How to Calculate Diminishing Returns

I had a fun debate with my kids recently about the Mega Millions lottery and at what point you hit diminishing returns on purchasing lottery tickets. Lottery tickets are inherently a bad mathematical bet; the ROI of many games is somewhere around -90%. For every $1 you spend on the lottery, you lose $0.90 of it.

That said, there is a very tiny case to be made that the purchase of a few tickets for an especially large prize is relatively harmless fun, as long as you don't need the money for something else. Arguably, if you spent $5 on the lottery versus $5 on candy, the lottery would technically be the healthier choice. While the odds of you winning the lottery are astronomically small, the odds of you winning with no tickets purchased is definitively zero.

How small? For the current iteration of the Mega Millions lottery, the probability of choosing the winning combination of numbers is 1 chance in 302,575,350. For context, the probability of you being hit by lightning twice in your lifetime is 1 chance in 100,000,000, so you're three times as likely to be struck by lightning twice as you are to win this particular lottery.

Calculating Diminshing Returns Exercise

With the understanding that your chances of winning are incredibly small comes the realization that increased numbers of tickets do technically cut your odds. But at what point is buying more tickets an exercise in futility? This is an exercise in calculating diminishing returns, and for this, we'll need to dust off our knowledge of derivatives. Let's start with the basics and lay out the odds.

Example Diminishing Returns Table

We see our total odds of winning shown in column B in the chart above, and the number of tickets purchased in column A. Column C, outlined in red, shows what the odds are for any given number of ticket purchases.

1 ticket purchased is 1 chance in 302,575,350.

2 tickets purchased is 1 chance in 151,287,675. That's a significant change between the two, cutting the odds in half. That's what started this whole line of questioning from my kids - if 2 tickets cuts your odds in half, more is better, right?

To understand how quickly that becomes futile, column D, highlighted in yellow, is the difference between the previous rows in column C, straight division of the current cell by the previous cell in column C. We see that when you buy two tickets, you cut the odds by 50%. When you buy three tickets, you cut the overall odds by 66.67% - you started with 1 chance in 302,575,350, and now you're at 1 chance in 100,858,450. You're still more likely to be struck by lightning twice in your life, but the odds are better. We can visualize that trend with the blue bar chart.

To understand what's happening, we look at the difference in change rate, shown in green in column E. This is a (new - old) / old calculation. Cutting our odds by 50% for 2 tickets but 66% for 3 tickets represents a 33% change in improving our odds. When we buy 4 tickets, that percentage change drops to 13%. When we buy 5 tickets, that drops again to 7%.

Pay attention to what happens after the 15th ticket. Our percentage change in improving our odds is effectively zero (it's a rounding error at that point). At that point, every additional ticket purchase doesn't really change our odds at all. So mathematically for this particular lottery, there would be no sense in buying any more than 16 tickets ever. We see this reflected in the blue bar chart - it basically levels off after the 16th ticket.

Incidentially, these are still astronomically high odds. 1 chance in 18,910,959 (the odds at 16 tickets) is still 10 times higher than the probability of you being struck and killed by lightning, which is estimated at 1 in 180,746 annually. So you're 10 times more likely to die being struck by lightning on the way to purchase your 16 lottery tickets than you are to actually win the lottery.

Practical Applications of Calculating Diminishing Returns

Now, this exercise is a fun way to teach kids about probability and diminshing returns, and emphasize that the math is firmly, firmly against you. Is there a chance you'll win? Yes. Is it a reasonable chance? Absolutely not. But how else might we use this mathematical exploration?

Diminishing returns is something overlooked when we spend money on anything. In business and marketing, some of us operate on the same naive principle that throwing more money at a problem will solve it, without looking for diminishing returns. If you were to assemble your data and examine it, you might find the same trend - at a certain point, adding more money just doesn't yield the same returns. They diminish, and your costs go up. You might have seen massive growth in a tactic early on, but as you reached more and more of that audience, your costs go up disproportionate to the results you're getting.

Here's an example from a recent campaign:

Diminishing returns in an ads campaign

I compared the rate of change in spend versus the rate of change in conversions; after about the 15th day of this campaign, the rate of change in conversions no longer outpaced the rate of change in spend. At that point, it was a linear trend - put $X in, get Y conversions. That worked for this campaign. If we had hit serious diminishing returns, the blue bars would go below the 0 line, where the rate of change of spending was increasing faster than the rate of change of conversions - and it's at that point we'd need to hit the brakes.

What's important about these calculations is that absolutely none of the advertising services provide this analysis to you. Not a single one will ever say, "hey, it looks like your performance is experiencing diminishing returns." Why would they? Once you know you're hitting diminishing returns, the logical thing to do is to stop spending as much money. Thus, you must perform these calculations on your own.

There are mathematical techniques and software that can do these sorts of things; to get into the hardcore stuff will require you to dust off your calculus books and refresh your skills on computing derivatives (or have beers with a colleague who remembers it better), but the benefit of those skills is saving yourself substantial amounts of money and headaches.


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