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Can I save my creatures from Day of Judgment with a Legion's Initiative ability of exile? Day of Judgment is on top of the stack for this example (I don't know if that will matter).

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You sure can.

The stack will also matter for this because otherwise your creatures will go poof. Basically in response to them casting DoJ, you pop your Legion's Initiative to exile your guys. There's no timing restriction on Legion's Initiative so as long as you can pay for it, you can do it.

Or in listed steps:

  1. Day of Judgement is cast and opponent passes priority (most likely onto an empty stack because it's a sorcery but there's ways to make it instant speed but that's irrelevant here.)
  2. You get Priority and pop your LI, putting the ability on the stack
  3. Priority Wave
  4. LI Ability resolves, all your dudes go away until the next combat
  5. Priority Wave
  6. DoJ resolves, blowing up the remaining dudes on the battlefield.
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    Correct, if it was instant, they could respond to you popping LI by casting it which would resolve first and blow up all your dudes. Any indestructible or regenerated guys would still be alive to be LI'd though so it's not entirely bad news. – Veskah Apr 20 '18 at 3:31
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    @Chris Your example is a case of stack matters. If you try to Shield and they respond with the bolt, your creature would die. If they try to bolt and you respond with shield, your creature will live. Put another way, your opponent will put a <Bad Thing> on the stack. You then get the chance to do something about it, which is what "Responding" means and is the driving force behind the priority wave business. This systems let you kill an opponent's creature before they buff it, or save your creature before your opponent can kill it. – Veskah Apr 20 '18 at 3:48
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    @Chris, Re "The stack does matter then", The stack doesn't matter; the stack is used. When you cast or activate in response to something, it means you cast or activate when that something is on the stack. By so using the stack, you cause your spell or ability to resolve before the spell or ability to which you are responding. – ikegami Apr 20 '18 at 5:37
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    @Chris Perhaps the best was to sum it all up is that "a card on the stack has no effect on the game until after that card resolves". This means that while Shielded Passage is on the stack, your creature does not have any protection, and it can be killed by a Lightning Bolt that is cast in response to it. While Lightning Bolt is on the stack, the creature has not taken any damage, and so can be given damage prevention by casting Shielded Passage in response. The spell cast second will resolve first, and that will often effect or nullify the first spell. – GendoIkari Apr 20 '18 at 13:49
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    "Basically in response to them casting DoJ" Or you can cast DoJ, exile your creatures, wait for your opponents' creatures to go away, then bring yours back. – Acccumulation Apr 24 '18 at 4:02

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