If person A asks person B to commit a crime, and actually intends that the crime be committed, then A is legally responsible. If the crime is actually committed, then person A is guilty of being an accessory to the crime; if it is not committed, then person A is guilty of solicitation.
So it just matters about person A's intent? If person A jokingly set out an elaborate plan by which to commit a crime, and person B followed that plan and actually commits teh crime, but person A was joking, is person A not responsible at all?
Could person B expect any leniency because person A convinced him to commit the crime? Would the punishment be any less for person B if they commited a crime they had thought to do themselves, versus if they had commited a crime person A had suggested/planned?
If it could be reasonably proven that the elaborate plan was said as a joke, or basically that A had no intent of the plan actually being carried out, then A would bear no responsibility.
In either case, B is just as responsible. However, in the case where A was serious, B might just get a plea bargain to lower their sentence if the conspiracy can't be proven as effectively before a jury without B's testimony. So, B has a higher chance of a harsher sentence if A was not serious.