I'm only familiar with PS4, but it's much, much quicker than that today.
Battlefield V [1] was released Nov 20 2018 for $60, in Dec 6 it was $36 (60% of original in 16 days), and in Dec 18 it was $30 (50% in 28 days) with PS Plus, which you need anyway to play multiplayer.
God of War [2], to take another example not plagued by BFV's troubled release and also being a single-player game, it took at bit longer to go down. It was released in April 20 2018, was down to $50 (83%) in June 8, $34 (58%) with PS+ in Aug 17, and $22 (37%) in Nov 16.
You really pay a premium to play at launch day or a couple of days early by pre-ordering. If you wait a month you usually have the first discount.
There's a nice community at Reddit called Patient Gamers [3]. You don't have to take vow to never buy new games, but people there usually wait at least six months before buying a game. Three super nice benefits of that:
1. Games are cheaper.
2. The game has been patched to fix bugs or add more content.
3. You can read reviews or check the Metacritic page to see if the game is actually as good as claimed or if it's hype based on the trailer. You can watch some real gameplay not carefully produced by the designers.
Still, a lot of people apparently pre-order games or buy mega-deluxe editions for $100, so game companies can get money from that group and then lower the prize a month later for the rest of the gamers.
[1] https://psprices.com/region-us/game/2347184/battlefield-v