三冠王的教诲：追寻难成之举的三大启示
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GLOBAL FINANCIAL STRATEGIES www.credit-suisse.com
三重冠的启示——追逐这难以企及成就的三个教训 2015 年 6 月 4 日
Triple Crown Teachings Three Takeaways from the Quest for the Elusive Feat June 4, 2015
Authors
Authors
迈克尔·J·莫布森 [email protected]
Michael J. Mauboussin [email protected]
丹·卡拉汉，特许金融分析师 [email protected]
Dan Callahan, CFA [email protected]
“当‘美国法老’在本周六于贝尔蒙特公园寻求完成三重冠横扫时，赛马迷们希望看到一场伟大的比赛，也许还能最终确认这匹冠军马在赛马史上赢得崇高地位。根据近期的历史，这不会发生。”
“When American Pharoah seeks to complete a sweep of the Triple Crown at Belmont Park on Saturday, racing fans will hope to see a great race and perhaps a definitive confirmation that the winner has earned a lofty place in the sport’s history. Based on recent history, it won’t happen.”
Andrew Beyer1
Andrew Beyer1
赛马三重冠的追求为投资组织提供了三个教训：
The quest for the Triple Crown of horse racing offers three lessons for an investment organization:
– 关注过去的结果 – 认识多样性的价值 – 价值 = 概率 × 价格
第一个教训是在评估未来结果时考虑基础概率。
Pay attention to the results of the past Recognize the value of diversity Value = Probability x Price The first lesson is to consider base rates in assessing future outcomes.
第二个教训是促进认知多样性，从而避免组织中出现相当于近亲繁殖的情况。
The second lesson is to promote cognitive diversity and thereby avoid the equivalent of inbreeding in your organization.
最后一个教训是聚焦投资中的错误定价。
The final lesson is to focus on mispricing in investing.
Introduction
Introduction
要跻身美国赛马万神殿，必须赢得三重冠——肯塔基德比、普瑞克内斯锦标和贝尔蒙特锦标。这三场比赛距离不同，且仅在五周内举行。对于一匹马来说，这是一项极其艰巨的任务。事实上，在过去一百年里，只有十一匹马做到了这一点。
To be part of the pantheon of horse racing in the United States requires winning the Triple Crown—the Kentucky Derby, the Preakness Stakes, and the Belmont Stakes. The three races cover different distances and occur over a span of just five weeks. It is an extraordinarily hard task for a horse. Indeed, only eleven have achieved it in the past century.
2015 年 6 月 6 日星期六，“美国法老”将成为第 32 匹尝试完成这一壮举的马。在此之前，这匹三岁枣色公马在参加的六场比赛中赢得了五场，包括在肯塔基德比中以一个马身的优势获胜，以及在普瑞克内斯中以更令人印象深刻的七个马身优势获胜。
On Saturday, June 6th, 2015, American Pharoah will be the 32nd horse to attempt to complete the feat.2 Up to that point, the three-year-old bay colt had won five of the six races he had entered including victories by a margin of one length at the Kentucky Derby and a more impressive seven lengths at the Preakness.
无论“美国法老”是赢是输，通往三重冠的道路为投资者和组织提供了三个教训。其中一些教训可能耳熟能详，但仍是很好的提醒。
Whether American Pharoah wins or losses, the path to the Triple Crown offers three lessons for investors and organizations. Some of the teachings may be familiar but are a useful reminder.
教训 1：关注过去的结果。赛马专家安德鲁·拜尔在谈到“美国法老”的获胜机会时说：“根据近期的历史，这不会发生。”为什么是这种悲观的调调？答案在于随时间变化的成功基础概率。
Lesson 1: Pay attention to the results of the past. Andrew Beyer, a horse racing expert, said of American Pharoah’s chances of winning: “Based on recent history, it won’t happen.” Why the downbeat message? The answer lies in the base rate of success over time.
在过去 100 年里，有 31 匹马有机会赢得三重冠，正如前文所述，其中 11 匹成功了。这给出了约 35% 的成功率。但这些数字掩盖了 1950 年前后显著的差异。1950 年之前，尝试的 9 匹马中有 8 匹获胜，接近 90%。自 1950 年以来，22 匹中只有 3 匹成功，而且自 1978 年以来一匹也没有。在过去 65 年中，超过 85% 的马未能赢得荣耀。
In the last 100 years, 31 horses have been in a position to capture the Triple Crown and, as noted, 11 of them succeeded. That places the success rate at about 35 percent. But those numbers belie a marked distinction before and after 1950. Before 1950, eight of the nine horses that tried won, or close to 90 percent. Since 1950, only 3 of 22 have succeeded, and none since 1978. In the past sixty-five years, more than 85 percent of the horses failed to capture glory.
此外，“美国法老”的速度并不特别出众。拜尔开发了拜尔速度指数，该指数衡量调整了赛道条件后的比赛结果。速度指数越高越好。
Further, American Pharoah’s speed is not particularly distinguished. Beyer developed the Beyer Speed Figure, which measures the results of a horse adjusted for the track conditions. Higher speed figures are better.
当你将“美国法老”在肯塔基德比和普瑞克内斯的表现与最近八匹三重冠挑战者的数据进行比较时——可靠的速度指数仅在 1990 年后才有——他几乎垫底。唯一速度指数更差的是去年挑战失败的“加州铬”，它在贝尔蒙特中并列第四。
When you compare American Pharoah’s figures for the Kentucky Derby and Preakness to those of the last eight Triple Crown aspirants—reliable speed figures have only been available since 1990—he falls near the bottom of the heap. The sole horse with worse speed figures was last year’s failed Triple Crown contender, California Chrome, who finished the Belmont tied for fourth place.
2014 年，“加州铬”需要面对 10 个挑战者，而“美国法老”将只与 7 匹对抗。这提高了“美国法老”的前景。但赢得贝尔蒙特在过去 36 年里让所有挑战者都无功而返。
In 2014, California Chrome had to deal with 10 challengers, whereas American Pharoah will run against just 7. That brightens American Pharoah’s prospects. But winning the Belmont has eluded all comers in the last 36 years.
这个教训是在评估未来结果时考虑基础概率。虽然找到合适的参考类别和相关数据可能很棘手，但心理学家已经表明，我们倾向于过于依赖自己的视角，而未能充分考虑他人的经验。
The lesson is to consider base rates in assessing future outcomes. While finding an appropriate reference class and the relevant data can be tricky, psychologists have shown that we tend to rely too much on our own point of view and fail to consider sufficiently the experiences of others.
教训 2：认识多样性的价值。在讨论成功率时，1950 年这个年份跳出了页面。为什么马在 1950 年之前有如此高的成功率，而在此之后却如此之低？
Lesson 2: Recognize the value of diversity. In the discussion of success rates, the year 1950 pops off the page. Why would the horses have succeeded at such a high rate prior to 1950 and at such a low rate after it?
答案有多个方面，但图表 1 是一个很好的起点。它显示了 1950 年到 2015 年三项赛事的获胜时间（当然，没有包括贝尔蒙特的结果）。这些比赛没有按时间顺序排列，而是按距离排序。普瑞克内斯是距离最短的比赛（1 又 3/16 英里），肯塔基德比居中（1 又 1/4 英里），贝尔蒙特最长（1 又 1/2 英里）。
The answer has many facets, but exhibit 1 is a good place to start. It shows the winning times for the three events from 1950 to 2015 (of course, it is without the Belmont result). Rather than listing the races in chronological order, they are ranked by distance. The Preakness is the shortest race (1 3/16 miles), the Kentucky Derby in the middle (1 1/4 ), and the Belmont the longest (1 1/2 ).
虽然普瑞克内斯的完赛时间自 1950 年以来略有下降，但你可以看到肯塔基德比和贝尔蒙特的时间趋势基本持平。例如，1950 年肯塔基德比的冠军“中间地带”的原始时间是 2:01.60，比 2015 年“美国法老”的 2:03.02 更快。
While the Preakness shows a modest decline in finishing times since 1950, you can see that the trend in times for the Kentucky Derby and the Belmont are essentially flat. For example, the winner of the 1950 Kentucky Derby, Middleground, had a raw time of 2:01:60, faster than American Pharoah’s 2:03:02 in 2015.
一位研究这个主题的学者写道：“尽管有培育更快纯血马的密集计划……比赛速度……在过去 40-60 年里并没有提高。”你能想到还有其他体育运动自上世纪中期以来成绩没有任何提高吗？
One academic studying the topic wrote, “Despite intensive programs to breed faster thoroughbreds . . . race speeds . . . have not increased in the last 40-60 years.”3 Can you think of any other athletic endeavor where the results have not improved since the middle of the last century?
图表 1：三重冠赛事获胜时间，1950-2015 年 普瑞克内斯锦标 02:02.69
Exhibit 1: Winning Times for Triple Crown Races, 1950-2015 Preakness Stakes 02:02.69
02:00.96
02:00.96
| 分钟：秒 | |||||||||||||
| 01:59.23 | |||||||||||||
| 01:57.50 | |||||||||||||
| 01:55.78 | |||||||||||||
| 01:54.05 | |||||||||||||
| 01:52.32 | |||||||||||||
| 1950 | 1955 | 1960 | 1965 | 1970 | 1975 | 1980 | 1985 | 1990 | 1995 | 2000 | 2005 | 2010 | 2015 |
| 肯塔基德比 | |||||||||||||
| 02:07.01 | |||||||||||||
| 02:05.28 |
Minutes:Seconds 01:59.23 01:57.50 01:55.78 01:54.05 01:52.32 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 Kentucky Derby 02:07.01 02:05.28
| 分钟：秒 | |||||||||||||
| 02:03.55 | |||||||||||||
| 02:01.82 | |||||||||||||
| 02:00.10 | |||||||||||||
| 01:58.37 | |||||||||||||
| 1950 | 1955 | 1960 | 1965 | 1970 | 1975 | 1980 | 1985 | 1990 | 1995 | 2000 | 2005 | 2010 | 2015 |
| 贝尔蒙特锦标 | |||||||||||||
| 02:35.52 | |||||||||||||
| 02:33.79 |
Minutes:Seconds 02:03.55 02:01.82 02:00.10 01:58.37 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 Belmont Stakes 02:35.52 02:33.79
Minutes:Seconds 02:32.06 02:30.34 02:28.61 02:26.88 02:25.15 02:23.42 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Minutes:Seconds 02:32.06 02:30.34 02:28.61 02:26.88 02:25.15 02:23.42 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
来源：www.preakness-stakes.info；www.kentuckyderby.ag；www.belmontstakes.com。
Source: www.preakness-stakes.info; www.kentuckyderby.ag; www.belmontstakes.com.
以人类为例。研究人员估计，在相同时间段内，从短跑到马拉松的精英人类赛跑获胜时间提高了 12% 到 13%。你可以通过更大的人口基数、更好的营养、更完善的训练方法、心理激励以及技术进步来解释这一进步。（这排除了其他可能影响表现增强的来源，这些来源对动物和人类都相关。）
Take humans as an example. Researchers estimate that winning times for elite humans running races from sprints to marathons have improved 12 to 13 percent over the same time period.4 You can explain this gain through a larger population, better nutrition, refined training methods, psychological incentives, and an improvement in technology. (This leaves aside other potential sources of performance enhancement that are relevant for animals and humans.)
一个合理的解释是，纯血马在 20 世纪中期达到了生理表现的顶峰。育种是这个论点中的关键部分。1791 年，詹姆斯·韦瑟比建立了纯血马育种记录，实际上创造了一个封闭种群。分析显示，在目前 50 万匹纯血马中，大约三分之一的基因来自四匹出生在 1688 年至 1725 年间的种公马。最大的十个贡献者负责大约一半的基因，而 31 匹马构成了当今马群基因组成的 80%。纯血马的近亲繁殖比率在所有马种中名列前茅。
One plausible explanation is that thoroughbred horses reached their physiological peak of performance in the mid-1900s.5 Breeding is a key part of this argument. In 1791, James Weatherby established a record of thoroughbred horse breeding, effectively creating a closed population. Analysis shows that about one-third of the genes in the current population of one-half million thoroughbreds come from four stallions born between 1688 and 1725. The top ten contributors are responsible for about half of the genes, and 31 horses account for 80 percent of the genetic makeup of today’s stock.6 Thoroughbreds have among the highest ratio of inbreeding of any horse breed.7
早期的赛马比赛距离往往比现在长，马主使用的马也更年长。在过去两个世纪里，比赛距离变短，马匹更年轻。因此，纯血马被培育成注重速度和早熟。这使得从 19 世纪初到 20 世纪中期，比赛成绩稳步提高。但除了在短距离上有所改善外，马的速度根本没有变快。自 1950 年以来，正常变异解释了大部分时间的起伏。
Early horse races tended to be over longer distances than today and owners ran horses that were older. In the last two centuries, the distances have gotten shorter and the horses younger. As a result, thoroughbreds were bred for speed and precocity. This allowed for a steady improvement in race performance from the early 1800s through the mid-1900s.8 But horses have simply not gotten faster, save for some improvement at short distances. Since 1950, normal variance explains most of the ups and downs in times. 9
事实上，近几十年来，近亲繁殖似乎变得更加严重。虽然多年来育种目标一直是追求速度，但如今的重点更多是培育在拍卖会上能卖出尽可能高价的周岁马。结果，自 1990 年代中期以来，近亲繁殖急剧增加。
Indeed, it appears that inbreeding has taken a turn for the worse in recent decades. Whereas the goal for years had been to breed for speed, the emphasis today is more on producing yearlings that fetch as much money as possible at auction. As a result, inbreeding has increased sharply since the mid-1990s.10
这个教训是避免在组织中发生近亲繁殖。研究表明，在面对复杂问题时，多样化的团队往往比同质化的团队表现更好。许多组织，包括投资银行、咨询公司和投资管理机构，倾向于从少数挑选出来的大学招聘，并非常看重个人匹配度。这种做法可能限制多样性。
The lesson is to avoid the equivalent of inbreeding in your organization. Research shows that diverse teams tend to outperform more homogenous ones when faced with complex problems. Many organizations, including investment banks, consulting firms, and investment managers, tend to recruit from a select number of universities and place substantial weight on personal fit. This practice can limit diversity.11
目标是建立一个具有高认知多样性和良好文化契合度的员工队伍。你可以把认知多样性的价值看作类似于遗传多样性的价值。记者马特·里德利在提到创新的关键是“思想的交配”时，使这种联系更加生动。关键是，你希望多样化的思想进行繁殖，以提高你组织的适应性。
The goal is to build a workforce with high cognitive diversity and good cultural fit.12 You can think of the value of cognitive diversity as similar to that of genetic diversity. Matt Ridley, a journalist, makes the link even more vivid when he suggests that the key to innovation is “ideas having sex.”13 The point is that you want diverse ideas to engage in reproduction so as to increase the fitness of your organization.
教训 3：价值 = 概率 × 价格。尽管尝试赢得三重冠的马的历史记录惨淡，但赌徒们似乎对每匹马的机会始终充满乐观。图表 2 显示，最近八位挑战者由赔率隐含的赢得贝尔蒙特概率在 40% 到 80% 之间。
Lesson 3: Value = Probability x Price. Notwithstanding the dismal record of horses trying to win the Triple Crown, bettors appear unfailingly optimistic about each horse’s chances. Exhibit 2 shows that the last eight contenders had probabilities of winning the Belmont in the range of 40 to 80 percent as implied by the odds.
图表 2：最近八位三重冠挑战者赢得贝尔蒙特的隐含概率 概率 银色魅力 49% 寂静行者 56 魅力四射 39 战争旗帜 44 搞笑同盟 50 精明琼斯 74 大棕 77 加州铬 63 来源：Equibase 公司。
Exhibit 2: Implied Probability of Winning the Belmont for the Last Eight Triple Crown Aspirants Probability Silver Charm 49% Real Quiet 56 Charismatic 39 War Emblem 44 Funny Cide 50 Smarty Jones 74 Big Brown 77 California Chrome 63 Source: Equibase Company.
“美国法老”也在同一范围内。最近 3-5 的赔率表明获胜概率为 63%。这个数字非常高，代表着很高的表现预期。赔率是通过彩池下注系统设定的，赌徒通过分配资金来表达他们的预期。在一匹马上押注的钱越多，它的获胜概率就越高。
American Pharoah is in the same range. The recent odds of 3-5 suggest a probability of winning of 63 percent. That figure is very high, representing a lofty expectation of performance. Odds are set through a pari-mutuel system, where the bettors express their expectations by how they allocate their dollars. More money bet on a horse drives up its probability of winning.
手相专家兼作家史蒂文·克里斯特强调，机会的吸引力取决于事件发生的概率和回报两个因素。在任何赌徒获得回报之前，会有一笔抽水用于支付奖金、赛道和州政府。这意味着赢奖池减少了 15% 到 20%。
Steven Crist, a handicapper and author, emphasizes that the attractiveness of an opportunity is a function of both the probability of the event occurring and the payoff.14 Before any bettors are rewarded, there is a takeout to pay the race purses, track, and state. This means that the win pool is reduced by 15-20 percent.
假设抽水是 20%。这意味着对一匹有 50% 获胜机会的马下注 4 美元，平均回报是 3.20 美元。
Assume the takeout is 20 percent. That means a $4 bet on a horse with a 50 percent chance to win pays, on average, $3.20.15
克里斯特的观点是，明智的手相专家需要概率和价格的组合才能产生正的期望值。知道哪匹马可能赢不会让你赚钱；你需要知道哪匹马是被错误定价的。几乎不用费力就能把它应用到商业和投资的世界里。知道哪家公司最好不是致富之路；知道哪家公司被错误定价才能让你赚钱。
Crist’s point is that the intelligent handicapper requires a combination of probability and price to yield a positive expected value. Knowing which horse is likely to win doesn’t make you money; you need to know which horse is mispriced. It takes little effort to translate this to the world of business and investing. Knowing which company is best isn’t the path to riches; knowing which company is mispriced makes you money.
三重冠竞争马匹吸引了许多不成熟的赌徒。在美国一个典型的周六，赛马总投注额约为 5000 万美元。去年贝尔蒙特锦标的投注额是这个数字的三倍。你可以放心地假设，增加的 1 亿美元中大部分来自沉浸在三重冠兴奋中的人。结果，三重冠竞争马匹几乎总是被错误定价。概率乘以价格会产生糟糕的价值。也许对某些人来说，押注潜在的赢家带来的兴奋价值超过了门票成本。但就金钱而言，这并非好赌注。
Triple Crown contenders attract a lot of unsophisticated bettors. On a typical Saturday in the U.S., wagers on horses are about $50 million. For last year’s Belmont Stakes, the bets were three times that amount. You can safely assume that most of the incremental $100 million was from people caught up in the excitement of the Triple Crown.16 As a result, Triple Crown contenders are almost always mispriced. Probability times price yields a poor value. Perhaps for some, betting on a potential winner has excitement value that exceeds the cost of the ticket. But that doesn’t make it a good bet in monetary terms.
最后一个教训是，你需要聚焦投资中的错误定价。三重冠竞争马匹有点像 1990 年代末的互联网宠儿：它们的过往成功吸引了那些在投资时不充分考虑价格与价值关系的个人。
The final lesson is that you need to focus on mispricing in investing. Triple Crown contenders are a little like dot-com darlings from the late 1990s: Their past success draws the attention of individuals who invest without sufficient regard to the relationship between price and value.
“美国法老”可能会赢得三重冠，如果他做到了，那将是一项非凡的成就，为这项运动带来当之无愧的兴奋。但无论结果如何，我们都可以说，他目前的概率似乎太高了。
American Pharoah may win the Triple Crown, and if he does it will be an extraordinary feat that will bring deserved excitement to the sport. But no matter the outcome, we can say that his current probability appears too high.
尾注
1 安德鲁·拜尔，“贝尔蒙特发生了什么？育种者弱化了耐力”，《华盛顿邮报》，2015 年 6 月 1 日。
Endnotes 1 Andrew Beyer, “What Happened to the Belmont Stakes? Breeders De-Emphasized Stamina,” Washington Post, June 1, 2015.
2 那匹马的名字叫“Pharoah”而非“Pharaoh”，显然是个拼写错误。马主 Zayat 马房曾举办一场征名比赛，一位名叫玛莎·鲍姆加特纳的女士提交了获奖名字。
2 The horse’s name is “Pharoah” instead of “Pharaoh” because of an apparent error. The owners of the horse, Zayat Stables, ran a contest to name the horse. A woman named Marsha Baumgartner submitted the winner.
Zayat 马房随后将这个名称提交给美国赛马会——负责登记纯血马的组织。目前尚不清楚究竟是鲍姆加特纳拼错了名字，还是 Zayat 马房在提交时犯了错。但由于该名称符合所有审批标准，申请便按提交内容获准通过。梅丽莎·霍珀特，《那么，是谁拼错了 American Pharoah？》，《纽约时报》，2015 年 5 月 22 日。
Zayat Stables turned around and submitted the name to the Jockey Club, the organization that registers thoroughbreds. It is not clear whether Baumgartner misspelled the name or whether Zayat Stables made the error when submitting it. But since the name met all of the criteria for approval, it was granted as submitted. Melissa Hoppert, “So, Who Misspelled American Pharoah?” New York Times, May 22, 2015.
3 马克·W·丹尼，《狗、马和人的奔跑速度极限》，《实验生物学杂志》，第 211 卷，第 24 期，2008 年 12 月，第 3836-3849 页。
3 Mark W. Denny, “Limits to Running Speed in Dogs, Horses, and Humans,” Journal of Experimental Biology, Vol. 211, No. 24, December 2008, 3836-3849.
4 大卫·S·加德纳，《纯血马与人类的赛跑性能历史演进》，《马科兽医杂志》，第 38 卷，第 6 期，2006 年 11 月，第 581-583 页。
4 David S. Gardner, “Historical Progression of Racing Performance in Thoroughbreds and Man,” Equine Veterinary Journal, Vol. 38, No. 6, November 2006, 581-583.
5 小罗杰·皮尔克，《纯血马已经跑到了极限速度》，FiveThirtyEight，2014 年 5 月 3 日。6 帕特里克·坎宁安，《纯血马的遗传学》，《科学美国人》，1991 年 5 月，第 92-98 页。7 杰西卡·L·彼得森、詹姆斯·R·米克尔森、E·格斯·科思兰、丽莎·S·安德森、珍妮特·阿克塞尔松、厄尼·贝利、丹妮卡·巴纳施、马修·M·宾斯、亚历山大·S·博尔赫斯、彼得·布拉马、阿图尔·达·卡马拉·马查多、奥特马尔·迪斯特尔、米凯拉·费利切蒂、劳拉·福克斯-克利普沙姆、凯瑟琳·T·格雷夫斯、杰拉德·格林、比安卡·哈斯、特尔希萨·长谷川、卡琳·赫曼、埃米琳·W·希尔、托索·勒布、加布里埃拉·林德格伦、汉内斯·洛希、玛丽亚·苏珊娜·洛佩斯、比阿特丽斯·A·麦吉夫尼、索菲亚·米科、尼古拉斯·奥尔、M·塞西莉亚·T·佩内多、理查德·J·皮尔西、玛利亚·拉克利奥、斯特凡·里德、克努特·H·罗德、毛里齐奥·西尔维斯特雷利、琼·斯温伯恩、户崎辉明、马克·沃丁、克莱尔·M·韦德、莫莉·E·麦丘，《全基因组 SNP 数据揭示现代马的遗传多样性》，《公共科学图书馆·综合》，2013 年 1 月 30 日。
5 Roger Pielke Jr., “Thoroughbreds Are Running as Fast as They Can,” FiveThirtyEight, May 3, 2014. 6 Patrick Cunningham, “The Genetics of Thoroughbred Horses,” Scientific American, May 1991, 92-98. 7 Jessica L. Petersen, James R. Mickelson, E. Gus Cothran, Lisa S. Andersson, Jeanette Axelsson, Ernie Bailey, Danika Bannasch, Matthew M. Binns, Alexandre S. Borges, Pieter Brama, Artur da Câmara Machado, Ottmar Distl, Michela Felicetti, Laura Fox-Clipsham, Kathryn T. Graves, Gérard Guérin, Bianca Haase, Telhisa Hasegawa, Karin Hemmann, Emmeline W. Hill, Tosso Leeb, Gabriella Lindgren, Hannes Lohi, Maria Susana Lopes, Beatrice A. McGivney, Sofia Mikko, Nicholas Orr, M. Cecilia T Penedo, Richard J. Piercy, Marja Raekallio, Stefan Rieder, Knut H. Røed, Maurizio Silvestrelli, June Swinburne, Teruaki Tozaki, Mark Vaudin, Claire M. Wade, Molly E. McCue, “Genetic Diversity in the Modern Horse Illustrated from Genome-Wide SNP Data,” Plos One, January 30, 2013.
8 米姆·鲍尔、比阿特丽斯·A·麦吉夫尼、迈克尔·G·坎帕纳、顾晶晶、丽莎·S·安德森、伊丽莎白·巴雷特、凯瑟琳·R·戴维斯、索菲亚·米科、弗劳克·斯托克、瓦列里·沃龙科娃、丹尼尔·G·布拉德利、艾伦·G·费伊、加布里埃拉·林德格伦、大卫·E·麦克休、加林娜·苏利莫娃、埃米琳·W·希尔，《纯血赛马速度的遗传起源与历史》，《自然·通讯》，第 3 卷，第 643 号，2012 年 1 月 24 日。9 赛克雷塔利亚特，有史以来最伟大的赛马，似乎是一个异例。这匹马的心脏估计重 22 磅，是纯血马平均心脏大小的 2.5 倍以上。研究人员认为，心脏大小的突变通过母系遗传。参见《布鲁斯·洛血统数字体系重新流行》，《马科兽医科学杂志》，第 14 卷，第 4 期，1994 年 4 月，第 200 页。10 M.M. 宾斯、D.A. 伯勒、E. 贝利、T.L. 李尔、J.M. 卡德威尔、D.H. 兰伯特，《纯血马的近亲繁殖》，《动物遗传学》，第 43 卷，第 3 期，2012 年 6 月，第 340-342 页。
8 Mim Bower, Beatrice A. McGivney, Michael G. Campana, Jingjing Gu, Lisa S. Andersson, Elizabeth Barrett, Catherine R. Davis, Sofia Mikko, Frauke Stock, Valery Voronkova, Daniel G. Bradley, Alan G. Fahey, Gabriella Lindgren, David E. MacHugh, Galina Sulimova, and Emmeline W. Hill, “The Genetic Origins and History of Speed in the Thoroughbred Racehorse,” Nature Communications, Vol. 3, No. 643, January 24, 2012. 9 Secretariat, the greatest racehorse of all time, appears to be an anomaly. That horse’s heart was estimated to weigh 22 pounds, more than 2.5 times the size of the average heart for a thoroughbred. Researchers believe that a mutuation for heart size is passed through the mare line. See “Bruce Lowe Figure System Coming Back Into Vogue,” Journal of Equine Veterinary Science, Vol. 14, No. 4, April 1994, 200. 10 M.M. Binns, D.A. Boehler, E. Bailey, T.L. Lear, J.M. Cardwell, and D.H. Lambert, “Inbreeding in the Thoroughbred Horse,” Animal Genetics, Vol. 43, No. 3, June 2012, 340-342.
11 劳伦·A·里维拉，《猜猜谁在工作场合格格不入》，《纽约时报》，2015 年 5 月 30 日。
11 Lauren A. Rivera, “Guess Who Doesn’t Fit In at Work,” New York Times, May 30, 2015.
12 伊丽莎白·曼尼克斯、玛格丽特·A·尼尔，《差异为何重要？组织中多样化团队的前景与现实》，《公共利益心理学》，第 6 卷，第 2 期，2005 年 10 月，第 31-55 页。
12 Elizabeth Mannix and Margaret A. Neale, “What Differences Make a Difference? The Promise and Reality of Diverse Teams in Organizations,” Psychological Science in the Public Interest, Vol. 6, No. 2, October 2005, 31-55.
13 马特·里德利，《思想的性结合》，Reason.com，2010 年 7 月。
13 Matt Ridley, “Ideas Having Sex,” Reason.com, July 2010.
14 史蒂文·克里斯特，《克里斯特谈价值》，载于拜尔等《与高手下注》（纽约：《每日赛马报》出版社，2001 年），第 61-75 页。
14 Steven Crist, “Crist on Value,” in Beyer, et al., Bet with the Best (New York: Daily Racing Form Press, 2001), 61-75.
15 以下是基于克里斯特文章的计算过程。假设一个 1000 美元的赢马彩池，因抽水及零头取整（向下舍入派彩）而缩减至 800 美元。一匹有 50% 胜率的马，会有 125 份 4 美元的投注押在它身上（125 美元 x $4 = 500 美元，即总投注额 1000 美元的 50%）。因此若该马获胜，派彩为 6.40 美元（800/125）；若败，则派彩为零。所以期望派彩是 (0.5 x $6.40) + (0.5 x $0) = $3.20。
15 Here’s the math, based on Crist’s essay. Assume a win pool of $1,000 is reduced to $800 because of takeout and breakage (rounding down payoffs). A horse with a 50 percent probability of winning would have had 125, $4 bets placed on him ($125 x $4 = $500, or 50 percent of the total $1,000 bet). So if the horse wins, the payoff is $6.40 ($800/125) and zero if he loses. So the expected payoff is (0.5 x $6.40) + (0.5 x $0) = $3.20.
16 埃里克·切米，《如果你赌普利克内斯锦标赛，别犯这些新手错误》，NBCNews.com，2015 年 5 月 15 日。
16 Eric Chemi, “If You Bet on the Preakness, Don’t Make These Rookie Blunders,” NBCNews.com, May 15, 2015.