Before the first game in any match, the two players in the game will use a random method to determine who chooses which player will go first. Usually this involves a coin flip or a die roll. If there is ever a dispute over randomly making this determination, call a judge.
The player that wins in this random determination chooses who will play first in the initial game. That choice determines the “starting player.” The starting player is the first to declare his or her choice of whether to keep his or her hand or mulligan each time a mulligan would take place (so the opponent knows that choice before declaring his or her own decision), and the starting player does not draw a card for his or her first turn. If the player with the choice of who plays first looks at his or her initial hand before announcing this choice, it is assumed that he or she is the starting player.
In subsequent games, the player that lost the previous game will choose who plays first. If a game ends in a draw, the same player that chose for that drawn game will choose for the following game.