原文 https://www.fangraphs.com/tht/rethinking-the-win-curve/
前言：这是去年的Saberseminar的presentation，我会在文内加一些自己的东西。
同时这篇是一个很好的范例，可以让大家思考一下为什么
“实际上根本没有数据派”这回事。
Transaction analysis has become one of the most important topics in all of
baseball research. The primary tool that is used in almost all public
transaction analysis is a dollars/WAR calculation. This tool has been
convenient in many applications. It is especially good for predicting how
much money an upcoming free agent will receive on the open market.
交易分析在这个时代变成棒球研究的显学之一，而最常为人所用的交易分析工具则是
每一WAR值得多少钱的算法，我们可以见到很多大大都用这个方便的工具在计算与预测
FA到底在市场上该拿多少钱！
However, dollars/WAR also has significant limitations. In many situations,
it simply does answer the questions we are interested in. As an example,
let's look at the Chris Sale trade to the Chicago White Sox. Last offseason,
the White Sox traded Sale to the Red Sox in exchange for Yoan Moncada,
Michael Kopech, Luis Alexander Basabe and Victor Diaz.
但是每一WAR值多少钱这个算法有很多明显限制，在我的看法来说，
他忽略每一个WAR的边际效益，也就是WAR并不是个个等值，
就跟意见并不是人人等值一样
有的意见就是没屁用or没建设性的意见，有的WAR就是根本没影响力 的WAR
所以忽略市场供需与球队偏好的算法，是不能作为准确估计薪水的方式！
这个算法也没有考虑volatility，所以不在风险中立机率测度下所估算的薪水并不是
真正的预期薪水，也就是这个算法没有综合老化、受伤风险、表现起伏、
以及WAR的时间价值等等，
也许0.5WAR/Y的衰退是一个经验上的估计，但实在是省略太多应该估计的内容，
也就是说每年预期衰退0.5WAR的算法乘上9M/WAR是一个相对不准确的公式。
By a dollars/WAR calculation, this was pretty close to an even trade,
seeming to be mutually beneficial. The Red Sox preferred the present wins
in the form of a starting pitcher, and the White Sox preferred the prospects.
A dollars/WAR calculation does not allow us to see how much each team
benefitted from this deal. I will present a new framework that can help
teams make decisions under uncertainty. We will be able to see how much
any player is worth to any team given a team's preferences and the
player's impact on the team's projections.
这个作者提出一些想法，并且使用去年的Sale交易案作为例子，
如果用每一WAR多少钱的算法，这看起来就像是一个相当公平的交易。
但是双方看来都会受益：
因为红袜想要顶级先发，南风城则是要重建所以需要好的菜逼八。
但是如果用每一WAR多少钱就不能知道这两队到底从这交易受益多少。
所以作者在这里建构一个新的方法，透过考虑不确定性与偏好来进行决策，
考虑不确定性之后可以看到在定下一个球队的偏好后球员的价值，
以及可以找出球员在球队预期表现上的影响。
About 10 years ago, Vince Gennaro, Nate Silver and others began writing
about the win curve. The win curve graphs the marginal value of an
additional win given a team's final record. Here is the win curve Silver
came up with.
大概是十年前开始，有不少人开始研究win curve，win curve可以描绘出
队伍赛季总胜场之下，每多一胜会有多少价值，以下是Nate Silver做的图，
https://i.imgur.com/inZ63r8.gif
As you can see, the value of a win peaks around win number 90. These wins
greatly increase a team's chances of reaching the playoffs, advancing in
the playoffs, and winning the World Series–many of the things teams really
care about. Focusing on a team's regular-season win total can serve as an
effective proxy for these goals.
你们可以看到，每多一胜的边际价值在90胜左右达到最高峰，
而也就是在队伍有机会进季后赛的胜场之后，每一胜的价值就会开始提升，
毕竟季后赛跟世界大赛冠军才是球队在意的事情！
I will distinguish Gennaro's win curve by calling it a roster value curve.
Gennaro graphed the total value of a roster over the number of wins it
produced. The win curve is simply a graph of the derivative of the roster
value curve. The key finding Silver and Gennaro found was that, for the
most part, not all wins are created equal. Teams in close contention for
the playoffs see much more marginal value from additional wins.
Gennaro做的图则是称为roster value curve，他画出总价值对应球队胜场数的图，
而win curve则显而易见的就是roster value curve的微分，两位的发现都是，
大多情况每一胜都不能创造同等的价值，一些更靠近季后赛竞争的队伍，
每一胜的边际价值就更高，如同最近这几天的天使，处在季后赛边缘的球队，
除了延长Upton的合约外，也成功地签下了哦他腻！
https://i.imgur.com/XX71dkM.png
哦齁！看看这张图，我勇果然可以很有效果的运用每一分钱，每增加一胜所需要的钱
其实都没有相去太远，相对于豪门基基而言，不但有效率而且要达到季后赛的花费
也比较低呢！
There has been a lot of debate about what the win curve actually looks
like. Many have argued that Silver significantly undervalued wins that did
not affect playoff odds. For this analysis, it is important to remember
the win curves are for the teams to decide. The curve simply shows how
much a team values each potential outcome.
有很多人也在争论win curve实际上应该长怎样，有的人说Silver明显低估不影响
季后赛机率的那些胜场，但是要切记，这个方法是拿来评估“一支球队”，
然后透过这个曲线呈现出来“一支球队”的“一场比赛结果的价值”是多少！
Silver and Gennaro focused on how much these win totals affected a season's
total revenue. However, other factors could shape a team's preferences,
such as how much an owner wants to win a World Series or how much one year's
win total impacts future revenues. The preferences of any team can only be
decided by the team itself. I will use a roster value curve as a team's
unique utility function.
Silver跟Gennaro都关注多少总胜场可以影响一个赛季的总收益，但有其他因素
可以影响球队的偏好，如老板有多想要世界大赛冠军（Hal 签下Ellsbury的偏好），
或者是一个球队单季的好战绩可以影响到未来收益（就跟队板跟风迷增加相似），
这些决策都是球队用他们各自的效用函数（utility function）而决定的，
而这篇文章则是以roster value curve作为球队的效用函数。
While a team could identify its preferences and build its own win curve,
the curve has limited ability in helping to guide its decisions. Obviously,
when teams have to make roster decisions, they are operating under
uncertainty. They do not know where they will land on the win curve or
exactly how their potential acquisitions will perform. Phil Birnbaum
wisely pointed this out when commenting on Silver's win curve:
虽然每支队伍都可以透过自己的偏好性来调整并且建构自己的win curve，但是
这在协助决策上还是有相当程度的限制，例如他们要决定roster名单的时候，
必须考虑不确定性，而他们也不会知道自己实际上应该是从几胜开始看win curve，
也不知道如果做了交易会怎么影响他们的表现，Phil Birnbaum就很聪明的嘴了
Silver的win curve一波：
"Silver's graph tells us how much an *actual* win is worth. But, before
the season starts, a team can't know how many wins it will achieve with
that kind of precision. Even if it's perfectly omniscient about how much
*talent* its team has, there's still a standard deviation of about six
wins between talent and achievement. A team that's created to be perfectly
average in every respect should go 81-81–but, just by random chance,
it will win fewer than 75 games about one time in six, and it'll win
more than 87 games one time in six."
“萧华的图告诉我们一场‘真正’的胜利值多少，但是在赛季开始前，根本没有队伍
会知道在某些决策下我们到底可以拿到多少胜，就算我们已经完美通晓这支球队多有
‘天份’，但实际上天份兑现的结果还是会有大约六胜的标准差，而这是一个很大
的差异，假设一支球队有五成胜率，我们有68%的信心水准胜场会落在75-87胜之间，
而这就已经是靠进季后赛，与低于五成胜率的差别，我们甚至有95%的信心水准胜场
会落在69-93胜之间，而这就已经是烂队跟季后赛球队甚至分区龙头的差别了。”
Birnbaum correctly concludes that this uncertainty will make the hump in
the graph wider and shorter. However, we can be a lot more precise and end
up with a much more useful result. We should leave the win curve as it is
and call it an ex post win curve. The ex post win curve will simply show
the value of each marginal win at the end of the season. From this we can
model a preseason, ex ante win curve, which will show the value of a
marginal projected win. This stochastic model will allow us to find
the marginal value of adding a given player to a specific team.
Birnbaum嘴的相当正确，不确定性会影响标准差，而标准差会影响到win curves，透过
峰形来做初步判定，越宽则标准差越大越窄则越小，而通常不确定性会增大标准差，
所以这里我猜win curves应该会动，标准差不会只有六场。作者提出两种新命名曲线，
ex post win curve 跟 ex ante win curve，前者就是原先的win curve，
可以在“球季结束”后呈现每多赢一场比赛的价值，而后者则是可以在“季前”做预测，
使用的是预测胜场数，所以我们不是用球季结束的“观察值（observations）”而是用
球季开始前的阵容“预测值（predictions）”，并且这样就可以透过加入球员
来估计所带来的影响，以及每一场胜利的边际价值。
To start, I will use a hypothetical ex post roster value curve.
And here is its corresponding ex post win curve. I chose to use a curve
that looked a lot like Silver's.
这里先用假设的ex post roster value curve做开场，而另一张图则是相对应的
ex post win curv，这里挑了一个跟萧华的图很像的曲线
https://i.imgur.com/47SmBiX.jpg
https://i.imgur.com/ZExBzHV.jpg
Next, I simply used probability mass functions to come up with
distributions of potential records given preseason forecasts. I created
normal distributions of win total projections with means between 60 and
100, all with a standard deviation of eight wins. The uncertainty in
these forecasts comes from three main sources: random variation, injuries,
and uncertainty of players’ true talent. Here is the probability mass
function for a team projected to win 81 games.
接下来我用了常态随机变量的机率质量函数（probability mass function, pmf，这里
是因为胜场数实质上不算小数点，所以是离散机率分配）来做为赛季前预测战绩使用，
此处使用的平均数是预测平均胜场为60-100之间的所有胜场，标准差为8胜，
这个标准差是透过各种不确定性得来，包含受伤、天份的不确定性、以及随机扰动等。
此处用81胜作为平均数与标准差8胜做一张常态分配图。
https://i.imgur.com/YfL6gAF.jpg
Using these distributions and the roster value curve, I found the expected
values of the rosters projected to win between 60 and 100 games. The value
of any projected record (or any asset in general ) is the sum of the
probabilities of ending up in every potential state multiplied by the value
of ending up in these states–our discounted expected payoff.
使用这些分配们，以及roster value curve，找出roster预期胜场在60-100胜之间的
资料。预期战绩（或是广义来说的任何资产）的价值就是把每一个可能的状态机率，
乘上这些状态最后的价值，也就是折现后的预期报酬。
其实这里偷偷藏一件蛮重要的事情：“胜场也是资产”，而且还引入折现discount，
这个假设我猜在后面应该必须用到，才能利用asset pricing来计算。
These expected values were easy to calculate because there are a discrete
number of outcomes for any season; a team can win between zero and 162 games.
Here is the formula I used.
这个就是离散机率分配的期望值算法
https://i.imgur.com/Yu2vq9e.jpg
x = 预期最后总胜场数, w = 实际最后总胜场数（0-162）,
p(w|x) = 在给定预期总胜场数下，实际上总胜场数为多少的条件机率
z_w = 给定总胜场数可带来的报酬
If the summation notation is unclear, here is a quick example:
E(Projected Win Total) = … + p(65 wins)*payoff(65 wins) +
p(66wins)*payoff(66 wins) +
p(67 wins)*payoff(67 wins) + …
这一段太囉唆，总之就是把每个情境的机率跟报酬相乘加起来。
Once we have the expected values of the projections, we can plot a preseason,
ex ante win curve. This win curve will show us the value of rosters given
their projected win total.
From here, we can easily build a new ex ante win curve.
有以上预期胜场算出来的期望值，就能画出ex ante win curve，也就是roster在给定
预期总胜场的价值应该是怎样的曲线，有这个也可以简单地做出新的ex ante win curve。
https://i.imgur.com/5TaIF4g.jpg
https://i.imgur.com/hbaBIb5.jpg
There is a lot to observe in these new ex ante curves. First, we can see the
win curve is much flatter and has a wider hump, just as Birnbaum predicted.
Any increase in uncertainty will continue to flatten the win curve. While
the marginal wins on the hump of the win curve (between 85 and 95 wins) are
most valuable, you do not know where you will end up on the win curve before
the season. By improving your projection from 82 to 83 wins, you may end up
getting yourself some of those most valuable wins.
我们可以从新的ex ante curves看出许多东西：
1. 这个win curve比起萧华的版本更平，峰更宽，如同Birnbaum嘴的一样，
只要增加不确定性就会使得win curve变得更平。
最有价值的胜场是落在85-95胜之间的那些胜场，你并不知道赛季结束后
会落在win curve的哪个位置，如果今天是打算从82胜进步到83胜，
到头来还是得透过roster调整尽可能的拿下那些最有价值的胜场。
Furthermore, we can see this team should never pay more than about $6.5
million to add a projected win to its preseason forecast. This means it
should not pay the going market rate for most free agents despite the fact
that this team has a $210 million payoff from winning 95 games. It is not
sensible for many teams to spend significant money in free agency, especially
when they are not in a high-leverage spot on the win curve. Empirically, we
see teams generally recognize this. The most valuable wins are
worth over $10 million to this team. However, it can’t go buy these wins
with certainty.
2. 此外，我们可以看到球队不应该付出超过6.5M来增加额外的一场预期胜场，
（ex ante win curve的最高点不超过6.5M）这也就表示预期会有高胜场的球队
不应该在市场上花大把钞票寻求自由球员，其实也蛮多队伍已经知道这件事，
而且有的时候最有价值的胜场还值超过10M！但我们仍旧无法确定多10M价值就能
真的多了一胜出来。
We can model the trade deadline by decreasing the amount of uncertainty in
the win curve. At the deadline, teams already have played over half of the
season. Therefore, they are much more certain of the value of the wins they
are acquiring. If we lower the standard deviation of the wins, we can build
a trade deadline win curve.
这里可以玩在交易大限前的win curve model，因为在交易大限前可以减少win curve
的不确定性，在Deadline的时候已经打了大半个赛季，有很多不确定性已经发生或经过，
所以追求每一胜的价值更清楚确定，这里透过降低胜场标准差建构交易大限的win
curve.
You can see this win curve clearly has a much larger peak than the ex ante
win curve. A team in contention may be willing to give up much more for a
projected win at the trade deadline than it would in the offseason.
(The win curve becomes a worse proxy for the outcomes that a team cares about
at the trade deadline, but this is a topic that requires a separate post.)
Adding a projected win at the trade deadline has a much higher probability
of adding the actual wins you are hoping for.
可以清楚地看到这次做出来的ex ante win curve有更高的峰，在竞争中的球队会愿意
在交易大限时付出比起在休赛季时更的多成本去取得额外胜场，
而且在交易大限时取得的预期胜场能转变为真实胜场的机率其实比起赛季出来的高，
就如同上一段所说，因为赛季打了大半，已经有些uncertainty发生过了。
但其实这个例子有点不好，如果有的球队他妈完全不在意交易大限，那偏好性就不同，
而win curves就不能这样画，但此处先不考虑这个case。
The final issue that needs to be tackled is making this single-period model
into a multi-year model. Luckily, this shouldn't be too difficult. We can
still use the same shaped win curves for every year in the future; they just
need to be discounted.
最后一个是把这个模型拓展到跨年度上，而这理论上并不困难，只要把未来每年所使用
的那些相同的win curves折现过就可以了。
There are three factors to consider when discounting these future wins:
baseball's continuing salary inflation, the interest rate, and impatience.
Baseball has seen consistent salary growth now for decades. We will label
the inflation rate as π, and the interest rate as r. The factor for
impatience will be β, where 0<β<1. We have seen many teams, most notably
Mike Illitch's Detroit Tigers, operate with very significant impatience.
Depending on this unique preference, it can be very rational to
sacrifice the future for an extra win now.
在考虑未来胜场折现的时候要考虑几个额外因素：薪资通膨 π，联邦政府利率 r ，
还有老板的不耐烦指数 β， 0<β<1。 文内说过去的老虎队很显然就是没耐心，
但我个人觉得比较近代的例子就是罗莉亚的马林鱼，跟他的房地产骗局（？
We can adjust the value of wins in future years by a factor of
[(1+π)β/(1+r)]^t, where t is the number of years we are in the future.
This equation is simply a scalar to adjust the win curve up or down. The
interest rate and β decrease the value of a future win, while (1+π)
increases how much we value a future win. In the current year, where t=0,
this scalar will just go to 1.
此处使用[(1+π)β/(1+r)]^t 来作为未来t年的胜场价值调整参数，
其中不耐烦指数β跟联邦利率会减损未来胜场价值，而通膨则会提升未来胜场价值，
当t=0的时候，就是未来0年，这个参数为1，也就是本年度的胜场价值。
Finally, we can now discuss using the model for transaction analysis.
We need to have a team's win curve and its projections with and without
a specific player. Given those two things, we can precisely calculate how
much that player is worth to a team. The fundamental concept of asset
pricing theory is that price equals expected discounted payoff. We can now
calculate the expected discounted payoff of any player for a given team.
把这个模型拿来做交易分析，我们需要一支球队的win curve，跟
“有/没有某个球员的两种预期胜场”来评估，因此我们就可以精确算出一个球员对
球队的价值，这里就真的用上了资产定价理论，也就是价格要跟预期折现报酬相同，
这里先来计算任何一个球员在某一队上产生的预期折现报酬。
For each year the player is under contract, we take the expected value of
the roster with the player and subtract both the expected value of the
roster without the player and the player’s salary. This will give us a
net present value evaluation of any player. In simplified mathematical
terms, it’s merely:
https://i.imgur.com/cADrtrr.jpg
当每个球员都身负合约时，可以用
（有这个球员的预期胜场报酬-没这个球员的预期胜场报酬-此球员薪水）＊调整参数
来决定此球员在第t年的净值为多少。
Going back to our original example of the Chris Sale trade, we can make
no declarations from here on how much each team benefitted from the deal.
But if we had a win curve for both teams and their projections with and
without the players, we could easily find the unique dollar value of each
player in the deal to each of the two teams.
数学式子我想大家看够多了，回来看一下文章开头所说那个Sale的交易，
我们不能在这里宣称两队到底在这笔交易里面获得了多少好处，
因为我们没有这两队的win curve跟他们的偏好以及预期胜场。
但如果有这些东西我们就可以轻易的算出在这笔交易中每个球员在两边队伍的价值了
（...那你举这个例子有屁用= =）
This model provides a simple framework to evaluate the payoff of every
potential transaction, though it does come with a few limitations.
The biggest issue is that it assumes a player will remain with the team
throughout his entire contract and for no longer. However, these sorts
of minor concerns can be accounted for manually. The question of how much
a player is worth to a given team no longer has to be a guessing game.
总而言之，这个模型提供一个简单的算法去评估每一笔可能交易的价值，
而非仅仅透过简单的WAR加减计算，而且这个算法的限制已经相对较少，
虽然这里最大的问题是假设了一个球员会完整走完他的合约，不会opt out等等，
但这些细节可以透过手动调整来解决，而使得这个方法让球员价值
不再是一场不知道答案的赌博！
后记：因为刚好是自己熟悉的两个领域，所以一边读一边翻觉得很顺畅，
如果有看不懂的地方请尽量提出！
但我认为有点缺陷的地方在于，偏好是属于动态的，例如最近的史棒棒交易，
是因为他有霸王条款、展现偏好，才得以让洋基队改变决策，
这样的动态过程他并没有展示要怎么调整，也许需要靠赛局/IO来解决。
另外一个缺陷在于他没有精确考虑前面的看法：aging跟伤病风险
如果能用其他统计方式如存活来算出hazard跟球员的生涯长度与对应时间受伤
机率，应该相对更准，或者是利用black-scholes来做定价等等，
但也许这样的解释力已经相对足够了！