Who goes first?
From the publisher’s rulebook

Who goes first in Dominion?

English second edition · 2021 rulebook revision

For a fresh game, choose the starting player randomly. If you just played Dominion and it was not a tie, the previous winner goes last; the player to their left starts.

Read the publisher’s rulebook

Rule details

The rematch exception applies only after a game without a tie; otherwise use the ordinary random selection instruction.

Random selection for a fresh game is an official instruction, not a house rule.

If there’s a tie

The cited starting-player instruction does not specify a tie-break. We don’t infer one.

Source & edition

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