Free Agency (883 views)

[eric]

This is going to take an extremely long time to figure out, so I'm going to post it in installments.

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The most important finding by far is that chances are not binned or rounded in any way, except (probably) to zero. This is brutal for data gathering, but that's how it goes.

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I created a league where teams had identical Win Ratings, then made players with identical attributes, hometowns, last teams, and greed/loyalty/playforwinner. I then set up several bid regimes to see what happened.

Bid 1
Team A - 1 year, $10m
Team B - 2 years, $9m, 1%
Team C - 2 years, $9.45m, 0%
Team D - 3 years, $7m, 0%
Team E - 5 years, $6m, 0%

Over 550 trials, the players went 261 147 142 0 0. That was a pretty big surprise for me, I thought the longer deals would get at least something but it seemed entirely governed by the first year.

Bid 2
Team A - 1 year, $9m
Team B - 2 years, $9m, 0%

Over 1216 trials players went 303 913, so later years definitely matter but it's pretty shocking how many players still went with Team A's offer.

Bid 3
Team A - 1 year, $4.5m
Team B - 1 year, $4,500,001
Team C - 1 year, $4.6m

It's possible that actual MLE offers are viewed differently by players than 1 year, $4.5m in cap space, but I still thought this would be helpful in determining how MLE-adjacent bids worked. Over 44424 trials players went 12956 12866 18602, or 29.2% ± 0.4%, 29.0% ± 0.4%, 41.9% ± 0.5%. It could actually be 29% 29% 42%, and those are the only integers those values all reach, but...

Bid 4
On this one I gave one team a one year $8m bid and the other a one year bid in increments of $0.1m higher up to $10.3. Here's what happened:
bid win loss pct ±
10.3 6482 0 100% 0%
10.2 4324 2 100% 0%
10.1 2164 6 100% 0%
10 2154 16 99% 0%
9.9 2143 27 99% 0%
9.8 2112 58 97% 1%
9.7 2103 67 97% 1%
9.6 2033 137 94% 1%
9.5 2000 156 93% 1%
9.4 2017 153 93% 1%
9.3 1941 201 91% 1%
9.2 1890 224 89% 1%
9.1 1867 303 86% 1%
9 1829 327 85% 2%
8.9 1790 370 83% 2%
8.8 1687 473 78% 2%
8.7 1658 512 76% 2%
8.6 1596 574 74% 2%
8.5 1546 610 72% 2%
8.4 1429 727 66% 2%
8.3 1383 787 64% 2%
8.2 1241 929 57% 2%
8.1 1175 995 54% 2%
So there's several things going on here.
1) The function is non linear.
2) 8m vs 10.2m has about a one in two thousand chance of winning, which is definitely not an integer percent.
3) There is probably a limit where a contract offer cannot possibly win. To go 6482 attempts without even one success suggests a maximum success rate of .00011, or just over one in ten thousand.
4) Fitting a second order polynomial to this function shows that bidding 8.000001 would give a 50.000045% chance of winning, or increasing your chances by about one in twenty thousand. Don't bid x + 1 dollar. It would take you sending 45 bids a year for 445 sim years for it to matter, and that's if it has any effect at all.

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Main takeaways:
1. First year dollar amount is by far the most important factor.
2. And if you're not close enough on that you have literally 0% chance.
3. Total dollar amount means nothing.
4. But adding years helps.

My plan for the next things to look at are:
a. Whether the important factor is raw difference (0.1m) or percent difference (1.25%).
b. Get a better bead on the additional years factor.
c. See if different player attributes change the results (age? experience? current rating?)
d. Get into loyalty and playforwinner.

Thoughts or requests are welcome.

[Citizen Cane]

appreciate the work. 1st year being most important was pretty well known, but its nice to know for certain now

[Odin]

fuck you tobi

[eric]

Three quick hits. I did the same thing with 7m going up in installments as I did with 8m, and found this:

bid win loss pct ±
7.1 1243 941 57% 2%
7.2 1351 819 62% 2%
7.3 2416 1196 67% 2%
7.4 2135 959 69% 2%
7.5 3732 1294 74% 1%
7.6 2496 808 76% 1%
7.7 3336 878 79% 1%
7.8 2367 503 82% 1%
7.9 3792 674 85% 1%
8 2310 0 100% 0%
8.1 5698 0 100% 0%
So it's going up faster than the vs. 8m bids, it hits a binary condition at 8m that is not explained by the continuous function preceding it, and we know that being 1m higher isn't the binary condition because an 8m bid can defeat a 9m bid. I found the exact break point to be $7,933,275 vs. an $8m bid: anything below that could never succeed (and by the same token $7,933,275 always beats $7,000,000 and $7,933,274), everything above that looked like 50/50 as we would expect from the very small dollar difference between the two.

And it gets worse! It turns out that break point depends on the salary cap number, which in the software defaults to $43,800,000. What's weird about this is $7,933,275 doesn't divide especially nicely into $43,800,000, so I checked for the breakpoint against a $50,000,000 cap and found $9,056,250. In both cases the number is pretty close to cap / 5.52105, and if you round to the nearest $5 you get it, but why anyone would want to use 5.52105 for any reason ever is beyond me.

And it gets even worse! Because the break point also depends on the player's greed.

So what I'm going to do is go back to a $43.8m cap and see if there are any other break points,
then I'm going to go back to a $50m cap and find them there,
then I'm going to see what the relationship between greed and break point is.

[Heebs]

Long live the $7,933,275 bid!

[eric]

Okay so here's how it works.

seed = cap / 381, rounded to the next highest multiple of 25. Why 381? Because heavyreign, that's why.
break salary = seed * f(greed)

The greed function is the same principle as resignings divided by two, so f(0) = 60, f(10) = 70, f(30) = 87.5, and so on up to f(41), which (I assume) because it would result in the first value above $12,500,000 is set to exactly $12,500,000, and this value holds for every greed value up to 100. What makes this really weird is that the player I'm testing has 9 years experience and so has a max salary of $15,000,000, but even for 100 greed will choose randomly between $15,000,000 and $14,999,999 options while always taking $12,500,000 over $12,499,999. Here it is in chart form, where "break" is the minimum salary you have to offer to have any chance at all if someone offers the breakpoint or more.

greed break
0 7875000
1 8006250
2 8137500
3 8268750
4 8400000
5 8531250
6 8662500
7 8793750
8 8925000
9 9056250
10 9187500
11 9318750
12 9450000
13 9581250
14 9712500
15 9843750
16 9975000
17 10106250
18 10237500
19 10368750
20 10500000
21 10598437
22 10696875
23 10795312
24 10893750
25 10992187
26 11090625
27 11189062
28 11287500
29 11385937
30 11484375
31 11582812
32 11681250
33 11779687
34 11878125
35 11976562
36 12075000
37 12173437
38 12271875
39 12370312
40 12468750
41+ 12500000
What It All Means

If you MLE someone who's getting any max deal, you cannot possibly win.

Because greed is assigned randomly, most (but not all) of the time you can't compete with any max deal without offering $12,500,000+ in the first year.

Everyone has a chance at 0/0/0 guys so long as they offer $7,875,000+.

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...but...

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I'm also pretty sure that it's the percentage difference in first years that is relevant rather than the raw difference. Here are two tables, the first is when the higher salary is $200,000 above the lower, the second is when the higher salary is 110% of the lower:

4.2 1439 689 68% 2%
6.2 1726 906 66% 2%
7.2 1351 819 62% 2%
8.2 1241 929 57% 2%

4.4 1928 494 80% 2%
6.6 1990 320 86% 1%
7.7 3336 878 79% 1%
8.8 1687 473 78% 2%
It's pretty clear there is an underlying phenomenon in the first table, while in the second everyone wants to be about 79% except for friggin' 6.6. It's such a huge outlier that I tested it again for another 3962 trials and got an identical 86% ± 1% measurement. This is good on the one hand because my measurements are repeatable, but it's bad because it makes no sense for ~$6m values to be so dramatically different when ~$4m and ~$7m are coherent.

So what I'm going to do is six equal percentage steps (because any more and I'd have to worry about rounding) and go from 1m starting to 7m starting and see what happens. I expect this to take a week or two.

[Odin]

The more you dig the more i realize what heavy reign means when he talks about the software having a ton of code he's ashamed of.

[eric]

I found another break point that acts slightly differently: players with 14 or more greed will never accept a deal below $4.5m if they receive an offer of $4.5m or higher. I'm reasonably sure lower greed simply has no break point. Players with 0 or 7 greed would rarely accept a $4m offer instead of a $5m offer, and players with 13 greed would sometimes accept a $4.3m offer instead of a $4.7m offer.

[eric]

Alright, so if we have one year salary offers A and B that don't cross any break points, and we express the difference between them as A - B = X, then we know that the success rate S for team A will be...

S(0) = 50%
S(infinity) = 100%

...which is a mathematical way of saying that if there is no difference between the two offers then players will choose randomly between them, and that no matter how much higher offer A is than B a player can't accept it more than 100% of the time. It's possible to construct all manner of equations that fit these parameters by going piecewise, and we know the coder has gone piecewise many times before, but from our data it looks like a smooth curve, so let's consider a certain curve...

S(x) = 1 - e^(-cx) / 2
S(0) = 1 - e^(0) / 2 = 1 - 1/2 = 50%
S(infinity) = 1 - e^(-infinity) / 2 = 1 - 0/2 = 100%

...where "c" is some yet to be determined constant. I have chosen this form of curve over other asymptotic curves like a hyperbola because it fits the following data set better:

x S
0.2 0.723839854
0.22 0.756221198
0.242 0.762672811
0.2662 0.788479263
0.29282 0.796846011
0.3 0.786175115
0.322102 0.812271062
0.33 0.802304147
0.363 0.828571429
0.3993 0.833179724
0.4 0.832380952
0.483153 0.861195542
0.484 0.868725869
0.5 0.861556982
0.5324 0.871333964
0.55 0.885954382
0.58564 0.893552876
0.6 0.88997114
0.605 0.896868885
0.644204 0.914366883
0.66 0.920506912
0.6655 0.919698314
0.7 0.92766151
0.726 0.94149318
0.73205 0.930029155
0.77 0.946246973
0.7986 0.939829193
0.805255 0.943235572
0.847 0.967105263
0.87846 0.959714286
0.9317 0.963492063
0.966306 0.962522046
1.02487 0.962279294
1.127357 0.94601569
I have rejected my previous hypothesis that % differences were what mattered because every one of these points was a 10% difference, and clearly the success rates vary far too wildly to be from a single input. Here's how it looks like in graph form:



The problem is that this function doesn't fit my prior data well for any value of c, and there really aren't any other parameters to change. The prior data also really doesn't match the idea that raw difference is the important factor. I don't know how to explain that, but I wanted to get this curve up.

[eric]

Greed and First Year Value

So I've been working on this part for a long time and it's just not resolving, so I'm going to draw a line under it and move on. Here's what we know and don't know:

1. First year value is very important.
2. It's not clear whether raw or % difference is the relevant parameter, but
3. Difference in first year value has an asymptotic function, meaning that the change in success for being $0.1m higher is larger than the additional change in success for being $0.2m higher, and so on, but
4. Certain values depending on greed override this function, making it impossible for any lower values to succeed.
5. Players with higher greed will see sharper changes in success given a set of different contracts than players with lower greed.

Hometown

A dropdown menu in players allows one to select a hometown team. For example, Kyrie Irving's hometown team is the Trailblazers. I tested this by making a player with 100 Greed, giving the hometown team a $6m bid and another otherwise identical team a $7.2m bid. The results for 0 and 100 loyalty were...
70% ± 3% (633 vs 267)
70% ± 2% (1281 vs 549)

Thus loyalty does not impact the player's desire to play with his hometown team solely on the basis of it being his hometown team. I then tested values of $7.4m, $7.5m, and $7.6m and found:
58% ± 2% (981 vs 714)
51% ± 2% (843 vs 801)
41% ± 3% (610 vs 889)

So the break even point is around $7.5m; or, hometown is worth an additional $1.5m on the first year offer. Unfortunately this is definitely not a constant with respect to first year value, because I also tested $8m vs $9.5m and found 21% ± 2% (257 vs 985), eventually settling on $8m vs. $9m as the break point with 50% ± 3%. Please also note how this is even less consistent if we compared percentage differences of 25% to 11%.

But it would get worse for math fans: it is also definitely not a constant with respect to greed, because when I tried the same $8m vs $9m with 0 greed the hometown team won 100% of the time.

If we try with 5 greed, from break point $8,531,250 we would expect the hometown team to win 0%, and this is in fact what happens. This is a rough graph of hometown success rate for $8m vs. non hometown $9m as a function of greed:



Ha! Ha!

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So bottom line, home town is a very significant factor and one that impacts even max deals, but breakpoints are stronger. Next up I'm going to focus on multi year deals.

[Heebs]

For multi-year deals, I am really interested in the following:

12.5M flat over 7 v. 12.5M w/ 3% increases over 7 v. 12.5 w 10% increases over 6.

[eric]

Apr 20, 2016 12:43:34 GMT -6 Heebs said:

For multi-year deals, I am really interested in the following:

12.5M flat over 7 v. 12.5M w/ 3% increases over 7 v. 12.5 w 10% increases over 6.

i won't be able to test 7 year deals for a long time because they require the player being on a previous team, which (presumably) means Loyalty will come into play as well. i am presently working my way up multi year deals though, which should give us some information.

[Deleted]

So do you or Souper Troopers fill in the hometown option or do we leave that blank?

[eric]

Another thing I just figured out, and we'll use the Nets as an example. Last year they signed Exum to a bid of $12.5m, 3%, 7 years. This resulted in a contract of...
$12,500,000
$12,875,000 ($12,500,000 * 1.03)
$13,261,250 ($12,500,000 * 1.03 * 1.03)
and so on

The year before, the SuperSonics signed Michael Beasley to a bid of $12.5m, 10%, 6 years. This resulted in a contract of...
$12,500,000
$13,750,000 ($12,500,000 * 1.10)
$15,000,000 ($12,500,000 * (1 + .10 + .10))
and so on

Note the difference! The reason this happens is because the maximum raises are calculated differently from the actual raises. The bid generator in Michael Beasley's case tries to do the same 1.10 ^ n as the Exum contract, but is truncated to the maximum raises that follow the 1.10 * n rule. If someone were to sign a player to a bid of $12.5m, 9%, 6 years, we would see...

$12,500,000
$13,625,000 (1.09 * the previous value)
$14,851,250 (1.09 *)
$16,187,862 (1.09 *)
$17,500,000 (1.081 *)
$18,750,000 (1.071 *)

Thus ending in the same place as the max deal of "10%" increases. This doesn't have any implications for prior research because that was all one year stuff anyway, and on the scales we're talking about it doesn't make a huge difference quantitatively, I'm really just posting it because it's confusing as heck to see without explanation.

[eric]

Apr 20, 2016 13:06:34 GMT -6 @dumptime said:

So do you or Souper Troopers fill in the hometown option or do we leave that blank?

I do not and cannot fill in the hometown (or greed, loyalty, playforwinner, or visible potential) scores because they are not included in the draft export. When I do a draft import the software I have assigns those values, and when I do a spot check of recent rookies they have hometown assigned. Although I can't check everyone, I did notice that neither Harrison had a hometown (there is an N/A option), so maybe when Odin was doing it he zeroed them out? You'd have to ask him.

With all that said, part of the create a player dump buck reward is setting hometown, and you can be darn sure I'm setting Jesse Epstein's.

[Heebs]

Apr 20, 2016 13:24:45 GMT -6 eric said:

Apr 20, 2016 13:06:34 GMT -6 @dumptime said:

So do you or Souper Troopers fill in the hometown option or do we leave that blank?

I do not and cannot fill in the hometown (or greed, loyalty, playforwinner, or visible potential) scores because they are not included in the draft export. When I do a draft import the software I have assigns those values, and when I do a spot check of recent rookies they have hometown assigned. Although I can't check everyone, I did notice that neither Harrison had a hometown (there is an N/A option), so maybe when Odin was doing it he zeroed them out? You'd have to ask him.

With all that said, part of the create a player dump buck reward is setting hometown, and you can be darn sure I'm setting Jesse Epstein's.

How will you get Mission Viejo? Can you do custom home towns, or will you just use Los Angeles?

[Odin]

I made all players manually in the software and hometown was always blank by default.

[eric]

Greed and Total Value

For 0 greed players contracts of even length with equal total value are a 50/50 shot. Here are the ones I tested:

2 yrs, $8m, 10% vs. 2 yrs, $8.4m, 0%
3 yrs, $8m, 10% vs 3 yrs, $8.8m, 0%
4 yrs, $8m, 10% vs 4 yrs, $9.2m, 0%
5 yrs, $8m, 10% vs 5 yrs, $9.6m, 0%
6 yrs, $8m, 10% vs 6 yrs, $10m, 0%

I also tested the first and last set against 100 greed, and found 43% and 6% for the first teams. As we already know greed makes first value difference more important. Later years can make up for it but only to a degree.

The Extra Year

As I mentioned on CYBER DUST APP it is hard to do seven year deals because they require a last team which requires loyalty. I actually found a way around this but not before I did all this other research so I'm going to post it anyway

Every contract will start at $12,500,000 and 0 greed. Let's start with 4 years vs. 5 years. The 4 year deal will use the faux 10% increases, so $13,750,000 $15,000,000 $16,250,000. The 5 year deals will have X% increases and here is how often the 4 year deal wins:

0% - 70%
3% - 53%
5% - 39%
7% - 27%
10% - 16%

Repeating the process with 5 years vs. 6 years:

0% - 81%
3% - 59%
5% - 43%
7% - 30%
10% - 17%

Thus the extra year definitely matters, but it's no guarantee and it means less and less the more years there are. If we check against 100 greed we see almost exactly the same changes. The rough break even point goes to 82% and 83%, 10% increases goes to 15% and 16%. It is reasonable to assume that the pattern will continue, and that 6 years vs. 7 years means you have to go all the way to 10% increases if you really want to keep your player...

...assuming that he has 0 Loyalty (and that 0 Loyalty means Loyalty has no effect).

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Because I did find a way to do 7 years without running an actual season and bringing dramatically more variables into play, what I'll do next is actual 6 years vs. 7 years as above, then add in hometown, then start playing with Loyalty.

[Citizen Cane]

This shit is fascinating.

[eric]

Seven Year Hitch

Every deal referenced in this post will start at $12.5m, and the team with bird rights will always be team A.

Because it has come up regarding a certain chosen member of this draft class, I first tested 0/0/0 players: team A offered seven years 10%, team B offered six years 10% and was the hometown team. The results: team B went 1512 and 0 for a player with all 50 attributes, 1582 and 0 for a player with all 100 attributes, and 112 and 0 against even a seven years 12.5% deal. Bottom line, hometown is critical for 0/0/0 players.

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Next I removed hometown and compared the same seven year 10% to six year 10% deals against a wider range of player builds of greed/loyalty. (Playforwinner is irrelevant because every team in this experiment had the same Win Rating, but for completeness' sake I will say it was always set to 0.) Team A won the following percentages:

81% for 0/0
80% for 100/0
100% for 0/50, 0/100, and 100/100
99% for 50/50

As before, a greedier player is less likely to want the extra year, but so long as you don't skimp on the raises the effect is small and Loyalty dominates.

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If we stick with the 50/50 player (because Greed and Loyalty will on average be 50 if selected randomly from 0-100) and look at X% raises for the seven year deal, we see Team A success rates of:

00% - 8%
03% - 38%
05% - 70%
07% - 91%
10% - 99%
MAX - 100%

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I'm going to test Greed/Loyalty on the 03% and 05% because we should be able to see the most pronounced effects there, and after that I think I'll be pretty well done with free agency. There is play for winner to consider but because I can't directly edit Win Rating it will be dramatically more complicated to construct a testing environment for it, and because of how it's calculated the differences are almost never all that extreme anyway. If anyone has specific questions they'd like me to look into feel free to post them, I'll probably be working on the 03%/05% until the end of the week.

[Heebs]

Team get rid of hometowns.

[Citizen Cane]

May 2, 2016 14:40:14 GMT -6 Heebs said:

Team get rid of hometowns.

[Deleted]

Team keep hometowns

[eric]

team hometowns for some, american flag bandanas for others

[eric]

So here are some results for 7 year 5% raises vs 6 year 10% raises:

L / G 0 50 100
0 44 24 16
50 95 70 46
100 100 96 79
So 100 loyalty 0 greed is 100%, 0 loyalty 100 greed is 16%. As you can see going diagonally to the right, it is not the case for this set of contracts that an equal amount of greed balances an equal amount of loyalty.

[eric]

Almost everything I've done so far has been two teams going head to head with bids. For example, a $12.5m 7 year 5% bid beats one $12.5m 6 year 10% bid 70% of the time. What if the seven year bid goes up against three six year bids? It will win less, and less and less as greater number of teams bid.

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Now, here's a huge amount of math to back that up. Think of it in terms of a four team single elimination tournament. There are three ways to do the seedings:
(A vs B) vs (C vs D),
(A vs C) vs (B vs D),
(A vs D) vs (B vs C)

and in each seeding there are eight possible outcomes. We'll go through the first seeding:
A beats B, C beats D, A beats C
A beats B, C beats D, C beats A
A beats B, D beats C, A beats D
A beats B, D beats C, D beats A
B beats A, C beats D, B beats C
B beats A, C beats D, C beats B
B beats A, D beats C, B beats D
B beats A, D beats C, D beats B

In our case, Team A will beat any of the other teams s% of the time, and every other matchup is a 50% outcome. Each possible outcome is a combination of three possibilities, so we can therefore get the chance of the given team winning as:
.5 * s ^ 2 - [team A wins twice, team C wins once]
.5 * s * (1 - s) - [team A wins once and loses once, team C wins once]
.5 * s ^ 2
.5 * s * (1 - s)
.5 ^ 2 * (1 - s) - [team A loses once, team B and C win once each]
.5 ^ 2 * (1 - s)
.5 ^ 2 * (1 - s)
.5 ^ 2 * (1 - s)
=
s ^ 2 + s * (1 - s) + (1 - s)
s ^ 2 + s - s ^ 2 + 1 - s
1

In other words, our eight outcomes cover 100% of the possible outcomes, which means we're not missing or double counting anything. This also means we can see the chance of each team coming out of this seeding with a win by summing, so Team A gets results one and three, Team B gets five and seven, Team C gets two and six, Team D gets four and eight:
a = s ^ 2
b = s * (1 - s)
c = .5 * (1 - s)
d = .5 * (1 - s)

And when we do the second seeding we get s * (1 - s) for c while b and d are .5, and in seeding three we get s * (1 - s) for d. To get the overall chance of winning we sum over all three possible seedings and divide by three:
a = s ^ 2
b = (s + 1) * (1 - s) / 3
c = (s + 1) * (1 - s) / 3
d = (s + 1) * (1 - s) / 3

And note that (s + 1) * (1 - s) can also be expressed (1 - s ^ 2). Every six year deal has the same chance of winning, and the seven year deal wins at its winning percentage against only one team squared. By extension of the same reasoning we see that the seven year deal will win at the cube of that percentage against seven other bidders, and at the fourth against fifteen other bidders: the more bids there are, the less likely it is to retain a player's services if and only if there is a chance that player would leave against only one bidder. Note that it is still in the best interests of six year deal teams that they are the only other one bidding, because even though the six year deal slice of the pie gets bigger it must be shared so many more ways that there is a net loss for any given six year deal team. Algebraically, (1 - s ^ n) / (2 ^ n - 1) -> 0 as n -> inf for all s.

.

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Empirically, the results we see for three on one for a 50 Greed 50 Loyalty player are 47% for a seven year five percent increase deal and 80% for a seven year seven percent increase deal against base chances of 70% and 91% respectively. These numbers are very nearly too low to a statistically significant degree, but are close enough that I'm putting it down to rounding if there is a phenomenon at all. Returning to the above algebra, we can look for how many six year deals need to be offered for a seven year ten percent increase deal to have a less than 50% chance of retaining such a player and we find that n = 70, or a league with a thousand million trillion teams, or the NHL

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This is the last thing I have thought of to research regarding free agency. If anyone has other requests regarding free agency or the software in general I am open.