When Eye of Doom enters the battlefield, each player chooses a nonland permanent and puts a doom counter on it.
, , Sacrifice Eye of Doom: Destroy each permanent with a doom counter on it.
10/17/2013 The active player chooses a permanent first, followed by each other player in turn order. Each player will know what previous players have chosen when making his or her own choice. Then the doom counters are put on all the chosen permanents simultaneously.
10/17/2013 None of the permanents are targeted. A player can choose a permanent with protection from artifacts, for example.
10/17/2013 If a permanent with a doom counter isn’t destroyed by Eye of Doom’s last ability (perhaps because it has indestructible or it regenerated), the doom counter will remain on that permanent.