From the Catacombs

The initiative is a designation a player can have. A player with the initiative designation is said to “have the initiative.” The initiative carries two inherent rules. First, whenever a player takes the initiative, and at the beginning of the upkeep of the player with the initiative, that player ventures into Undercity. Second, whenever one or more creatures a player controls deal combat damage to the player who has the initiative, the first player takes the initiative. Also, some abilities will refer to having the initiative and provide other benefits.

Only one player can have the initiative at a time. As one player takes the initiative, any other player that had the initiative ceases to have it.

A player who currently has the initiative may take the initiative again. This causes that player to venture into Undercity again, but does not cause them to have multiple initiative designations.