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Does the last few players in the first round really get to catch up by picking first in the second round?

Background

In fantasy football, different players have different projected values that correspond to points that they are likely to earn throughout the season. The person who picks first in the draft gets to choose the player they want from the pool of all players, second pick gets to choose the player they want from the pool of all players excluding the first pick's player, and so on. This obviously gives the owner of the first pick an advantage.A common way to minimize this advantage is to use a serpentine draft. In this draft, the player that picks first in round one picks last in round two, first in round three, last in round four, and so on. The player that picks second in round one picks second to last in round two, second in round three, and so on.

Giving this a brief amount of thought, you can spot the obvious issue that the serpentine draft is only balanced if player value declines linearly with pick number. Is this the case? It seems unlikely as you'd expect the top one or two players at each position to be significantly more valuable than the rest.

#### Projection Analysis

There's no obvious (to me at least) function to use to model player value vs pick. Anything that decays seems reasonable. I'd guess that 'C1 - C2*ln(pick)' and 'C/(pick^C2))' where C1 and C2 are constants are reasonable models.To check, I took a variety of projections from ESPN. One example is that taking the top 40 WRs gives me 'value = 261/(pick^0.2)' as a reasonable fit. Combining the top 14 QBs, 28 WRs, and 28 RBs gives me 'value = 381 - 57*ln(x) as a reasonable fit. The specific constants don't really matter for this so I'll choose nice values.

Potential top 70 picks with a rough fit to check plausibility of the model |

Potential top 40 WRs with a rough fit to check the plausibility of the model |

I'll consider two models:

- Value = 1/(pick^(0.2))
- Value = ln(168) - ln(pick)

How badly do people get screwed in each of these with a typical serpentine draft? An easy way to find out is to calculate the expected value of each person's picks using the models and compare the extremes. You get the following for the two models:

- Model 1: First pick gets ~8.7% of the value and eleventh pick gets ~8.2% of the value; first pick has a projected advantage of ~6% in value head-to-head; all other picks are in-between these
- Model 2: First pick gets ~9.2% of the value and tenth pick gets ~8.0 % of the value; first pick has a projected advantage of 15% in value head-to-head; all other picks are in-between these

#### One Pick Per Person Per Round

Given the restriction that each player gets one pick per round, can we do better? If so, how much?We can run the numbers to minimize the difference between projected values. I'll set the first two rounds to what they are with the standard serpentine described above. The basic algorithm is to evaluate at each pick which person needs to pick next, let them pick if they haven't already for this round, and move to the next person if they have. You check when you're done, and rearrange as needed. Writing a quick program to optimize the ordering you get:

P stands for person |

Doing the same thing for Model 2, you get:

P stands for person |

It looks like the first pick gets an insurmountable lead after the first round still as he picks last in every subsequent round. To check if this fixed the fairness issue, we look at the range of scores again:

- Corrected Model 1: First pick gets ~8.6% of the value and pick gets ~8.3% of the value; last pick has a projected advantage of ~4% in value head-to-head; all other picks are in-between these
- Corrected Model 2: First pick gets ~8.7% of the value and last pick gets ~8.3% of the value; first pick has a projected advantage of ~5% in value head-to-head; all other picks are in-between these

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Can Pick In Any Order; Every Player Gets Same # Of Total Picks

What if you drop the concept of rounds? Can you make this completely fair? The code for this got a bit tricky and blew up because there's a huge number of permutations and a large number of them have the same projected fairness. The idea for this one is similar to the above one except that you don't have to wait for the next round. Each person could draft all 12 slots in round 3 if that made sense for this (it doesn't).Many of the later picks are very close so tiny changes in the model change the results, but to give an idea of what solutions look like, I've posted a run from each.

P stands for person; this is the free for all post round 2 using Model 1 |

P stands for person; this is the free for all post round 2 using Model2 |

How fair do these end up being?

- Both models: Every person ends up with 8.3% of the value (values identical down to 0.01%).

#### Where Does This Break Down?

Everywhere.- This doesn't factor in number of players at each position
- This doesn't factor in bye weeks
- This doesn't factor in optimal matchups
- I've only known anything about fantasy football since last Friday

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