Keep a fixed order of players across rounds and display players in the correct order

Currently, the players scores are stored in an unordered hashmap, meaning that there is no actual order of players across rounds. Also, because that map is used to display the players in the UI, the order of players there randomly changes.

Locate the unordered hashmap that stores player scores and the UI code that uses it to display players. Verify how players are carried between rounds, then ensure their order remains fixed across rounds and matches the displayed order; done means the same player order is preserved and no longer changes randomly.

Written by the indexing model from the issue text.