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Ellysia Silvertongue is the noble guildmaster. You can find her in her home in North Cyan.

Ellysia is currently engaged to Sir Becket, and so obviously she is planning to vote for him in the election. To get her to change her mind, you'll have to make her think less highly of Becket.

To do that, buy the Haunted Mansion from Serah Blackhand after you've completed her goblin quest. While exploring the mansion, you should learn that Becket is having an affair with Ellysia's maid, and you might even be able to pick up Becket's Ring as proof.

When you give this information to Ellysia (using the ring if you're not persuasive enough), she'll change her vote to you, and you'll also receive 1000 experience and a ring of protection +1.