If you want to turn Lewis into a Poodalist you need to challenge Akarion and win without transforming him, then flirt with Rihana until she propose you to swap court, at the second challenge you must lose either to her or Mazareth. By doing so you or both of you will have enought influences for turning Lewis into a Poodalist.
If you want to turn Rihana into a Oleogarch you need to challenge Akarion and win without transforming him, then flirt with Rihana until she propose you to swap court OR use your influences to make her fall for Lewis immedialty, then you win the second challenge against her or anyone to gain enought influences to turn her.
If you have use your influences to make her fall for Lewis, she will gladly accept the transformation, if not she will consider it as a betrayal.
You can still use your influence after to make her think she always was an Oleogarch.
Depending of what you pick the transformation is different.