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?
你可曾在候診室枯等數小時 儘管早已預約指定看診的時間? 訂了房,但旅店以客滿為由 而拒絕你住宿? 買了票,卻無法登機? 這些都是「超額預定」的症狀 亦即商家或機構超賣的行為 儘管這行為常常激怒客戶 超賣往往能增加收益 讓商家更有效地運用資源 明知不是所有的人都會準時赴約 看醫生、投宿旅店或搭機 所以儘管額滿,仍釋出額外預定名額 航空公司是最典型的例子, 部份因這種情況頻繁發生 每年約五萬名乘客 因超售機票而無法登機 航空公司對此數字一點也不意外 因為他們用統計數字 來決定要賣多少機票 這是個精準的操作 賣少了,浪費座位 賣多了,可能要付出代價 補償金,免費機票、住宿 和惱怒的客戶 這是個簡化了的計算方法 航空公司累計了多年的資料 知道誰會、誰不會準時出現 在哪些航班的登機門 例如,他們會知道某個特定航線 旅客有 90% 的機率會準時出現 為了簡化運算 我們假設每個乘客都獨自旅行 沒攜家帶眷,也沒參團 如果航班有 180 個座位 賣出 180 張機票 結果最可能是 162 個乘客登機 當然,登機人數可能多於或少於此數 每個數值出現的機率 呈「二項分佈」 機率最高處是最可能出現的人數 現在看收益 航空公司收取每個乘客的機票錢 但為每個被擠下飛機的乘客花錢 假設每張機票二百五十元, 不能改為較晚的航班 為每個被擠下飛機的乘客花八百元 這些只是舉例的數字 真正的數字各異 所以若不超賣,賺四萬五千元 超賣十五張,且至少十五人未出現 賺得 48,750 元 這是最好的情況 最壞的情況,每個人都出現 十五名乘客不幸被擠下飛機, 收益只餘 36,750 元 比當初只賣出 180 張票的收益更少 但重要的不只是 每個情況的收益多好或多差 而是多容易發生 每個情況發生的機率是多少? 可以用二項分佈得到答案 就此例而言 恰好195 個乘客出現的機率 接近零 恰好 184 個乘客 出現的機率為 1.11% 以此類推 這些機率乘以各個情況的收益 加總起來 用賣出 195 張票的進帳 減去那個總額 就得到賣了 195 張票的 期望收益值 依相同的方法計算 每種超賣機票的期望收益值 航空公司就能找出 可能得到最大收益值的方案 本例是賣 198 張票 航空公司可能獲益 48,774 元 比完全不超賣,多了將近四千元 那僅是一航班而已 乘以每一航空公司每年的百萬次航班 超賣肯定急速地聚少成多 當然,實際的運算要複雜得多 航空公司會加入其他因素 以得出更精確的模型 但是他們應該這樣做嗎? 有些人主張超賣是不道德的 以同一資源向兩個人收費 當然,如果你百分之百確定 有人將會缺席 賣掉他們的座位無妨 但是倘若你只有 95% 的把握呢? 75% 呢? 可有一個分辨不道德與務實的數值呢?