Initiative rolls do not determine surprise!
I've read every post on this thread, and joined this forum just to point this out! On here, people are seriously saying that because your initiative score is reached that you are no longer surprised!
That is not how surprise works. The DM determines surprise. How? He can decide, by fiat, that creature X is surprised. Or, he may use some other method. The most common being opposed Stealth/Perception rolls, although opposed Deception/Insight rolls would be more appropriate in situations where the creature was already seen but you had no reason to believe that it was going to attack, such as being suddenly stabbed by your dentist for no reason.
What are the effects of surprise? Well, there are two game mechanics (that I know of) that apply when a creature is surprised:-
* in the first round of combat, a surprised creature cannot move or act on its turn, and cannot take reactions until its first turn is over
* if an assassin hits a surprised creature, then that hit is a critical hit!
Both are true, and both are true independently. Whether your turn has come up yet in the initiative count has absolutely no impact on whether or not you are surprised; it only impacts when you can act and/or take reactions.
When does a surprised creature stop being surprised? Is it when his first turn is over? No. The initiative count/your initiative total do not determine surprise.
The DM determines surprise (by fiat or by an appropriate check), and the DM determines when a surprised creature stops being surprised (by fiat, or by an appropriate check).
Conceptually, a creature is surprised when it doesn't realise that it's in danger, and stops being surprised when it (finally) realises that it is in danger! So, if that sniper's arrow (critically) hits you, the DM will (probably) determine, by fiat, that you are no longer surprised on the grounds that this arrow sticking out of you (along with the Sucking Chest Wound) counts as clear evidence that you are in danger.
What if the sniper missed? The arrow might pass unseen and unheard into the bushes, or it might clatter loudly off the stonework. If there is a doubt, that's what rolls are for!
Also, it doesn't have to be the first round of combat per se, it could be the first round for a particular creature. For example, if you are in combat when a hidden enemy snipes from the bushes, this would not auto-crit. You are in battle, on the lookout for danger, you are not surprised. So, that attack from hidden may have advantage (and Sneak Attack), but no auto-crit. But imagine that your combat has been going for some time, when you suddenly burst into a room. The occupant may very well be surprised! It is his first round of combat, and he may not have been aware of danger. Contested rolls (or DM fiat) to determine if that creature is surprised.
So, back to the example earlier in the thread: the target is standing in the open (on his smartphone or whatever) when the hidden assassin says he's going to attempt to assassinate the target. At this point the DM determines surprise, either just by saying that the target is surprised or (more likely in this case) having that assassin make a Stealth check which beats the target's passive Perception.
Let's say the target is surprised. Now, initiative is rolled. Lets say that the vagaries of the dice result in the victim having a higher initiative total.
Initiative counts down, and reaches the score of the target. He is surprised, so cannot act. After this, he can take reactions.
Does the ability to take reactions now mean that he is no longer surprised? No! There is no rule to support this absurd idea! If you look at the reactions you have available, I'd be shocked if you can take it against an attack that you know nothing about! You take reactions to respond to things you can see (or are aware of), and the target in this little scenario is still blissfully unaware of the assassin! Remember, surprise is determined by opposed Stealth/Perception scores (in this scenario), not by initiative (which determines how fast you react, not what you perceive).
So our victim is still surprised, because he doesn't know of any danger and the assassin has done nothing (yet) to change that.
Next, the assassin's turn. He shoots from hidden, so has advantage. If that attack hits, then because the victim is still surprised then it is an auto-crit because the assassin class feature says it is. If that attack hits and crits, and the target is still conscious, is he still surprised? I doubt it! I think that most DMs would rule by fiat that he now realises that he is in danger. Subsequent attacks may or may not have advantage, but will not auto-crit from the assassinate ability.
What if the assassin misses? The DM will determine (by fiat or by rolling) if the target has stopped being surprised, based on things that the target could perceive (like noticing an arrow zip by his nose).
What if the target doesn't notice the miss; is he still surprised? Yes, if the DM determines that he is still surprised in round 2, then the game mechanics of surprise are still in play. Granted, his first turn has come and gone, so the target can act normally on his turn in round 2, but if he has no reason to believe he's in danger then he is still surprised and still vulnerable to auto-crits from the assassinate ability. When the target takes his action in round 2, he cannot attack the assassin simply because he has no clue that there even is an assassin. He could, in theory, use an action to roll a Perception check, and if he beats the assassin's previously rolled Stealth check then he will notice the assassin and as a consequence no longer be surprised. However, in theory, the target my stay surprised for many rounds.
In practice however, this is unlikely. If an arrow has been shot, this is combat and we are already in combat rounds. If the assassin did not even shoot, then there is little reason to be in combat rounds.
