Are we decentralized yet?

TIL about the Herfindahl–Hirschman Index and I wanted to test it with a weird corner-case that I remember.

At one point in the late 1980's Microsoft had a GREATER than 100% market share of the Macintosh spreadsheet market.

How is this possible?

Market share (for a given period) is the participant's sales in the market divided by total sales. It just so happened that Lotus had more returns than sales of their failed spreadsheet, Lotus Jazz. So Lotus, had a negative market share and Microsoft had more sales of Excel than total sales in the market, resulting in a greater than 100% market share.

I don't remember the exact numbers and I believe there was at least one other competitor in the study. But let's just say the numbers were:

Microsoft: 102% Lotus: -2%

In that case the Herfindahl–Hirschman Index would be 102^2 + (-2)^2 = 10404 + 4 = 10408.

So, in this pathological case it is possible for the HHI to exceed 10,000.

Edited: Added (for a given period) above, for clarity.

> Market share (for a given period)

That's not a thing. Market share has no time period. It's an instantaneous measure of the state of things right now. How many of your company's units are out in the wild right now being used, vs. how many total units (sold by any company) are out there in the wild right now being used.

That number can and will be different at different points of time, as people buy and return your products, and buy and return your competitors' products[0]. You can certainly say that your market share increased by 300% or by -40% over a given period of time, but your actual market share is always a number between 0% and 100%.

> Market share (for a given period) is the participant's sales in the market divided by total sales.

No, that's a company's share of sales as compared to the industry/product category as a whole. Not market share.

[0] You also should take into account people who throw your product in the trash (or for software, delete it) without returning it. Depending on context, you might even want to take into account people who put your product in a box in their basement and never use it again. Assuming you could actually divine those numbers, which of course you likely can't.