There's a lot of strong dynasty analysis out there, especially when compared to five or ten years ago. But most of it is so dang practical-- Player X is undervalued, Player Y's workload is troubling, the market at this position is irrational, and take this specific action to win your league. Dynasty, in Theory is meant as a corrective, offering insights and takeaways into the strategic and structural nature of the game that might not lead to an immediate benefit but which should help us become better players over time.

Estimating the Impact of Win-Now Trades

Last week, we talked about the power of thinking carefully about a topic to find answers when you don't have the data or the resources to do research. I also discussed one of my favorite thought experiments-- estimating the value of a regular-season game relative to a playoff game in fantasy.

Based on my thought experiments, I calculated that in a typical 12-team league, posting the top weekly score in Weeks 16 or 17 would boost a team's naive title odds by over 8% (from 8.3% to 16.6%). The top weekly score in Week 15 would increase title odds by over 4% (from 8.3% to 12.5%). And I estimated that the top weekly score during a regular season week would likely result in a 1-2% boost to a team's title odds.

Then I built and shared a model to estimate championship odds based on expected weekly winning percentage so we could test these questions directly and found that with our baseline assumptions (including that all teams naturally had a 50% chance to win each week), a guaranteed regular-season win boosted title odds by 2%, to 10.3%.

It was nice to validate the original estimate, but I noted that the most powerful part of a model like this is we could easily change the default assumptions and test the impact of a host of factors. I ended last week with a few questions for you to carefully consider. Here's the list again:

1. With our assumption that we had a 50% chance to win each game, taking the genie's offer (of a guaranteed win) in the wildcard round increased our title odds by 2.2% more than taking it in the regular season, while taking it in the finals was 6.3% better. If our default chances of winning each game were 70% instead, would the wildcard round become more advantageous, less advantageous, or remain the same? How about the finals?
2. What if there were no bye weeks and only four teams made the playoffs? Would the finals week become more or less valuable compared to a random regular season week, or would its value remain the same?
3. If the genie offered to guarantee a victory in two regular-season weeks, would that be twice as valuable?
4. How many regular-season victories would he have to guarantee for it to become more valuable than a guaranteed high score during the championship week?
5. Is it better to improve your chances of winning every game by 5% if you're starting with a 50% chance to win or if you're starting with a 60% chance to win? At what starting winning percentage would an extra 5% chance be most impactful?
6. Who is more likely to win a championship, a team with a 50% chance to win every game or a team that alternates between a 30% chance to win and a 70% chance to win?

Now we're going to answer those questions. And then we're going to generalize those answers to address the big, important question: when is the most impactful time to make win-now trades in dynasty?

Answer #1

• With our assumption that we had a 50% chance to win each game, taking the genie's offer (of a guaranteed win) in the wildcard round increased our title odds by 2.2% more than taking it in the regular season, while taking it in the finals was 6.3% better. If our default chances of winning each game were 70% instead, would the wildcard round become more advantageous, less advantageous, or remain the same? How about the finals?

For a team with a 70% chance of winning each game, taking a guaranteed win during the regular season increases title chances by 1.67% (from 41.72% to 43.39%). Taking the same guaranteed win in the wildcard round increased title odds by 3.72%. The relative edge of the wildcard round went from 2.2% to 2.05%, which is a decline, but only a very small one, even though this team already has about a 2-in-3 chance of earning a bye and skipping the wildcard round entirely.

On the other hand, taking the guaranteed win in the championship game increases expected title chances by a whopping 17.88%, nearly 10% better than in the original "50% to win" scenario. For teams that are already great, the last two weeks of the season become significantly more valuable relative to the rest of the year.

Answer #2

• What if there were no bye weeks and only four teams made the playoffs? Would the finals week become more or less valuable compared to a random regular season week, or would its value remain the same?

In leagues with a playoff bye, the guaranteed regular-season victory increased title chances by 2.01%. In leagues without a bye, the guaranteed regular-season victory increased title chances by 2.21%. The guaranteed playoff victory remained equally valuable in both leagues, but the fewer teams make the playoffs, the more important each regular-season game becomes.

Answer #3

• If the genie offered to guarantee a victory in two regular-season weeks, would that be twice as valuable?

This is a deceptively tricky one. It might be tempting to use one of my favorite problem-solving strategies-- reductio ad absurdum. Comparisons that are difficult in the middle of the distribution often become simple at the extremes.

If the value of wins increased linearly, then going from one guaranteed win to two guaranteed wins would be just as good as going from 13 guaranteed wins to 14 guaranteed wins. But whether a team finishes 13-1 or 14-0, it is certainly going to earn the #1 seed and a first-round bye in the playoffs; the marginal value of that fourteenth win is zero.

Because the first guaranteed win has some marginal value and the last guaranteed win has no marginal value, we have demonstrated that not all wins are equally valuable. And since the last marginal win has no value, we might be tempted to assume that the marginal value of each additional win is less than the value of the win before.

But it's not! While the first guaranteed win increases your title odds by 2.01%, the second increases it by 2.21%, and the third provides an even bigger boost-- 2.31%. Here's the marginal value of each additional guaranteed win in the regular season.

What's going on here? It's the bye week. Earning that bye automatically doubles your chances of winning it all (assuming you are otherwise 50/50 to win each game). Based on historical rates, a team needs an expected record of 9-5 to have a 50/50 shot at the bye. 

For a team with a 50% chance of winning each game to have an expected record of 9-5, it needs to start with four guaranteed wins already in the bank. So it makes sense that each guaranteed win that gets us closer to that goal is more valuable than the one that came before, and also that once we've hit that mark, each extra win becomes less valuable than the one before.

(Technically, because an 8-6 team is very slightly more likely to earn a bye than a 10-4 team is to lose the bye, the breakeven point is more like 8.9-5.1. This is why the fourth guaranteed win is imperceptibly less valuable than the third one, it pushes us just a hair past the true tipping point for the bye.)

Answer #4

• How many regular-season victories would he have to guarantee for it to become more valuable than a guaranteed high score during the championship week?

 We already established that a high score in Week 17 doubles your season-long title odds (from 8.3% to 16.6%, or one in six). How many guaranteed regular-season wins would it take to top that? Well, we know that a guaranteed ticket to the wildcard round would leave us with 1-in-8 championship odds (the chances of winning three straight coin flips), while a guaranteed bye would put our odds at 1-in-4.

Given that 8 wins makes it overwhelmingly likely you make the playoffs and 10 wins makes it overwhelmingly likely you earn a bye, I might expect the breakeven point to be around a 9-5 expected regular-season record. Which, assuming our chance of winning in non-guaranteed games was 50%, would require four guaranteed wins to achieve.

Is that right? More or less! Three guaranteed regular-season wins increase our title odds by 6.54%, which is well short of the 8.33% boost granted by a championship game high score. Four guaranteed regular season wins increase our odds by 8.85%, which is just marginally better than the title game. So four games is the tipping point where the regular-season boost inches past the playoff boost.

Answer #5

• Is it better to improve your chances of winning every game by 5% if you're starting with a 50% chance to win or if you're starting with a 60% chance to win? At what starting winning percentage would an extra 5% chance be most impactful?

This is a really cool one. We already saw from question #3 that earning the bye is one of the biggest determinants of our overall title chances. Given how the marginal value of extra wins increases until you get up to 3 or 4, you might think that the marginal value of a 5% boost to our expected win% likewise increases until we're more likely to earn a bye than not and then starts to decline. That "more likely to earn the bye than not" mark comes at around 65% expected win%. (Actually closer to 64%, but we're rounding.)

And all of that is correct as far as it goes. But there's a second factor at play-- our chances of winning the title contingent on making the playoffs. If we assume we have the bye already, then a team with a 50% chance to win each game has a 25% chance at the title, a team with a 60% chance has a 36% chance at the title, a team with a 70% chance has a 49% chance at the title, and a team with an 80% chance has a 64% chance at the title. Those odds increased by 11%, then 13%, then 15%. These are accelerating returns-- the better our winning percentage is to start, the more valuable it is to boost it further.

So in this question we start with accelerating returns on our bye chances up to a 64% win%, then we have diminishing returns. But we have accelerating returns on our chances of winning in the playoffs across the whole distribution. Which creates a situation where our chances of winning a title rise, then peak, then fall... and then eventually start rising again.

Weird, huh?

One huge caveat: it is cheaper and easier to improve your odds of winning by 5% when you're starting at 50% than when you're starting at 60%. First off, the better your team is, the harder and more expensive it will be to find upgrades for your starting lineup. Second off, fantasy football scores tend to follow a bell curve, which means it takes fewer points per game to move from the 50th to 55th percentile than to move from the 70th to 75th percentile. (You can easily verify this for yourself by looking at your league's history. Scores are much more tightly clustered among teams ranked 5th-8th than among teams ranked 1st-4th or 9th-12th.)

So while a better team benefits more from a given boost to its weekly odds, a worse team can buy that boost much more cheaply.

Answer #6

• Who is more likely to win a championship, a team with a 50% chance to win every game or a team that alternates between a 30% chance to win and a 70% chance to win?

Clever readers probably noticed I was being tricky here and answered "It depends on whether the second team is 70% to win in the wildcard and championship vs. 30% to win in the semifinals, or 30% to win in the wildcard and championship vs. 70% to win in the semifinals".

First, let's look at the impact on the regular-season record. Alternating a high and a low chance of winning decreases variance. (At the most extreme, imagine a team that had a 100% chance to win in one week and a 0% chance to win in the next. Such a team would finish every year with a 7-7 record.) For fantasy, decreasing variance is bad; if the previous answers have impressed one thing upon you, hopefully that one thing is the importance of earning the bye.

Both teams have an equal chance of making the playoffs, but the "50% every week" has a 1.87% better shot at earning that critical bye. If the 70/30 team gets that unlucky "30% in the wildcard and finals, 70% in the semis" pattern, the 50/50 team pulls even further ahead; its chances of winning the league are 3.01% better than the up-and-down squad's.

On the other hand, if the 70/30 squad gets the more favorable "70% in the wildcard and finals, 30% in the semis" pattern, the 50/50 team... still has a 0.05% better shot at winning the title! The only trick here is that there was no trick; the lower-variance team has better odds either way.

The 70% win rate in the wildcard round more than offsets the reduced chances of earning a bye and makes this team slightly more likely to reach the semifinals (39.44% vs. 33.33%). But the consistent team has a 25% chance of winning it all once it reaches the semis, while the inconsistent team has just a 21% chance (regardless of which playoff pattern it gets), which puts the consistent team back in front again.

The takeaway is that low-win% weeks in the playoffs hurt your title chances a heck of a lot more than high-win% weeks help them. (Again, at the most extreme, a team that is 100% to win in one week and 0% to win in the next can never possibly take home a title in a system that requires winning two consecutive games.)

Applying These Lessons

From the questions above, we learned that the three playoff weeks are several times more valuable than any regular season week, though the rarer playoff spots are in your league, the more valuable the regular season becomes. We learned that in leagues with playoff byes, earning that bye is of critical importance. We learned that teams can never really be "too good" to benefit from further increases to their title chances. We learned that variance in final season-long outcomes is useful, and ironically the best way to get that variance is by minimizing the variance in our week-to-week odds. And finally, we learned that shoring up a weak matchup in the playoffs is more impactful than further improving a strong one.

So when should managers make win-now trades? I would still argue... maybe never. I favor building the best team possible and letting the chips fall where they may.

But if you're going to make a win-now trade, these guidelines will ensure that it is as impactful as possible.

1. It is better to make a move early than late. Each extra regular-season win you add is more valuable than the last until you get to around 10 total wins.
2. It is better to make win-now trades when you're in a tight race for a bye than when you're in a tight race just to make the playoffs. Look to make such trades after you start 3-0 (even with a relatively weak team), avoid them after you start 0-3 (even with a relatively strong team).
3. Good teams do, in fact, benefit more from an increase in chances of winning. (Though bad teams can buy those increases much more cheaply.)
4. It is better to make a trade that improves your performance in a week when your team is looking weak rather than in a week when your team is already looking strong.
5. Even in the most implausibly optimistic scenarios, there's a 90+% chance that a win-now trade will not change your end-of-season outcomes and whatever you spent on it will be wasted.