 RBs and a flex). Based on recent rankings, a league-winning team’s lineup might look something like this:

QB: Pat Mahomes
RB: Melvin Gordon
RB: Alvin Kamara
WR: Adam Thielen
WR: Tyreek Hill
FLEX: James Conner
TE: Eric Ebron

In a league like mine with 3 flexes, you would need two more flexes from the top-14 WRs and RBs. The lineup above would need to be rounded out with players such as Julio Jones or James White.

So ask yourself, as you look to bid on a player after Week 7 or 8: Does the player I am bidding on belong with the players listed above?

This is not to say you should never bid on a player who is not Week 16 material. You need to get to Week 16 to play in Week 16. Sometimes you really need to fill a spot or to beef up your team, especially after mid-season when the competition is getting more intense. By all means, spend $20 to get a backup RB thrust into a leading role by an injury to the bellcow. Just keep Week 16 in mind, and keep an eye on the competition’s FAAB.

Summary

The average team in a guillotine league drafts one player who will be started on Week 16. By Week 9, the average surviving team will have two Week 16 starters on its roster. By Week 12, three. By Week 14, four. and by Week 15, six.

A FAAB dollar spent after Week 9 is worth four times as much as a dollar spent in Week 1. After Week 14, it is worth 36 times as much. After Week 15, it is worth 96 times as much.

After Week 15, if your opponent has x FAAB dollars and you have nx + n FAAB dollars, you can acquire the top n players, Alternatively, if you have nx + 2n FAAB, you can get 2n – 1 of the top 2n players.

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