Posted April 6, 2010 By Presh Talwalkar. Read about me , or email me .



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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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I’m not a fan of gift cards, but a recent promotion got my attention. A store was giving a bonus for gift card purchases.

It worked like this: if you purchased a $50 gift card, then you would get a $10 gift card for free. I thought the 20 percent bonus made it a decent deal.

As I was mulling the offer, I got to thinking and a curious idea struck me. I thought: the 20 percent bonus is nice, but what would happen if I repeated the process?

What would happen if I used the bonus gift card towards another gift card purchase and redeemed another free bonus? Would the deal be any sweeter?

I set to calculation and confirmed the potential for gaming. Repeating the promotion does in fact increase the bonus!

To see why, let us go through some mundane calculation. The first time you do the offer, you end up with $60 in gift cards for spending $50 in cash, which translates to a 20 percent bonus (60/50 – 1 = 20 percent).

The second time you do the offer you have a leg up. You can use the free $10 gift card bonus from the first round towards the next purchase. This means you only need to spend an extra $40 in cash to qualify for the promotion and you end up with another $60 in gift cards.

All in all, you have netted a total of $110 in gift cards ($60 first round, $50 net from second since you spend the 1st round’s free $10 bonus) while spending only $90 in cash ($50 first round, $40 second). The bonus is therefore 22.2 percent (110/90 – 1) — just a little bit higher than before!

I was excited to see a sort of multiplier effect when repeating the promotion. So it got me thinking about several questions:

How big can the bonus get if repeated infinitely?

What is a general formula for other gift card bonus structures?

Does the formula depend on the scale of the gift card or only the percent bonus (i.e. 20 percent bonus versus $10 bonus from $50 versus $20 bonus from $100)

How do the bonus offers compare? Does a 20 percent bonus scale any differently than a 10 percent bonus?

This post is a bit of mathematical excursion, but I hope at least a few of you will enjoy this!

How big can the bonus get if repeated infinitely?

It was fun to see the gift card bonus actually increased when the promotion was iterated.

How does the bonus grow when repeated even more?

It is useful to calculate a few terms to see the general formula. The first time you have to spend $50 to get $60 in gift cards. But the second time you only have to spend an extra $40 in cash, plus the free $10 bonus from the first round, to end up with an additional $60 in gift cards. You end up with $110 in gift cards for spending $90 in cash.

The third time is much like the second. You again have to spend an extra $40 in cash, plus the free $10 bonus from the second round, to end up with an additional $60 in gift cards. All in all, you end up with $160 in gift cards for spending $130 in cash. This is a bonus percentage of just over 23 percent.

Here is a table with a few more of the terms:

We can see the bonus is approaching something like 25 percent. But what is it exactly?

To figure that out, we need to deduce the general formula for the gift card bonus, which is the ratio of the gift cards you get divided by the cash you spend.

Think about the general process to deduce the formula. What is happening each round is that you have to spend an extra $40 in cash to net an extra $50 in gift cards (it’s net since you get $60 minus the free $10 gift card you have to use). The general bonus can therefore be written as:

Now we can see what happens in the infinite repetition.

We can prove the limiting discount is 25 percent:

Pretty cool, but how does this work in general?

What is a general formula for other gift card bonus structures?

To write a general gift card formula, we will need a little bit of extra notation.

Let’s say the free gift card bonus is F and the gift card purchase amount is G. Now we will rephrase the logic we went through symbolically. Remember that in the above example:

–in the first round, you end up with $60 but you have to spend $50 in cash

–in the second and ongoing rounds, you end up with a net $50 for spending an extra $40 in cash

To rewrite that using variable notation:

–in the first round, you end up with F + G but you have to spend G in cash

–in the second and ongoing rounds, you end up with a net G for spending an extra G – F in cash

I will skip some of the trivial algebra and simply remark the general formula takes the expression:

Does the formula depend on the scale of the gift card or only the percent bonus (i.e. 20 percent bonus versus $10 bonus from $50 versus $20 bonus from $100)

There is an interesting question here about whether it’s the percent bonus (20 percent) or the exact terms that matter.

So let’s calculate an example to get an idea. Instead of giving a $10 bonus on $50, imagine the promotion was $20 on $100. How would the bonus change?

We can use the general gift card formula to see. In the second iteration, we can see that you get $220 in gift cards for spending $180 in cash. This translates to a 22.2 percent bonus–exactly the same as before with the $10 bonus from $50!!

You can verify with a few more terms to see that it must be the percentage and not the specific terms.

How can we prove this?

Let us define a new variable as the percent bonus. In symbolic terms, P = F/G.

Now, in the general formula, let’s divide each term by G and see what happens:

As you can see, the terms F and G “disappear,” and the resulting formula depends only on P and n— that is, we have proven it is the percentage that matters and not the scale of the promotion.

How do the bonus offers compare? Does a 20 percent bonus scale any differently than a 10 percent bonus?

We can take the limit to see how bonus formulas scale in general:

The resulting formula shows that the gaming effect is larger for bigger initial bonus.

An example is that a 10 percent initial bonus only grows to a meager 11 percent bonus, but a 20 percent to a 25 percent bonus, and a 50 percent bonus translates into a whopping 100 percent bonus!

A closing note



Obviously it would be impractical to buy so many gift cards. Luckily the bonus converges decently quickly, so this could be a strategy for reselling gift cards or for giving gifts to friends at a decent discount.

In writing this post, I also learned some companies restrict using gift cards to buy gift cards. Be sure to check if you want to try this out!