1. If I crack a fetch land, would Opposition Agent allow my opponent to get an appropriate land from my deck & exile it instead? Or if I skip the search & “fail to find”?
2. If multiple players controls an OA, who would search? Is it time stamp or APNAP?
3. While searching, can my opponent see my hand & other hidden information only known to me?
1) You can't skip the search on something like Polluted Delta because it is not optional. If the ability resolves with an opposing Agent on the field, they will get to control you during the search and exile one of your lands. If they want.
2) It is Timestamp for the "control" effect. That is, whichever Agent enters last will "win". For the "exile" ability, this is a replacement effect and you can choose how they apply. But since you are currently being controlled when that choice is made, it is pretty likely the person controlling you is going to choose to have their exile effect win.
3) Yes. Any information you can see, they can see too. The only exception is your sideboard.
1. e.g. Flooded Strand. Since the fetchland's ability has no 'you may' clause, you need to start searching, and since you start searching, the opponent with an OA controls you from there, so failing to find is not your call. If the spell or ability that would make you search had a 'may' clause, as triggered abilities more often do, you could opt not to search since they won't control you then, but fetchlands (at least, what most people call by that name) don't give you the option, so you can't stop them from stealing your card.
2. Timestamp. There is a replacement effect involved for "exile the card instead of putting where you would put it normally" part, which means it's the affected player who chooses... BUT since the affected played is being controlled by the player with the OA which has the latest timestamp, unless that player wants to give the card to the other OA player, they probably get it.
3. Yes, that's part of controlling a player, no matter how limited the duration of the effect is.
Private Mod Note
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I'm a former judge (lapsed), who keeps up to date on rules and policy. Keep in mind that judges' answers aren't necessarily more valid than those of people who aren't judges; what matters is we can quote the rules to back up our answers. When in doubt, ask for such quotes.
In regards to the time stamp question, what if two agents enter at the same time via an effect like Living Dead?
613.7k. If two or more objects would receive a timestamp simultaneously, such as by entering a zone simultaneously or becoming attached simultaneously, their relative timestamps are determined in APNAP order (see rule 101.4). Objects controlled by the active player (or owned by the active player, if they have no controller) have an earlier relative timestamp in the order of that player's choice, followed by each other player in turn order.
So it depends on whose turn it was when Living Death resolved.
If it's Player A's turn and they cast Living Death, with Player A and Player B both getting an Agent on the battlefield, then Player A's Agent will have the earlier timestamp (as the active player's), and Player B's Agent gets the later timestamp, meaning Player B's Agent 'wins out' when both would apply to the same search.
2. If multiple players controls an OA, who would search? Is it time stamp or APNAP?
3. While searching, can my opponent see my hand & other hidden information only known to me?
2) It is Timestamp for the "control" effect. That is, whichever Agent enters last will "win". For the "exile" ability, this is a replacement effect and you can choose how they apply. But since you are currently being controlled when that choice is made, it is pretty likely the person controlling you is going to choose to have their exile effect win.
3) Yes. Any information you can see, they can see too. The only exception is your sideboard.
2. Timestamp. There is a replacement effect involved for "exile the card instead of putting where you would put it normally" part, which means it's the affected player who chooses... BUT since the affected played is being controlled by the player with the OA which has the latest timestamp, unless that player wants to give the card to the other OA player, they probably get it.
3. Yes, that's part of controlling a player, no matter how limited the duration of the effect is.
613.7k. If two or more objects would receive a timestamp simultaneously, such as by entering a zone simultaneously or becoming attached simultaneously, their relative timestamps are determined in APNAP order (see rule 101.4). Objects controlled by the active player (or owned by the active player, if they have no controller) have an earlier relative timestamp in the order of that player's choice, followed by each other player in turn order.
So it depends on whose turn it was when Living Death resolved.
If it's Player A's turn and they cast Living Death, with Player A and Player B both getting an Agent on the battlefield, then Player A's Agent will have the earlier timestamp (as the active player's), and Player B's Agent gets the later timestamp, meaning Player B's Agent 'wins out' when both would apply to the same search.