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In the previous video, we
used linear optimization

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to allocate ads.

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In this video, we're going
to use a greedy approach

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to allocate ads to queries.

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The approach is called greedy
because we allocate the ads

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sequentially and
we prioritize which

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combinations of ad
and query to use,

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based on the average
price per display.

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So here we have a
spreadsheet set up

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in the same way as the one
from the previous video

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with some minor tweaks.

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If we scroll down, we
have this table here,

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which has cells corresponding
to combinations of advertisers

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and queries, which tells us
basically how many times we

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display a particular
advertiser's

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ad with a particular query.

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But to the side here,
we have another table

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which we will use to
keep track at every stage

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of our allocation process,
how much we can allocate--

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how many displays we can
perform of a particular ad

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with a particular query,
based on the budget,

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and based on how many
displays of each query

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remain unallocated.

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Further down, where we
have our constraints,

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we've added some new
cells here, which

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are labeled with the
word "Remaining,"

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to indicate how much of the
budgets of each advertiser

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remaining, and how many
displays of each query

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remain unallocated.

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So let's get started with
our greedy allocation.

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So to start, we go up to our
Average Price Per Display

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table, and we find
the combination

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of advertiser and
query which gives us

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the largest average
price per display.

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So, in this case, the largest
average price per display

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is five.

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And this happens when we display
Verizon's ad with Query 3.

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So this is the first
variable that we're

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going to change in
our greedy allocation.

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These are the first,
basically the first displays

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that we're going to use.

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So if we scroll down, we
add Verizon in Query 3.

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And to figure out
the budget limit,

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we go to Verizon's remaining
budget which is $160.

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And so we take that
$160, and we divide it

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through by the average
price per display of $5.

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So, we get this number
32, and so what this means

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is that we can display
Verizon's ad with Query 3,

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32 times based solely on the
remaining budget of Verizon.

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Now, of course, we also have
to respect how many displays

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are unallocated of each
query, and for query 3,

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we have 80 displays.

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So the query limit
at this stage is 80.

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So now how many
times do we actually

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display Verizon's
ad with Query 3?

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Well, the most that we can
display Verizon's ad with Query

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3 is going to be the smaller
of these two numbers.

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And so in this case the smaller
of the two numbers is 32.

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So, we allocate Verizon's
ad to Query 3, 32 times.

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So now after we've changed
the value of this variable,

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you can see that the budget
of Verizon has changed.

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So now it is 0.

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So we've completely
extinguished Verizon's budget,

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and the remaining unallocated
displays for Query 3

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has also changed.

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So this number used to
be 80 and now it is 48.

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So, since we've extinguished
Verizon's budget,

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we go back up to our Average
Price Per Display table.

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And so now we have to
select a new combination

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to use in our allocation.

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Since we've used up
Verizon's budget,

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we're going to highlight these
cells corresponding to Verizon.

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We're going to
highlight them in red

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to indicate that we can't
use any of those combinations

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anymore.

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And from the remaining
cells in the table,

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we want to find the combination
of Advertiser and Query

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that gives us the largest
average price per display.

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So in this case,
the next highest

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is for T-Mobile and
Query 3, and in this case

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the average price
per display is two.

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So this is the next
combination that we'll

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use in our allocation.

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So if we scroll down, we
add T-Mobile and Query 3.

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So now we have to
calculate the budget limit.

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T-Mobile's remaining
budget is $100.

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We take $100, and we divide
it through by $2 per display,

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and we get this value of 50.

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And the query limit for
T-Mobile and query 3

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is the remaining number
of displays for Query 3,

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which is 48.

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So we scroll up and we add 48.

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And so now, how many
times do we actually

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display T-Mobile's
ad with Query 3?

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Well, that number
is 48, because 48

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is the smaller of
the two numbers.

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So we go ahead and we add that.

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And so now T-Mobile's
budget has changed.

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So it's dropped
from 100 to four,

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and the remaining
displays of Query 3,

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that number has dropped to 0.

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So there are no more
displays of Query 3

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remaining that we can use.

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So now we move on to the
next stage of our allocation,

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and we go back to the Average
Price Per Display table.

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And since we've used up all
of the displays of Query 3,

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we're going to highlight the
remaining cells corresponding

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to Query 3 in red,
just to remind us

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that we can't use them.

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And so now from the
remaining cells,

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again, we want to find
the highest average price

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per display combination.

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And so from the remaining
cells, the highest average price

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per display occurs when we use
T-Mobile's ad with Query 1.

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So that's the next variable that
we'll use in our allocation.

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So we scroll back down and
we add T-Mobile with Query 1.

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To compute the budget limit,
we take T-Mobile's remaining

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budget which is four,
$4, and we divide it

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through by the average
price per display of $1.

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So, we get four displays
according to the budget.

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And the query
limit, in this case,

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is 140, because there are
140 displays of Query 1

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that haven't been
used towards any ad.

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Now how many times
do we actually

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display T-Mobile's with Query 1?

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Again, we just take the minimum
of these two quantities, which

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is four.

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We go ahead and we enter
that into our table.

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So now if you look
at the budgets,

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we've completely extinguished
T-Mobile's budget

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and the remaining number
of displays of Query 1

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that haven't been allocated
has dropped to 136.

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So now we move on to the
next stage of our allocation.

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Again, we've eliminated
T-Mobile's budget.

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So now we highlight
those cells in red.

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And so now, we want to pick
the highest average price

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per display from
the remaining cells.

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Now in this case, the only two
combinations that remain to us

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are to display AT&T's
ad with Query 1

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and to display AT&T's
ad with Query 2.

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Both of these have the same
average price per display,

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so it doesn't matter which
one we really choose.

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But for the purpose
of this solution,

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let's just go with AT&T
displayed with Query 1.

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So the average price
per display is 0.5.

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So we go down, we add that
entry to our side table here.

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And to compute the
budget limit, we

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take AT&T's remaining
budget of 170,

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and we divide it through by 0.5.

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To get the query
limit, we just look

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at the number of remaining
displays of Query 1,

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which is 136.

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So we add that.

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And so now, obviously, the
smaller of the two quantities

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is 136.

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So we'll display AT&T's ad
with Query 1, 136 times.

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So now if we look at the
state of our allocation,

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we've basically used up all
the displays of Query 1.

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And we've used up some
part of AT&T's budget,

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though there's still a lot left.

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So now we go back up
and we now proceed

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to make the next allocation
in our greedy solution.

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We've eliminated
AT&T in Query 1.

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So now the only combination that
remains is AT&T with Query 2.

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So we go down.

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We add that entry
to our side table.

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And so now to get
the budget limit,

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we take AT&T's remaining
budget of $102,

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and we divide it
through by 0.5, which

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is the average price
per display to get 204.

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And the query limit
now is 80, because 80

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is the number of remaining
unassigned displays of Query 2.

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And now the smaller of
the two numbers is 80.

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And so we add 80 to our table.

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And so at this point, we've
gone through all the entries

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in our average price
per display table,

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and we've eliminated
basically all of them.

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And in terms of
the allocation, we

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know that we can't make any more
allocations, because if we look

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at our constraints, basically,
there are no more displays

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of any query that have not been
assigned to any advertiser.

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So all of these
entries here are 0.

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And in this case, AT&T's budget
is still some positive value.

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So we still have $62 of
ATamp;T's budget remaining.

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But since we don't have any
more displays of any query

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that we can use, we
are basically done.

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So this is our greedy solution.

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And so there are a couple
of interesting things

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to note about this
greedy solution.

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So the first is that the
actual combinations that

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are used by the
greedy solution are

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different from
those that are made

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by the linear optimization
based solution.

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So for example, in the
optimization solution,

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if you recall, we only
used Query 3 with AT&T.

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So we only displayed-- whenever
we displayed an ad with Query

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3, it was only ATamp;T's ad.

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But in this case,
we don't actually

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display AT&T's ad
with Query 3 ever.

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And in fact, we display only
T-Mobile's ad and Verizon's ad

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with Query 3.

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We don't use AT&T. So that's--
so the actual allocation

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changes and so as a result,
the revenue that we get from

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the greedy solution is different
from the revenue that we get

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from the optimization solution.

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If you recall, the revenue
from the optimization solution

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was $428.

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Now, this may not seem like
a very large difference.

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So this is a difference of $60.

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But in relative terms, this is
actually a rather large amount,

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368 relative to 428
is roughly 14 percent.

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And so, hopefully this
illustrates the fact

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that an optimization
based solution

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can provide a significant
difference in performance

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relative to a basic
common sense solution.

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So this concludes our
construction and our discussion

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of the greedy solution.

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In the next video, we will go
back to our linear optimization

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model, and we will consider
some of the sensitivity analysis

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that goes along with that model.