To help users out with ordering cabs Yandex released an app that actually let's you order cabs and shows you pre-negotiated prices, you can pick whichever one you like (economy, premium, luxury). They include user ratings and some sort of due-diligence for taxis to get accepted to their network.
But apart from that, a Lyft-like system has existed in many cities for years. In Moscow just holding up your hand on the side of the street prompts anyone to just drive over to the side of the street and offer you a ride. Most of these are full-time or part-time drivers, but also people getting home from work who don't mind a small detour for a bit of extra cash. At a fraction of the price this is the most popular way for people to get around, although some cars look like they date back into the USSR era.
This is a perfectly acceptable scenario and I think with modern technology the market will start to regulate itself.