TLDR:
Look at the character's facial expression as you scroll through each of the four emotions. There will always be two good emotions (characterized by smiles) and two bad emotions (characterized by frowns). These will not change during the mini game.
What you want to do, is make sure the smallest wedge is on the two bad emotions every time. This is possible since the wheel always moves in a clockwise direction. So look at the way the wedges spawn, and determine which ones to click such that you will be able to get the smallest wedges on the two bad emotions each time.
This method will always work. It will always increase disposition, and if repeated enough times, will max disposition out. No need to waste money on bribes! Once you get the hang of this, it only takes a few seconds longer.
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