Eye of Doom — ruling 1

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 their own choice. Then the doom counters are put on all the chosen permanents simultaneously.

Eye of Doom — ruling 2

None of the permanents are targeted. A player can choose a permanent with protection from artifacts, for example.

Eye of Doom — ruling 3

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.

Eye of Doom : règles, fonctionnement et interactions (MTG)

Eye of Doom is one of those cards players keep asking about. Here is, in plain words, what its official text says, the rulings published by Wizards of the Coast, and the combos players actually run.

Official card text

When this artifact enters, each player chooses a nonland permanent and puts a doom counter on it. , , Sacrifice this artifact: Destroy each permanent with a doom counter on it.

How it works : Eye of Doom

Original English text — Wizards of the Coast / Scryfall

#1October 17, 2013· Wizards of the Coast
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 their own choice. Then the doom counters are put on all the chosen permanents simultaneously.
No other card is named in this ruling.
#2October 17, 2013· Wizards of the Coast
None of the permanents are targeted. A player can choose a permanent with protection from artifacts, for example.
No other card is named in this ruling.
#3October 17, 2013· Wizards of the Coast
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.
No other card is named in this ruling.