Psychic Battle
Whenever a player chooses one or more targets, each player reveals the top card of their library. The player who reveals the card with the greatest mana value may change the target or targets. If two or more cards are tied for greatest, the target or targets remain unchanged. Changing targets this way doesn't trigger abilities of permanents named Psychic Battle.
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Official rulings (4) · Scryfall
You can't change the number of targets, choose an illegal target, or make something get targeted more than once.
An empty library will have no card to reveal, so a player with an empty library has no card to compare. The effect looks for the highest cost card of the cards that actually are revealed.
You can change all, some, or none of the targets.
The mana value of a split card is determined by the combined mana cost of its two halves.