Have you ever sat in a doctor's office for hours despite having an appointment at a specific time, has the hotel turned down your reservation because it's full? Or have you been bumped off a flight that you paid for? These are all symptoms of overbooking, a practice where businesses and institutions sell or book more than their full capacity. While often infuriating for the customer, overbooking happens because it increases profits while also letting businesses optimize their resources. They know that not everyone will show up to their appointments, reservations and flights, so they make more available than they actually have to offer. Airlines are the classical example, partially because it happens so often, about 50000 people get bumped off their flights each year. That figure comes at little surprise to the airlines themselves, which used statistics to determine exactly how many tickets to sell. It's a delicate operation, sell too few and they're wasting seats, sell too many and they pay penalties, money, free flights, hotel stays and annoyed customers. So here's a simplified version of how their calculations work. Airlines have collected years worth of information about who does and doesn't show up for certain flights. They know, for example, that on a particular route, the probability that each individual customer will show up on time is 90 percent. For the sake of simplicity, will assume that every customer is traveling individually rather than as families or groups, then if there are 180 seats on the plane and they sell 180 tickets, the most likely result is that 162 passengers will board. But of course, you could also end up with more passengers or fewer. The probability for each value is given by what's called a binomial distribution, which peaks at the most likely outcome. Now let's look at the revenue. The airline makes money from each ticket buyer and loses money for each person who gets bumped. Let's say a ticket costs 250 dollars and isn't exchangeable for a later flight and the cost of bumping a passenger is 800 dollars. These numbers are just for the sake of example. Actual amounts vary considerably. So here, if you don't sell any extra tickets, you make 45000 dollars. If you sell 15 extras and at least 15 people are no shows, you make forty eight thousand seven hundred fifty dollars. That's the best case. In the worst case, everyone shows up, 15 unlucky passengers get bumped and the revenue will only be thirty six thousand seven hundred fifty dollars, even less than if you only sold 180 tickets in the first place. But what matters isn't just how good or bad a scenario is financially, but how likely it is to happen. So how likely is each scenario? We can find out by using the binomial distribution in this example, the probability of exactly 195 passengers boarding is almost zero percent. The probability of exactly 184 passengers boarding is one point one one percent and so on. Multiply these probabilities by the revenue for each case, add them all up and subtract the sum from the earnings by 195 sold tickets and you get the expected revenue for selling 195 tickets. By repeating this calculation for various numbers of extra tickets, the airline can find the one likely to yield the highest revenue in this example. That's 198 tickets from which the airline will probably make forty eight thousand seven hundred seventy four dollars, almost 4000 more than without overbooking. And that's just for one flight. Multiply that by a million flights per airline per year. And overbooking adds up fast. Of course, the actual calculation is much more complicated airlines apply many factors to create even more accurate models, but should they? Some argue that overbooking is unethical. You're charging two people for the same resource. Of course, if you're 100 percent sure someone won't show up, it's fine to sell their seat. But what if you're only 95 percent sure, 75 percent. Is there a number that separates being unethical from being practical?
你有没有曾经等医生几个小时看诊的经历? 尽管你有提前预约特定的时间? 你有没有经历过被酒店拒之门外因为客满? 或者你有没有经历过被航空公司拒载尽管你买了票? 这些都是超额预定的现象, 一个公司或者机构 卖出或者预定大于他们容量的行为。 尽管这行为使客户很愤怒, 超额预定的存在因为它可以增大利益 这还使公司更好的优化资源。 它们知道不是所有的客户都会出现, 预约的时间, 和预约的航班, 所以它们会提供比它们实际容量更多的空位。 航空公司就是一个经典案例,一部分因为它发生地太过频繁了。 每年大约有50,000人被拒载。 这个数字让航空公司也大吃一惊, 因为它们有利用统计数据去分析售票情况。 这是一个精准的运算。 卖了少了,它们浪费了空位。 卖了多了,它们有会遭到惩罚。 钱,免费的机票,酒店和恼怒的客户。 这里呢,有一个简化版的公式,告诉你这个计算是如何运行的。 航空公司累计了多年的数据和信息 关于谁会或者不会出现在一个特定的航班。 举个例子,它们知道在一个特定的航线, 每一个客户的按时出现率是90% 为了简化运算 我们假设每个人都在独自一人旅行 并没有家人或者团队 那么,如果航班有180个空位,它们就会卖180张机票 结果会是162个乘客出现在飞机上 当然, 也有可能更多或者更少的乘客出现 每个数值出现的几率是通过 二项分布得出的 在最有可能的情况下达到最高点 现在我们来看一下收益 航空公司在每个乘客身上赚钱, 在被拒载的乘客上损失钱 如果每张票买$250美元,并且不能改签 拒载一位乘客会花费航空公司$800 这些数字只是为了举例子而已 真正的数字根据情况会有不同 所以,如果你不卖出更多的票子,你会赚$45,000 如果你多卖15张票,并且这些乘客都没有出现 你会赚$48,750 这会是最好的情况 最坏的情况,每个人都出现 15个不幸的乘客会被拒载,收益也只会是$36,750 比卖出180张票要来的更少 但是重要的并不只是财政上的好与坏 重要的是这有多么容易发生 每个情况发生的几率是多少? 我们可以通过使用二项分布来得出答案 在这个例子中,正好有195个乘客出现的几率 大约为0%。 184个乘客出现的几率为1.11%,以此类推。 用这些几率乘以每个情况下的收益, 然后把它们加起来, 再减去卖出195张票的收益, 你会得出卖出195张票的预计收益。 在多卖出的票上重复使用这个计算, 航空公司可以找出一个最后可能得到最大收益的方案。 在这个例子中,为卖198张票, 航空公司可以以此得到$48,774的收益, 比没有超额预定多赚了$4,000。 这仅仅是一架航班而已。 用这个数字乘以每个航空公司每年上百万的航班, 超额预定就聚少成多了。 当然,实际的运算要逼着复杂很多。 为了得出更精确的模型,航空公司会增加其他因素。 但是它们这样做应该吗? 有些人主张超额预定是不道德的。 你用同一个东西在两个人身上收取了的费用。 当然,你如果能百分百确定有人不会出现, 出售他们的座位也是允许的。 但是如果你只是95%确定呢? 75%? 能否有一个数字能用来区分不道德行为和面对实际?