I believe it is least influence, with the tiebreaker being rank of the official (higher number=least senior breaks the tie). From the rules, p. 17: If more than two players have influence in the ministry, then the player with the least influence must temporarily give all his or her influence to another player. The player with the least influence is the one with the least markers. In the case of a tie for least markers, look at each tied player's most senior official in the ministry; the tied player with least influence is the one whose senior official has the least seniority, shown by the highest number next to the official space.

Thanks for the file, I totally missed the bit about not being able to bribe more officials than someone who gave you a gift.