Does an Initiative Roll imply combat?

Question

I am wondering if a initiative roll means combat started, for the purpose of perks like the Assassin rogue's Assassinate which states

You have advantage on attack rolls against any creature that hasn't taken a turn in the combat yet.

So if initiative is rolled but the target hides or move away, or just does nothing, do I still get advantage after their turn has passed?

Lets say the assassin enters a big room, he sees his target on the end of the room, initiative is rolled, and the assassin rolls higher, going first. So we have 2 options now:

1. The target sees the assassin as a threat.

2. The target does not see the assassin as a threat.

The assassin is not going to move with stealth in either case, so no surprise rolls.

1. Target run away or comes closer but the room is big so combat will happen on the second round but target got a turn and moved, but not in combat.

2. First turn assassin closes the gap to strike on the next turn. No surprise but it will be the first combat round.

I don´t see a difference between (1) and (2) for the assassinate option. Can I use it in both scenarios?

3. Target runs away and hides but fails the ability check, does that change anything?

Answer 1

Initiative is Step 3 of 5 in the “Combat Step-by-Step” section of the rules for combat.

The rules for combat have a handy summary of running combat:

Combat Step-by-Step

1. Determine surprise. The DM determines whether anyone involved in the combat encounter is surprised.
2. Establish positions. The DM decides where all the characters and monsters are located. Given the adventurers’ marching order or their stated positions in the room or other location, the DM figures out where the adversaries are — how far away and in what direction.
3. Roll initiative. Everyone involved in the combat encounter rolls initiative, determining the order of combatants’ turns.
4. Take turns. Each participant in the battle takes a turn in initiative order.
5. Begin the next round. When everyone involved in the combat has had a turn, the round ends. Repeat step 4 until the fighting stops.

Additionally, we see this in the beginning of the section “The Order of Combat”:

The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative.

Therefore, rolling initiative unambiguously signifies that combat has already begun. The rules above state: “everyone involved in the combat encounter rolls initiative”, so if you rolled initiative, you’re involved in the combat encounter. So when the Assassin’s Assassinate feature states:

You have advantage on attack rolls against any creature that hasn’t taken a turn in the combat yet.

It means that once the target’s turn has passed, this feature does not give you advantage on attacks against them, no matter what they chose to do or not do with their turn, because combat began before initiative was rolled.

If combat has not begun, the DM should not call for initiative rolls.

As explained in the previous section, initiative is not rolled until combat has begun. So if combat has not begun, the DM should not call for initiative rolls.

Unfortunately, the rules do not give much guidance about when exactly combat has begun, leaving the DM with much latitude in interpreting what it means to begin combat, and consequently, when to roll initiative.

This Q&A may have some helpful guidance for handling encounters like the one you describe: What is the proper use of initiative for a cautious encounter?