Has a hotel turned down your reservation because it's full?

旅館是否曾以客滿為由，拒絕了你的預約？

Or have you been bumped off a flight that you paid for?

或者你曾否付費買了機票，卻被告知需搭乘下一個航班？

These are all symptoms of overbooking, a practice where businesses and institutions sell or book more than their full capacity.

這些都是超賣現象，某些企業或機構販售或提供預訂超過滿額數量的一種做法。

While often infuriating for the customer, overbooking happens because it increases profits while also letting businesses optimize their resources.

雖然這常讓消費者火冒三丈，但超賣的存在是因為它能增加營收，同時也能讓公司充分利用資源。

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.

商家知道不是所有人都會履行預約、預訂和班機，所以他們選擇賣出比所擁有還多的量。

Airlines are the classical example, partially because it happens so often.

航空公司是典型的例子，部分原因是有鑑於它發生的機率較高。

About 50,000 people get bumped off their flights each year.

每一年約有五萬人都被迫換航班。

That figure comes at little surprise to the airlines themselves, which use statistics to determine exactly how many tickets to sell.

這個數字並未讓航空公司太過驚訝，因為他們是運用數據來決定究竟該賣多少張票。

It's a delicate operation.

這是需要謹慎處理的作業。

Sell too few, and they're wasting seats.

賣太少會浪費機位。

Sell too many, and they pay penalties—money, free flights, hotel stays, and annoyed customers.

賣太多，航空公司則要付出代價，如罰錢、免費航班、免費住宿，還要面對惱火的乘客。

So here's a simplified version of how their calculations work.

以下是航空公司計算模式的簡化版。

Airlines have collected years worth of information about who does and doesn't show up for certain flights.

航空公司蒐集了多年的資料，紀錄哪些人遇到哪些航班時的出現機率。

They know, for example, that on a particular route, the probability that each individual customer will show up on time is 90%.

舉例來說，他們知道某一特定航線的單一旅客準時出現機率是 90%。

For the sake of simplicity, we'll assume that every customer is traveling individually rather than as families or groups.

為了簡單計算，我們假設每一位旅客都是單獨旅行，而不是家族或團體旅遊。

Then, if there are 180 seats on the plane and they sell 180 tickets, the most likely result is that 162 passengers will board.

那麼，如果飛機上有 180 個座位，他們全數賣光，最可能的結果是有 162 位乘客登機。

But, of course, you could also end up with more passengers, or fewer.

當然，最後的乘客人數可能更多，也可能更少。

The probability for each value is given by what's called a binomial distribution, which peaks at the most likely outcome.

每一位乘客帶來的價值由所謂的二項分佈計算，並顯示出最有可能的結果。

Now, let's look at the revenue.

我們現在來看看營收。

The airline makes money from each ticket buyer and loses money for each person who gets bumped.

航空公司可以從每位買票者身上賺錢，但在延遲班機旅客身上會賠錢。

Let's say a ticket costs $250 and isn't exchangeable for a later flight, and the cost of bumping a passenger is $800.

我們假設一張機票價格是 250 美元，且不可更換至後續航班，而讓一名乘客延遲班機的成本是 800 美元。

These numbers are just for the sake of example.

這些數值只是做為例子參考用。

Actual amounts vary considerably.

真正的數值會有很大的差異。

So here, if you don't sell any extra tickets, you make $45,000.

在這裡可以看到，如果不超賣機票，能賺 45,000 美元。

If you sell 15 extras, and at least 15 people are no-shows, you make $48,750.

如果超賣 15 張票，且至少有 15 位乘客未出現，你能賺 48,750 美元。

That's the best case.

這是最理想的狀況。

In the worst case, everyone shows up.

最差的狀況是，每位乘客都出現了。

15 unlucky passengers get bumped, and the revenue will only be $36,750, even less than if you only sold 180 tickets in the first place.

15 位不幸的乘客被延班機，營收只有 36,750 美元，甚至比當初如果只賣 180 張還少。

But what matters isn't just how good or bad a scenario is financially, but how likely it is to happen.

但重點不是財務計算上情況有多好或不好，而是發生的機率有多高。

So, how likely is each scenario?

那麼，每種情況發生的機率為何？

We can find out by using the binomial distribution.

我們可以藉由二項分佈來找出答案。

In this example, the probability of exactly 195 passengers boarding is almost 0%.

在這個例子中，195 位乘客全數登機的機率接近 0%。

The probability of exactly 184 passengers boarding is 1.11%, and so on.

184 位乘客登機的機率為 1.11%，以此類推。

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.

將每一個可能結果的營收乘上機率，把它們相加然後扣掉 195 張票的收入，就能得到販賣 195 張票的預期營收。

By repeating this calculation for various numbers of extra tickets, the airline can find the one likely to yield the highest revenue.

藉由同樣方式計算不同的超賣張數，航空公司就能找出獲得最高營收的方案。

In this example, that's 198 tickets, from which the airline will probably make $48,774, almost 4,000 more than without overbooking.

在這個案例中，198 張是最理想的數字，航空公司能從中獲取 48,774美元，比起不超賣多了近 4,000 美元。

And that's just for one flight.

這只是一趟航班的數字而已。

Multiply that by a million flights per airline per year, and overbooking adds up fast.

將這個數字乘上每年每間航空公司共一百萬次的航班，超賣的營收會竄升更快。

Of course, the actual calculation is much more complicated.

當然，真正的計算方式要複雜多了。

Airlines apply many factors to create even more accurate models.

航空公司採用許多其他因素來獲得更正確的模組。

But should they?

但他們應該這樣做嗎？

Some argue that overbooking is unethical.

有些人辯稱，超賣是不道德的。

You're charging two people for the same resource.

同樣一張機票賣給了兩個人。

Of course, if you're 100% sure someone won't show up, it's fine to sell their seat.

當然，如果你確定某個人百分百不會出現，賣了他們的座位是沒問題的。

But what if you're only 95% sure?

但如果你只有 95% 的確定性呢？

75%?

或 75%？

Is there a number that separates being unethical from being practical?

是否有一個數值能區分不道德做法以及切合實際做法？