Smallest in a range without picking same person twice

I have two columns as below below I need to find the lowest value from each person First score and they find the lowest value of the second score without including the second value of the person of which we used their first value.
Name Score
Fred 5
Fred 4
Joe 4
Joe 3
Mary 6
Mary 5
Sue 5
Sue 3

In the example about, I would want to pick Mary's score of 3, then look that the other scores and not pick Mary's score of 5. ie I would pick Joe's score of 4

What are the exact results expected from this dataset? And why?

Nevermind. Need to work on the second half.

Are you the same person as OP?

Me? Nope.