I live in a different but still expensive Germany city and rent a apartment from a city-owned public housing company. The rent of the landlord's apartments is 20-30% lower than what you would pay in the free market, and in my neighbourhood I can get a pretty good idea of what people do to get one of those highly desired apartments. It's just crazy. I have heard everything short of bribing the person responsible for selecting renters. Calling her daily, flowers, pity stories.. another way to get into a highly coveted apartment is to get a recommendation from a renter who's leaving, basically taking over their contract. Often that involves paying absurdly high sums to the previous renter for floors and furniture left behind, or doing renovations for them.

By the way, getting into the best kindergartens and schools appears to work like that as well. Many German cities are so overcrowded that you need to network like in socialism to get what you want, as there are things that money can't buy (or at least only a lot more money can buy). Those who are not well connected have a hard time to get the city's services, or at least will end up with worse facilities.

The demand is just higher than the supply, and if the demand can not be controlled by increasing the price, people will find other ways to fight for what they want...