So the villain has an evil plan and crosses paths with the hero. However, the hero's antics drive the villain up the wall, to the point where he calls the police and begs for them to arrest him.
Another reason why the criminal might prefer jail to the hero is if the hero is extremely scary rather than annoying.
Can be the result of Pity the Kidnapper, Unintentionally Notorious Crime or Mugging the Monster. An interesting subversion is Cruel Mercy if the hero realizes that jail is too easy for the villain. Contrast The Mob Boss Is Scarier, where a criminal is so fearsome, that no one will inform on them and will take jail time or other legal penalties rather than risk their wrath.
Inversion in Blake and Mortimer. When Olrik (who had been working for a rogue Soviet doctor) is arrested, he's his usual flippant self when Blake and Mortimer visit his cell, but when he learns he's being exchanged by the British police to the Soviets, he begs for them to prevent it.
Many criminals let themselves be imprisoned so as to evade The Punisher (and depending on their status may end up in a Luxury Prison Suite). When that happens, Frank turns himself in, always getting sent to the same prison, and takes out the criminal (and the criminal's bodyguards, tough guys who want to show off, rival prison gangs, prisoners he runs into on the way, and his cellmate) before breaking out of the prison.
In The Muppets Take Manhattan, a phony Broadway producer is about to be arrested for fraud when he takes Gonzo and Camilla hostage. After an attack from Animal, he begs to be arrested.
John Woo's The Killer has Wong Hoi doing this at the end of the movie to get away from Inspector Li Ying, who wants him dead for killing Ah Jong, the protagonist of the film. This results in Li gunning him down right in front of his fellow officers and then getting arrested.
In the Honor Harrington short story "What Price Dreams" the control officer of an assassination attempt was detected and mobbed by treecats, to such a degree he screamed for help and made a full confession more or less on the spot to the rangers and bodyguards coming to the scene.
In an episode of The Fresh Prince of Bel-Air, the Banks hire an ex-con as a servant while Jeffery's away. Eventually he decides to violate his parole and go back to jail rather than continue to work for Phil Banks.
A subversion occurs in the iCarly episode, "iFind Lewbert's Lost Love". Even though Chuck Chambers was responsible for stealing all the TV remotes in the Bushwell Plaza, Lewbert tells the police that he stole the remotes in order to get away from Marta Trundel, his ex-girlfriend. Just to ensure he's away from her for a long time, he slaps one of the officers to extend his sentence.
In one episode inveterate gambler Nick comes in crowing about a big score he had made: he picked all the bowl games, then bet it all on a hockey game and won there too. Then his bookie comes into the detective squad room and turns himself in for illegal betting: turns out everyone had had the same sting of luck and he can't cover his bets. And not everyone is as nice as Nick is.
In another episode a burglar had been caught and tried in a makeshift community court (which had been designed for small civil disputes) and had been serving his "sentence" locked up in the basement. He escapes and is picked up by the real cops; he's grateful to be in a real jail.
At the end of one The Adventures of Superman episode, one villain tells another he can call the police. The villain's response? "Better a hundred of those guys than Superman!"
The Looney Tunes short, "Bugs and Thugs", has Bugs Bunny trick Rocky and Mugsy into hiding in a stove, then later impersonate a policeman and light the stove with them still inside. When the real policeman shows up, Rocky and Mugsy beg for him to take them to jail.
The Tiny Toon Adventures episode, "Take Elmyra Please" has Arthur Jabba's henchmen, George and Leonard, kidnap Elmyra Duff and hold her for ransom in exchange for her father's fuel formula. Elmyra thinks their hideout is a TV studio and her antics drive George and Leonard up the wall, to the point where by the time her younger brother, Duncan, and the police show up, George and Leonard beg for the police to take them to jail. Of course, since the kidnapping was Mr. Jabba's idea, they rat him out and he is arrested as well.
The Action League Now episode, "Flesh and Blood" has two criminals impersonating The Flesh's parents and tricking him into robbing a mansion. However, the Flesh's inadvertent injuries to them get them to confess they're not his real parents and beg for the police to take them to jail.
The episode, "Child Fearing" has Mojo Jojo being released from prison to babysit Blossom, Bubbles, and Buttercup while the Professor is at a meeting. At first, he wants them to help him take over Townsville, but taking care of them is too much for him to handle, and he calls the police so they can take him back to jail.
The Amoeba Boys avert this - they simply want to go to jail to prove they're as much of criminals as the other villains in Townsville.
In the episode, "Sun Scream", two criminals are disgusted by the girls' sunburnt skin peeling and beg to be taken to jail. However, the girls decide to leave them stranded on an island with only a tube of sunscreen instead. The criminals get sunburned as a result of not using the sunscreen, claiming it's only for nerds.
In the The New Woody Woodpecker Show episode, "Baby Buzzard", Buzz disguises himself as a baby so he can rob Woody's house when Woody starts his babysitting service. When Woody finds out that Buzz is robbing his house, he makes him take a nap with a live, ferocious badger, who mauls him while Woody calls the police. When the Policeman arrives, Buzz begs for him to arrest him. Subverted at the end, in which the badger also gets arrested and mauls Buzz while in the paddy wagon.
In Teen Titans 4th season, Dr. Light faces Raven using her dark powers, whereas exposure to that in the 1st season reduced him into a shivering wreck. Upon seeing that thing again in the 4th season, Dr. Light goes from "No one defeats Dr. Light!" into "I'd like to go to jail now, please."
Variation in The Batman. In one episode, Penguin tries to hand himself over to Batman because an even more dangerous vigilante named 'Rumor' has begun attacking Gotham's criminal community. Everyone knows that criminals who cross Batman end up handed over to the police and arrested, but criminals Rumor targets are abducted and ever seen again, so Penguin decides a jail cell is the safest place he can be.
In the Tuff Puppy episode, "Moms Away!", Snaptrap plans to get rid of all the Moms in Petropolis by shipping them to a deserted island and stranding them there. At the end of the episode, Dudley foils Snaptrap's plan and decides to have his Mom punish him. Snaptrap begs to be taken to jail instead.
An episode of Mister T titled "Mystery of the Stranger" deals with abducted kids. Mr. T and company track down a pair of kidnappers. Eventually, Mr. T catches one of them and with a Death Glare, offers the kidnapper a choice: face him or go to jail. The kidnapper begs to be thrown in jail.
An Accidental Hero example: In the 1936 Porky Pig short The Blow Out, Porky, here a young child, is trying to earn spending money by returning objects that grown-ups have dropped. He thus innocently chases a Mad Bomber around, repeatedly handing his activated Time Bomb back to him. The increasingly frazzled bomber finally runs voluntarily into an open police wagon, only for Porky to toss the bomb in after him.
A variant in Darkwing Duck: Megavolt and Quackerjack for once don't want to break out of the St. Canard Jail, given that they're in an alternate future and that future's Darkwing has gone Knight Templar, they have a good reason.
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