This question comes from a modified white elephant gift exchange game (and is a tip of the hat to the many swapping puzzles out there lately).
This version of 'White Elephant' is rather simple: There are six gifts and five people. Each gift is a check worth either $10, $100, $1,000, $10,000, and $100,000. No one knows which check is which as they are hidden in sealed envelopes. One person gets to select first. They open the gift and reveal the prize to the group. The person who goes second has a choice, they can take the gift from the first person or select another mystery envelope. If they select a new envelope, then like the first person they reveal the gift to the group and allow the next person an option to take any gift previously opened or open a new envelope. If a person elects to take an opened envelope from a previous person then that item is now permanently theirs and no one can take it from them. The person who lost their envelope now must select a new envelope from the unopened pile. Each person continues until they have an envelope (with one envelope left unopened). The order of selecting the gifts must be finalized before any picking starts.
What would be the best position to go in if you get to decide when you pick? What would be your expected winnings?