Yep. This whole thing is very closely related to that game. But embedded in that game is almost always a rule where the "return" in terms of customer value relative on "investment" in terms of effort/discounts/perks is better for new customers relative to old ones.
Generally, it comes down to a choice between losing/spending money on churners or losing/spending it on advertising. I'd wager some are and others aren't doing their long term math right, but regardless that dynamic emerges, and it recurs.
Typically it emerges where "customer acquisition" is hugely expensive or important and this typically emerges when either (1) it is a winner-take-most-game or (2) the product is an ironic "commodity:" A product that is 90% standardized under the hood but has a sales oriented differentiation layer. EG, the underlying product is electricity, cable TV or money market stuff like insurance or loans. The company you deal with as a customer is a customer services, sales & marketing layer. Guess how they compete.
Hence the whole cat-and-mouse churning game.