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Have you ever sat in a doctor's
office for hours
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despite having an appointment
at a specific time?
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Has a hotel turned down
your reservation because it's full?
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Or have you been bumped off a flight
that you paid for?
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These are all symptoms of overbooking,
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a practice where businesses
and institutions
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sell or book more
than their full capacity.
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While often infuriating for the customer,
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overbooking happens because
it increases profits
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while also letting businesses
optimize their resources.
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They know that not everyone
will show up to their appointments,
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reservations,
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and flights,
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so they make more available
than they actually have to offer.
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Airlines are the classical example,
partially because it happens so often.
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About 50,000 people get bumped
off their flights each year.
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That figure comes at little surprise
to the airlines themselves,
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which use statistics to determine
exactly how many tickets to sell.
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It's a delicate operation.
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Sell too few, and they're wasting seats.
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Sell too many, and they pay penalties -
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money, free flights, hotel stays,
and annoyed customers.
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So here's a simplified version
of how their calculations work.
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Airlines have collected years worth
of information
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about who does and doesn't show up
for certain flights.
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They know, for example,
that on a particular route,
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the probability that each individual
customer will show up on time is 90%.
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For the sake of simplicity,
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we'll assume that every customer
is traveling individually
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rather than as families or groups.
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Then, if there are 180 seats on the plane
and they sell 180 tickets,
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the most likely result is that 162
passengers will board.
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But of course, you could also
end up with more passengers,
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or fewer.
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The probability for each value
is given by what's called
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a binomial distribution,
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which peaks at the most likely outcome.
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Now let's look at the revenue.
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The airline makes money from each
ticket buyer
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and loses money for each person
who gets bumped.
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Let's say a ticket costs $250
and isn't exchangeable for a later flight.
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And the cost of bumping
a passenger is $800.
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These numbers are just for the sake
of example.
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Actual amounts vary considerably.
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So here, if you don't sell
any extra tickets, you make $45,000.
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If you sell 15 extras
and at least 15 people are no shows,
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you make $48,750.
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That's the best case.
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In the worst case, everyone shows up.
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15 unlucky passengers get bumped,
and the revenue will only be $36,750,
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even less than if you only sold 180
tickets in the first place.
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But what matters isn't just how
good or bad a scenario is financially,
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but how likely it is to happen.
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So how likely is each scenario?
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We can find out by using
the binomial distribution.
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In this example, the probability
of exactly 195 passengers boarding
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is almost 0%.
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The probability of exactly 184 passengers
boarding is 1.11%, and so on.
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Multiply these probabilities
by the revenue for each case,
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add them all up
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and subtract the sum from the earnings
by 195 sold tickets,
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and you get the expected revenue
for selling 195 tickets.
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By repeating this calculation
for various numbers of extra tickets,
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the airline can find the one likely
to yield the highest revenue.
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In this example, that's 198 tickets,
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from which the airline will probably
make $48,774,
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almost 4,000 more than without
overbooking.
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And that's just for one flight.
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Multiply that by a million flights
per airline per year,
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and overbooking adds up fast.
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Of course, the actual calculation
is much more complicated.
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Airlines apply many factors
to create even more accurate models.
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But should they?
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Some argue that overbooking is unethical.
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You're charing two people
for the same resource.
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Of course, if you're 100% sure someone
won't show up,
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it's fine to sell their seat.
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But what if you're only 95% sure?
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75%?
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Is there a number that separates being
unethical from being practical?
Tomoyuki Suzuki
00:03:28,738 I think 'revenue' should be replaced by 'loss'.