As competition grows bribery costs may increase. This may be caused by "global events" cards or by the actions of other families (e.g. "all dockworkers are now requiring an extra $100/rd in light of the risks they take to smuggle narcotics" or "the Montague Family are now paying an extra $300 to their magistrates. As a result your judge wants a briberise...").
An opponent may double-bribe you by paying the corrupt official twice his bribe-cost. If the corrupt official fails an integrity check then this tactic works and the official's affilitations are shifted to the opposition.