For the first game of a match, a designated player – the winner of a random method (such as a die roll or coin toss) during Swiss rounds, or the player ranked higher at the end of Swiss rounds during playoff matches – chooses either to play first or to play second.
For non-Playoff matches, both players must agree on the random method. This can get a little tricky when the players don’t agree on the method (such as rolling a spindown dice) so feel free to step in and help players select a random method like odds and evens or something similar. Higher ranked choosing is now just the default rule for any single elimination portion of a tournament. This only applies to the initial choice during the match. After the first game, the loser of the previous game makes the choice. This rule was created to give more value to those who did better in the Swiss portion of the event, giving some advantage to “playing it out” rather than intentionally drawing in, which means people playing more Magic!
They must state this choice before looking at their hand. If they state no choice, it is assumed that they are playing first.
Players are supposed to decide whether they want to play or draw before looking at their opening hands, since the contents could influence their decision greatly. If someone does not announce their choice before seeing their opening hand, they gain information would could impact their decision. This is a fairly common and innocent mistake, so there is a simple solution; the player who sees their opener before declaring plays first by default.
The player who plays first skips the draw step of their first turn. This is referred to as the play/draw rule.
For more information on this, check out CR 103.7, which covers the beginning of the game.
After each game in a match, the loser of that game decides whether to play first in the next game. They may wait until after sideboarding to make the decision. If the previous game was a draw, the player who decided to play or draw at the beginning of the drawn game chooses.
Games that have ended in a draw have functionally not happened. There is thus no need to rechoose, as we are instead merely beginning the act of playing again. Thus, if the previous game ended in a draw, the player who chose last chooses again.