Bribes are allowed. Any player may offer to give another player any number of resources as part of a deal, even if those resources aren’t required to solve the Problem.
Personality cards may never be offered as a bribe. However, you can threaten to play Personalities from your hand if you think it will help your case. For example, you could threaten to play a Coinshot to discard a crucial Advisor. Or you could threaten to play a Lurcher to steal the resources needed to make a deal and solve a Problem.