In 1931, American economist Harold Hotelling published the seminal paper The Economics of Exhaustible Resources. Harold described a problem many firms face: how much of a non-renewable resource should they sell at any given time? This problem is more apparent when thinking about managing an oil supply but just as relevant when considering how to manage match-3 levels.
The oil firm needs to compare the rate at which the oil price increases against the interest rate. The more the interest rate rises, the more sense it makes to sell oil now and invest the money rather than waiting for the price of the oil to increase. As such, the cost of oil is strongly correlated with the interest rate increase over time.
The tools of the field help us solve tricky now or later dilemmas. Hotelling’s approach and intuition helped form what is called dynamic programming economics. More wrinkles have been added to the model to solve everything from cake-eating to managing how many fish should be caught.
Match-3 Player Level Management
Whereas increasing difficulty in delaying this from occurring has its own cost.
The solution to this system of equations is what we might call the maximum sustainable difficulty. And the intuition is similar to what was first developed for the oil firm.
A bunch more stuff falls out of the model. For instance, King should significantly increase the difficulty of levels just near the exhaustion point, just as the oil firm would dramatically increase the price of the last bits of oil. There’s a minimal downside to doing so, as the player will be in a high-probability space soon enough. Furthermore, since King is always stockpiling levels, new players face an exhaustion point further away than players who downloaded the game at the first launch. It’s in King’s interest to ease up on the difficulty for early levels since the area underneath the churn curve expands as the exhaustion point moves rightward.
There’s much more room for expansion of the model, and I encourage data scientists to play a vital role in systems design.