A few days ago at work, I found myself in one of those situations where numbers behave in a counterintuitive way. A situation where an apparently logical reasoning leads to incorrect conclusions—what I call the cake paradox.
I brought up cakes because I used them to explain (with difficulty) to my colleagues where the trick was.
Let’s go over everything with a similar example (made-up data for illustrative purposes):
- “Leggo” is an Italian publishing company active in two markets: Italy and China.
- In Italy, “Leggo” is the market leader, with a 50% market share in 2015 (meaning, for every 100 books sold, 50 are published by “Leggo”).
- In China, “Leggo” has a 5% market share in 2015.
- Given the size differences of the markets, “Leggo” has a total market share of 9.1% (aggregating data from Italy and China, with calculations available in the table below).
- In 2016, “Leggo” increases its sales and manages to steal market share from its competitors in both Italy and China.
- In Italy, it goes from 50% to 52% market share.
- In China, it goes from 5% to 6%.
Even though “Leggo” performed better than its competitors in both markets, the overall market share (China + Italy) decreased from 9.1% to 8.8%.
Does something not add up?
How is it possible that even though “Leggo” is doing very well, increasing its sales in both markets, stealing market share both in Italy and China, its total market share is decreasing?
Try it for yourself; all the calculations are in the table:
| Italy | China | Total | |
| 2015 Book market | 100 m | 1,000 m | 1,100 m |
| 2015 “Leggo” sales | 50 m | 50 m | 100 m |
| 2015 Market share | 50% | 5% | 9.1% |
| 2016 Book market | 102 m | 1,600 m | 1,703 m |
| 2016 market growth | 2% | 60% | 55% |
| 2016 “Leggo” sales | 53 m | 96 m | 149 m |
| 2016 market share | 52% | 6.0% | 8.8% |
The paradox is explained by the two different markets: Italy and China. Italy is much smaller than China, and China is growing at a much higher rate than Italy.
Forget the example of books with realistic numbers; let’s abstract the concept by thinking of the cakes mentioned earlier.
We have two cakes of similar size, one white and one black. Initially, we own 80% of the white cake and only 20% of the black cake.
In total, we own the equivalent of one entire cake (meaning our market share will be 50%, one cake out of two).
Suddenly, the black cake rises and becomes 100 times the size of the white cake.
Our total market share drastically drops from the previous 50% to a number close to 20%, the market share of the huge cake (exactly 20.6%).
Now, it should be easier to understand how our market share, which was previously an average of 20% and 80% (since the cake were the same size), will now be much closer to 20% of the black cake, which is 100 times larger than the white cake.
In practice, our dominant position in the white cake, which used to balance out the black cake (since they were of similar size), now barely matters in a situation where the black cake represents almost the entire market.
The Cake Paradox shows us how it can be detrimental for a company to hold a leadership position in a slow-growth market. Even if it outperforms its competitors in emerging markets, it can still lose market share overall!