In my first article on guillotine leagues, which set out the Bid Late, Bid Low strategy, I justified the “bid late” part of the strategy, in part, based on the fact that competition is laxer early in the season and intensifies as the season goes on. In this post I go into more detail.

Early on in the season, a bad team is more likely to survive by sheer luck than late in the season, because when there are more teams, there are more chances that at least one of them will have a low outlying score that is lower than the bad team in question. This reduces the disadvantage of having a bad team but also reduces the advantage of having a good team. In a guillotine league, it makes no difference if you are the highest scoring team or the second-lowest scoring team, until Week 16.

This suggests that all else being equal, player acquisition should wait until later in the season. This fits nicely with other dynamics favoring late season acquisition, including that the later it is in the season, the better you understand which players will be healthy and perform well in the important matches at the end of the season; that more top players become available; and that FAAB deflation increases the value of your FAAB dollars as the season proceeds.

### Quantifying Probabilities

The low risk of losing in the early season is easy to see, but can it be quantified? Well, yes. Given the number of teams remaining and a distribution of possible scores for each team, the probability of your team’s survival could be calculated. Now, I’m not saying *I* can calculate it. I am no statistician. But it can be calculated in principle. Perhaps I will learn how to do it sometime and write an article about it.

What is within my competency is to code a simple simulation. Python has a modular function called “random.gauss”, which given a mean and a standard deviation, returns a random point under a Gaussian probability curve, basically a normal bell curve.

In simple terms: the team’s mean describes how good a team is by stating the average score it would get if the team played a large number of times. Standard deviation is a measure of variability: the smaller the number, the closer together the scores will be bunched. A team with mean 120 can be expected to score, on average, 120 points. A team with mean 120 and a standard deviation of 50 will have very high variability of outcomes, meaning that very high and very low scores should be expected with high probability. Regardless of the value of the standard deviation, scores are more likely to be closer to the mean than further from it. Typically about 68% of the data points fall within 1 standard deviation of the mean. The bell curve is symmetric, so half of 68%, meaning 34%, of points fall between the mean and one standard deviation above the mean, and 34% of points fall between the mean and one standard deviation below the mean. Sixteen percent will be more than one standard deviation above the mean, and 16% will be more than one standard deviation below the mean.

The simulation I created uses one set of figures (for mean and standard deviation) to represent my team, and then separate figures for the competition. To keep things simple, in these simulations all the teams other than mine have the same mean and standard deviation as each other, regardless of how many of them there are. Given these figures, the program runs the random.gauss function on each team, mine and the competition, and checks whether I beat the lowest score. I repeat this 5000 times, count up the number of times I survive, and divide by 50 to get my probability of survival expressed as a percentage.

Disclaimer: I am assuming that the range of scores for a given lineup can be described in a Gaussian distribution, which is not a fair assumption, but should be close enough for the rough job I’m trying to do here, to put some numbers on the role of probability at different points in the season. And I am taking mean and standard deviation values as givens without having any account of how they would be arrived at if we were aiming for a high degree of accuracy. Basically, I’m not trying to make any predictions here. I’m just trying to illustrate general patterns and broad principles using rough figures.

### First Set: One Standard Deviation Difference Below Mean

To get started with a rough sense of the implications of being significantly worse than everyone else, I first ran the numbers assuming a mean score of 90 with standard deviation of 10 for myself, with everyone else in the league one standard deviation above me, at mean 100 and standard deviation of 10.

So if my lineup’s production curve has a mean one standard deviation below the mean of the competition, it means that there’s about a 16% chance that a randomized output for my lineup will be greater than the mean of the competition. It also means that for any given randomized output for any of the other lineups, there’s a 16% chance that it is less than my mean. There are also probabilities that I don’t know that (1) I score more than 100 but the other team also scores more than 100 and more than my team; (2) I score less than 90 but the other team scores less than 90 and less than my team; (3) I score between 90 and 100 and the other team scores less than me. All of these represent winning situations for me.

Here are the probabilities that at least one team out of the competition in the given week scores less than me, i.e., the odds that I survive the guillotine league competition that week. n represents the number of teams in the competition.

Week 1 (n = 16): 74%

Week 2 (n = 15): 74%

Week 3: 73%

Week 4: 72%

Week 5: 72%

Week 6: 69%

Week 7: 67%

Week 8: 65%

Week 9: 64%

Week 10: 62%

Week 11: 59%

Week 12: 54%

Week 13: 49%

Week 14: 45%

Week 15: 37%

Week 16 (n = 1): 24%

So heading into Week 10, for example, if the whole league is at mean 100, standard deviation 10, and your team is at mean 90, standard deviation 10, your odds of surviving the week are about 62%.

### Second Set: Two Standard Deviations Below Mean

We can change the numbers to figure out other distributions of mean and standard deviation.

For example, what if everyone else’s mean was 100 with standard deviation 20, and your mean was 90 with standard deviation 20? Then the numbers would look like this:

Week 1 (n = 16): 87%

Week 2 : 87%

Week 3: 86%

Week 4: 85%

Week 5: 83%

Week 6: 82%

Week 7: 82%

Week 8: 80%

Week 9: 79%

Week 10: 75%

Week 11: 75%

Week 12: 72%

Week 13: 68%

Week 14: 62%

Week 15: 51%

Week 16 (n = 1): 36%

As both of the above result sets show, there is a slow and steady decline from Week 1 to Week 11 or 12, and a sharp dropoff afterwards. In both cases, being a worse team than average in Week 10, when you face 7 competitors, only increases your chances of losing about 10%-12% over being similarly positioned in Week 1, when you face 16 competitors. This suggests that if you are following a strategy of delaying improvement of your team in order to get more players for cheap, around Week 12 might be when you are aiming to no longer be significantly worse than the rest of the league.

### Third Set: One Standard Deviation Above Mean

What if your team is one standard deviation better than the rest? The following are the results for mean = 110 and standard deviation = 10, when the rest of the league has mean = 100 and standard deviation = 10. We get the following results:

Week 1 (n = 16): 99%

Week 2 : 99%

Week 3: 99%

Week 4: 99%

Week 5: 99%

Week 6: 99%

Week 7: 99%

Week 8: 99%

Week 9: 98%

Week 10: 98%

Week 11: 97%

Week 12: 96%

Week 13: 96%

Week 14: 93%

Week 15: 89%

Week 16 (n = 1): 76%

What this shows is that if you are a standard deviation above the rest up to about Week 14, you are virtually guaranteed to survive any given week. In Week 15 you have a 9 in 10 chance of winning. In Week 16 it’s a three in four chance.

### Fourth Set: Slow Decline Followed by Sharp Increase

Let’s finish out this article with something a little more sophisticated, by doing a simple simulation of a league where you follow the Bid Low, Bid Late strategy, and nobody else does. Let’s assume this means you draft well and end up one standard deviation above the rest, but then gradually decline relative to everyone else until Week 10, because you do not overbid on free agents. Starting Week 11, you start ascending compared to everyone else, as you are able to outbid them for the top players.

We will assume that you start the season at mean 110, standard deviation 10, while everyone else is at mean 100, standard deviation 10. Through Week 10, your team stays constant, while the other teams’ average increases by 2 points per week, and standard deviation increases by 0.5 per week. That means the following values for other teams through Week 10:

Week 1 (n = 16): mean 100, SD 10

Week 2: mean 102, SD 10.5

Week 3: mean 104, SD 11

Week 4: mean 106, SD 11.5

Week 5: mean 108, SD 12

Week 6: mean 110, SD 12.5

Week 7: mean 112, SD 13

Week 8: mean 114, SD 13.5

Week 9: mean 116, SD 14

Week 10: mean 118, SD 14.5

Assume that starting Week 10, other teams continue to increase by the same amounts, but your team is able to increase its mean by 6 points and standard deviation by 1 point per week.

That means the other teams will have the following:

Week 11: mean 120, SD 15

Week 12: mean 122, SD 15.5

Week 13: mean 124, SD 16

Week 14: mean 126, SD 16.5

Week 15: mean 128, SD 17

Week 16 (n = 1): mean 130, SD 17.5

Your team’s statistics starting Week 11 would be:

Week 11: mean 116, SD 11

Week 12: mean 122, SD 12

Week 13: mean 128, SD 13

Week 14: mean 134, SD 14

Week 15: mean 140, SD 15

Week 16 (n = 1): mean 146, SD 16

As a reminder, these are just rough numbers only meant to give a flavor of what to expect.

What would the weekly odds of survival look like? Here are the results:

Week 1 (n = 16): 99%

Week 2 : 99%

Week 3: 98%

Week 4: 97%

Week 5: 96%

Week 6: 94%

Week 7: 93%

Week 8: 90%

Week 9: 85%

Week 10: 82%

Week 11: 84%

Week 12: 87%

Week 13: 88%

Week 14: 88%

Week 15: 84%

Week 16 (n = 1): 75%

Several dynamics interact here: (1) the initial increase in mean and standard deviation of the competition while your team’s stays constant up to week 10; (2) the subsequent rapid increase in your team’s mean, which catches up with the competition’s mean by Week 12, and standard deviation, which never quite catches up; and (3) the decreasing number of teams in the competition. (1) and (2) together describe a pattern of decreasing chances of winning through Week 10, followed by an increased chance through Week 16. Dynamic (3) decreases your team’s (any every other individual team’s) chances of surviving, first slowly, then very quickly.

As the result of the interaction of these dynamics, your odds of survival drop from virtually guaranteed in the first few weeks to about 4 in 5 by Week 10, as the competition improves in quality while declining in number. By Week 14, dynamic (3) slows, then reverses, the improved chances of winning caused by dynamic (2), so that by Week 16 your chances are only 3 in 4. These are good odds because your mean score is much better than the other team’s – by about one standard deviation – but they are lower than the odds in any previous week because there is only one team that needs to get lucky and beat you in order for you to lose.

The numbers above tell you the odds of your team winning week-to-week. How does that translate to the odds of your team winning the league? Multiplying the percentages above shows that the chances of winning the league are about 17.5%. If that doesn’t sound like much, consider that every other team in the league only has a 5.2% chance of winning, so 17.5% is very good.

### Conclusion

Probability plays such a big role in the early season that bad teams are more likely to survive the beginning of the season than good teams are to win at the end of the season. This suggests the strategy of saving your money during the early season to spend later.

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