I am pretty huge fan of Mafia series (Minus Mafia 3, one of the most dissapointing release from a non-Czech company) and I took a long interest about how the police works in all series. I will create 3 threads that are related about the police. Their A.I, behavior, tactics and all that, are not really mentioned in the Police pages for some reason but these threads on forum will explain them. Well, I did a pretty big and long research about how police works in all Mafia series, because, hey, that's my hobby, I like it.
Starting from Mafia: The City of Lost Heaven.
The police is heavily corrupt and bribed by the mob (Similarly to Chicago in 1920 where 90% of the police force was on payroll by Al Capone). That means, they will act hostile against you for whatever you do and ignore completely your enemies (Gangsters in that case).
The police regularly patrol in the streets in blue uniforms and most of them are routinely armed with Revolvers, a standard weapon. All foot patrol cops will always-always be armed with a standard weapon, which is Revolver. Patrol officers in police cars are initially armed with revolvers but they will upgrade their weaponry when you get WANTED status.
The police can and will ticket the player for 3 traffic law violations: Speeding, vehicle colliding and running red lights. For strange reasons, the tickets are completely useless in-game and nothing to worry about as they will never affect negatively on your player. In story mode, they do ticket but nothing actually happens if you get a ticket (You dont lose money for example). However, in free-ride mode, that may be a bit hassle for you because the ticket costs 1000$ regardless of traffic violations and regardless how many traffic law you violated in a row and that price may be a bit hassle to you because you won't be able to purchase weapons and/or ammo if you don't have enough money. But again, this does not heavily affect you.
On the other hand, if you get signalled by a patrol car/foot officers to pull over and stop (Aka, pull over) but you refuse do that, then from "ticket" you will go to "handcuffs", as in, 2 stars (For GTA and Watch Dogs players). At that point, patrol officers in police cars will attempt to ram into your car and try pit and box you, completely blocking your vehicle from moving if 3 cop cars have cornered you. Having a weapon in the open, refusing to pull over, reckless driving (As in, causing a serious vehicle collision and/or driving above 90MPH), punching a civilian, carjacking will cause the police to arrest you. If a patrol officer is next to you, your character will lay on the ground and get himself arrested by the cop. If the player had a gun in the open, he will drop it before laying on ground. After that, you get game-over screen and that's it.
Finally, this is where the shit hits the fan, if you done a very serious crime, the "handcuffs" will go to "Gun" and this is where you will start regret doing that serious crime. Unlike Mafia 2 and maybe Mafia 3, the police can actually finish you off much faster and easier in this game. All patrol officers in police cars (Drivers and passengers) will shoot at your and at your car through their window. If you have punched a cop, opened fire (Hit a person or on a property, does not matter), the police will start shoot back. Initially, all cops will still be armed with revolvers and all patrol cars will keep ramming to your car as they drive-by shoot at you, making escaping very difficult.
On the worst cases, if you are being chased by the police (Handcuffs and Gun only) for at least one minute or two minutes, you will become "WANTED" by the police and all cops will be alerted about you. In that case, at least 3-4 patrol cars will spawn (More likely on "Gun" level) and give chase as they aid the other cops. If you got "Gun' level, those officers will be armed with Shotguns, capable to kill you in 1-2 shots. Also, in that case ("Gun" only), police will form roadblocks around the city where there will be about 3-4 cop cars and 3-4 officers armed with shotguns and thompsons which warrants a suicide and instant death as they can just explode your vehicle in seconds. To lose the wanted status, one must hide for several seconds.
Foot patrol officers making more presence on the streets than patrol cars so a care must be taken.
Now, this is a 2003 game so it's obvious the A.I might have flaws. However, the excuse of the cops being bribed is good and genuinely for the story and real-life based events. However, this is not fair when they even ignore the ordinary civilians and your friendly gangsters. What do I mean? They totally ignore any enemy gangsters, sure, but they also ignore every civilian (Most of them crash into police cars and they wont react) and even your fellow gangsters. They are programmed to respond to your actions and only you, no matter what. A gangster can accidentally kill them but cops will ignore them. If gangsters chase you, it's best to avoid them and run away because the cops will kill you if you fight back at gangsters. This issue is not very very persisent but a care must be taken as gangsters can be triggered very easily (Colliding their cars and so on).
On interesting note but I do not have proof, I, with the help of mafiascene, managed to crack the Police A.I code and managed to get them to respond the gangsters' violations.
What did I do? Well, there is a program where you can execute orders in that program (as the game is running). I placed 2 gangsters ID in Police Watchlist and pushed "DONE". I then tried to get gangsters to attack me, they did, a patrol car passed by and they immediatelly went on alert and attacked the gangsters, killing them.
I tried to get police to arrest the gangsters instead and well, I met strange results. As you know, if you get arrested, a sequence starts which is scripted to end after the arrest, right? Well, it's same for gangsters. If a cop approaches a gangster, a sequence scene starts (With the camera focused on the duo as if you were getting arrested) and after the gangster gets handcuffed, the game crashes or simply freezes.
If a gangster engages a shootout, the police usually win. And if police wins, the patrol cops return back to their patrol cars and stay idle and they seem to freeze afterwards as I can not draw their attention anymore. On the other hand, cops may just stand on the streets idle.
In conclusion, this might be unprogrammed A.I when it comes to other A.Is but I can no way rule it out as "rushed" development on A.I behaviors, patterns and scripts.
That's all for Mafia 1. The police is simple and follows basic patterns and all that. If it happens, do that, if that happens, do that, if the player does that, do that too and so on. I hope I helped!