Dynamic personalized pricing is a real thing. Has been for ages. The old-fashioned techniques are coupons and loyalty cards, or just having higher prices in higher-end stores. Competition isn't nearly as perfect as you say. It's very common for high-end stores to sell identical items at higher prices and still sell plenty of them.

These days you can do a much better job if you have data about your prospective customer. This is not a hypothetical. For example, Target was found to charge higher prices in their app if your location was close to one of their stores. Orbits and Delta have both been found to offer personalized prices as well.

https://retailwire.com/discussion/will-targets-dynamic-prici...

https://www.fastsimon.com/ecommerce-wiki/personalization/dyn...

https://www.pbs.org/newshour/economy/personalized-pricing-ha...

Price fixing where everybody uses the same software is a real thing. RealPage recently settled a lawsuit over this.

You seem to be taking a very Libertarian approach where you assume economics 101 wins out over anything more complex, but if you look at what's actually going on in the world this is not the case.