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Mind Tools: Applications and Solutions 
Video
Games:
Reason
and Revenue in a Blockbuster Promotion
Part 1
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Blockbuster Video stores in my area ran an earlyreturn promotion on selected New Releases: Return designated titles before 8:00 P.M. on their due date, they advertised, and get a dollar off the rental of an Old Release.
The promotion was a mechanism for altering the return curve of popular titles. Ordinarily the return curve for videos is out of phase with the demand curve. The return rate rises just before closing, when customer demand is waning; the demand rate peaks earlier. As a result, popular titles can be out of stock during the highdemand period, but back in stock during the subsequent, lowerdemand period.
If Blockbuster could redistribute the returns to approximate the demand curve, more of these titles would rent again (i.e., turn over) the same night, which would have a positive effect on revenue. On the other hand, the earlyreturn discount necessary to stimulate the turnover would have a negative effect.
Was this promotion likely to increase revenue? Even with the promotion up and running, it's hard to judge its impact. Many variables randomly affect the periodtoperiod change in a store's revenue—variables like the weather, the competing entertainment options, and the popularity of new titles entering the store's inventory. The indeterminate influence of such things obscures the promotion's net effect.
But there is a way to evaluate the promotion: Examine it in the abstract to find its underlying mathematical structure; then use that structure to predict the promotion's influence on revenue.
Here's how we'll proceed. We'll create a simplified situation, ignoring all variables unaffected by the promotion's presence or absence. In our simplified situation, we'll require a store to collect the same revenue with the promotion as it collects without it. When that happens, what are the controlling variables and what values do they take on?
The two controlling variables are additional turnovers and exercised discounts. Let's define these terms.
Additional turnovers. With a promotion in effect, more highdemand titles will turn over again on the day of their return. Subtract the number of highdemand turnovers when there isn't a promotion from the number of highdemand turnovers when there is. The difference is the additional turnovers.
turnovers
with promotion  turnovers without promotion
= addiitional turnoovers
Exercised discounts. With the promotion in effect, all customers who return their promoted titles early will receive a dollar off the rental price of an Old Release. Their use of that discount is an exercised discount.
Unearned discounts. Some customers return their videos before 8:00 P.M.—with or without the promotion—and their returns always turn over again on the due date. These are regular early returns: They don't generate new turnovers (and new revenue) because they don't increase the store's inventory of available cassettes. Nonetheless, they too are awarded a discount. Their discount is unearned—a free ride.
Earned discounts. Other customers return their videos before 8:00 P.M. because the promotion induces them to. These are additional early returns: They generate new turnovers (and new revenue) because they increase the store's inventory of available cassettes, thereby supplying previously unmet demand. Their discount is earned.
For the promotion to succeed, the store's revenue from the additional turnovers must be sufficient to offset both the unearned and the earned discounts.
What we need to know is this: What is the breakeven ratio, the ratio of additional turnovers to exercised discounts at which a store's revenue with the promotion will equal its revenue without the promotion? With this critical information, we can assess the likelihood that a store can exceed the ratio's value and increase its revenue.
The required ratio of additional turnovers to exercised discounts is equal to a fractional expression that relates the fee changes in two subsets of renters. Subset A, represented in the faction's numerator, generates a lower average fee during the promotion. Subset B, represented in the faction's numerator, generates a higher average fee.
Renters in Subset A. Subset A consists of all patrons who, during the promotion, rent a highdemand title on Day 1 and return it early on Day 2.
After the patrons in Subset A return a highdemand title, they'll rent another video—their Next Rental. (They may rent their Next Rental immediately or later.) What their Next Rental will be depends on the absence or presence of the promotion.
If there is no promotion, the Next Rental will be either an Old Release (which rented for $2 when I encountered the promotion) or a New Release (which rented for $3). The collective choices of all the patrons in Subset A will result in some average fee for the Next Rental—a fee between two and three dollars.
If there is a promotion, the Next Rental will be a Discounted Old Release—generating a fee of one dollar. The promotional discount reduces A's average Next Rental fee from what it otherwise would have been.
Subset A's average fee decrease per rental is the difference between an average Next Rental fee and a discounted Next Rental fee. It can be as little as one dollar—the $2 Old Release fee minus the $1 discount. Or it can be as much as two dollars—the $3 New Release fee minus the $1 discount.
The numerator of the fraction is, then, the average fee decrease on Subset A's Next Rental as A exercises the promotional discount.
average
Next Rental fee  discounted Next Rental fee =
average fee decrease in Subset A
Renters in Subset B. Subset B consists of all patrons who on Day 2 want to rent a New Release and who do find an acceptable New Release in stock if there is a promotion, but don't find an acceptable New Release in stock if there isn't a promotion.
When there is no promotion, there will be an inadequate supply of New Releases. Subset B renters, finding no acceptable New Release in stock, will have two options to fall back on:
• trade down to an Old Release (which rents for $2),
• forgo the rental altogether (and pay no rental fee).
On the other hand, when there is a promotion, the supply of New Releases will increase (as videos that previously were returned too late to rerent now become available earlier in the evening). Subset B patrons, now finding acceptable New Releases in stock, will rent them. These additional turnovers will increase the average fee that Subset B patrons generate. How much?
The average fee increase per rental is the difference between a New Release Rental and an average Fallback Rental. It can be as little as one dollar—the $3 New Release minus a $2 trade down Old Release. Or it can be as much as three dollars—the $3 New Release minus a $0 forgone rental.
The denominator is, then, the average fee increase on Subset B's rental as B trades up to a New Release.
New
Release fee  average Fallback fee =
average fee increase in Subset B
Summary of the breakeven ratio. To recap, the fractional expression is the average fee decrease per rental in Subset A divided by the average fee increase per rental in Subset B. And the value of this expression is the required ratio of additional turnovers to exercised discounts—the value at which the revenue with the promotion is the same as the revenue without the promotion.
(average
Next Rental fee)  (discounted Next Rental fee)
(New Release fee)  (average Fallback Rental fee)
=
average
fee decrease in Subset A
average fee increase in Subset B
=
additional
turnovers
exercised discounts
=
the breakeven ratio
The Breakeven Ratio's Range of Values
Rental fees. A store's rental fees determine the breakeven ratio's range of values. During this promotion, Blockbuster was renting New Releases for $3, Old Releases for $2, and Discounted Releases for $1.
We'll use these prices to find the range's highest and lowest values. We'll also calculate two intermediate values.
Average rental fees. We start by assuming that the average fee for a Next Rental is either the maximum, $3 (i.e., everybody's Next Rental is a New Release), or the minimum, $2 (i.e., everybody's Next Rental is an Old Release). Likewise, we assume that the average fee for a Fallback Rental is either the maximum, $2 (i.e., everybody's Fallback Rental is an Old Release), or the minimum, $0 (i.e., everybody's Fallback Rental is No Rental at all).
Fee combinations. The various pairings of these maximums and minimums give us four combinations:
1. A $3 Next Rental for Subset A with a $2 Fallback Rental for Subset B.
2. A $2 Next Rental for Subset A with a $2 Fallback Rental for Subset B.
3. A $3 Next Rental for Subset A with a $0 Fallback Rental for Subset B.
4. A $2 Next Rental for Subset A with a $0 Fallback Rental for Subset B.
We insert a combination of values into the fractional expression
(average
Next Rental fee)  (discounted Next Rental fee)
(New Release fee)  (average Fallback Rental fee)
putting the fees from Subset A in the numerator and the fees from Subset B in the denominator. Then we do the calculation (keeping in mind that a Discounted Next Rental always equals $1 and a New Rental always equals $3). There is a separate calculation for each combination. See Table 1.
Table 1. Breakeven Ratio for Maximum and Minimum Fee Combinations
Subset A Renters 
Subset B Renters 

Average 
minus 
Discounted Next Rental Fee 
divided 
New 
minus 
Fallback 
equals 
Breakeven 
($3 
 
$1) 
/ 
($3 
 
$2) 
= 
2/1 
($2 
 
$1) 
/ 
($3 
 
$2) 
= 
1/1 
($3 
 
$1) 
/ 
($3 
 
$0) 
= 
2/3 
($2 
 
$1) 
/ 
($3 
 
$0) 
= 
1/3 
The range of values. Each calculation gives the breakeven ratio for one pairing of Next and Fallback Rental fees. The breakeven ratio is the factor by which a store's turnovers must increase for it to recoup the revenue it will lose on the discounted rental. Our results show that the ratio's range extends from a high of 2/1 to a low of 1/3.
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