I read this in the UK, can't recall if paperback. Sometime between 1965 - 1975.
The premise was a city of (I think) ten million inhabitants surrounded by (I think) wastelands. There were robotic law-keepers who are always wandering about through the streets.
One point of interest is that this is meant to be a utopian society where everyone is healthy and thus there are no medicines. If anyone gets injured the nearest robot gathers them up in its arms and marches them off to hospital. However the truth is much darker; they're actually processed in a disposal unit. (I want to say like Soylent Green but I'm not certain, they might simply be killed.)
The population is unaware of this, except for a few of the high level bureaucracy. The bureaucrats tend to have secret stashes of old medicines and first-aid kits hidden away in their rooms.
The protagonist is bumbling about one day, living the life of a spoiled brat; his father was high up in the Council. He's at some kind of function with canapés and drinks. The wall screens suddenly flash up with a Mandatory Vote, everybody stands in orderly lines to register their vote at terminals.
Suddenly he starts cold sweating; he realises he's skipped the last two Mandatories. You're only allowed to miss three in a set time period and you lose your privileges. He does mental arithmetic and panic mode sets in; he's right on the deadline!
He is almost at the booth but the seconds are running away so he asks the woman ahead to skip. She looks at him and the line and waves him past, he hastily slams on a button with his wrist tag, just in time. However, in his panic, he fails to check Pro or Con or the issue being voted on, he simply votes. This causes his dad to get a notification that his son is voting against the ruling council, baffling the older gent, who has carefully taught his son the relevant catch words to ensure you vote with the Council.
While the protagonist is calming down, and chatting to the girl who let him past, an announcement comes over the speakers that a dangerous disease carrier is in the building, please report this unfortunate person for help. The description matches her and he is about to report her, but she shows him a long healed scar (I think her arm) and explains that is all that's wrong with her.
She had an accident months ago but ran from the robots because she was aware. She then leads him into a secret world of people who avoid the hospital, where he learns about the real fate of the injured and joins their rebellion.
Eventually there are riots with people attacking the robots. A bad guy on the Council is controlling and co-ordinating the crackdown on the rebels but gets involved in a scuffle. The robots break it up and he starts giving one some orders about the pursuit.
It then says "You are bleeding." (He'd been punched on the nose.) He replies "Oh this? It's nothing" and continues giving his deployment order. The robot suddenly says "You must be taken for hospitalisation immediately." and grabs him. A lot of screaming and shouting of "No! No! I'm on your side" as it marches into the service tunnels and takes him to the whirring blades.