What the law prohibits
Competition law does not prohibit high prices. This is the single most important thing to understand about the legal position, and it surprises most people.
The prohibition targets agreement. Section 1 of the US Sherman Act reaches contracts, combinations, and conspiracies in restraint of trade. Article 101 of the Treaty on the Functioning of the European Union reaches agreements between undertakings and concerted practices. Both require a meeting of minds between competitors.
Charging a high price alone is legal. Charging the same price as a competitor is legal. Watching a rival's prices and matching them is legal. What is illegal is agreeing with the rival to do so.
The reason is practical. In a concentrated market, firms can reach high prices simply by each recognising their interdependence, without any communication. Economists call this conscious parallelism or tacit collusion. It is economically harmful and, in most jurisdictions, lawful, because there is no agreement to prohibit and no conduct a court could order a firm to stop.
Algorithmic collusion lands directly in that pre-existing hole.

