From the start of the meeting, the prime minister had been tapping into the guildmaster's fears. He didn't make an excessively logic-driven argument or take a data-driven approach to try and convince the guildmaster that siding with them was objectively the best decision.
No.
He simply asked questions that focused on the guildmaster's fear of three Sage-level nations.
He asked him whether he thought that the Kandrian Empire could protect him from the alliance.
He asked him whether he thought Rui could protect him from the alliance.
He simply asked whether it was truly a decision he wouldn't come to regret.
These were questions with absolutely no rational foundation and were simply designed to place the guildmaster in a negative state of mind.
Of course, the guildmaster was cognizant of the prime minister's machinations. However, it did not stop them from working due to the fact that they were based in some truth.