I've seen some posts in the OSR blogosphere about initiative that are interesting in their takes on the idea, but to which I want to respond ...
In Dungeons and Dragons, of course, "initiative" is the roll that one makes at the beginning of combat (either once, or every round of combat, depending on editions ...) which determines the order of combatants and their actions.
Noisms at Monsters and Manuals talks about it directly in "Seizing the Initiative", in which he points out that "initiative" is a thing that one can actually observe "being seized" in sports matches or in historical accounts of battle, even if the precise ... Platonic ideal, one could say ... cannot be directly observed. In a match between two teams, one of them suddenly takes charge seemingly of everything and destroys the other team. This is not modeled well in the D&D "roll for initiative" rule, he argues, and might better be modeled by some action that could be taken by players or NPC enemies to "seize the initiative" for their team and ensure that they act first (until the initiative is taken back).
Emmy Allen at Cavegirl's Game Stuff has an excellent post about how RPGs fail to model the realities of skirmishes and combat when one has actually partaken in modern recreations of melee combats with swords, spears, shields, etc ("Shit games don't get about combat situations"). She doesn't talk about initiative in particular, so much as "reach" weapons (spears), missile weapons, shield walls, and formations in general, and their effects on the people involved in a combat. But these things are basic parts of "initiative" ... some D&D games use "reach" as the deciding factor for "initiative", i.e. spears always strike before swords ... AD&D complicated things by having smaller/faster weapons strike first after the initial round, though this is not unreasonable when considering the effect of the Spanish rodelleros (using sword and buckler) against pike formations in the 16th C.
And just recently, Patrick Stuart at False Machine has written an idea for initiative in which it is literally a physical thing carried by the characters which can be seized back and forth, OR as a mechanic related to how much information the player/characters are trying to ascertain before they act, this post being, "Physical Initiative and Query Initiative". In a sense, I quite like the Physical Initiative--you could tie it to a standard bearer-character, or a drum/pipes character keeping time for the march, and if they are captured or killed, they can no longer provide initiative for their own team (Patrick has even more bizarre and interesting versions that are worth checking out).
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So these are all interesting ways to think about what D&D or its descendants are modeling, and about how one might model things in one's own game to better match what one thinks "initiative" is (or more broadly, just to remember that any game mechanic can be fiddled with, changed, or forgotten, to fit what you want to happen in your game!).
But, I think this quibbling over "initiative" as a mechanic actually misses the initiative of those playing the game. Sure, in my game as in most "OSR" games, a d6 is rolled each round, one for the player/characters, one for the monsters they're fighting, and the higher result wins. There are lots of fiddly particulars depending on the referee--Aleksandr Revzin has the players win on a tied roll; I ask magic-users to declare spells before initiative is rolled, and if their side loses initiative and they get struck by an attack, their spell fizzles; also, concerning tied rolls, I have all attacks happen simultaneously, so that even those "killed" in the round still get their attacks off.
All of this is largely cosmetic, though.
Initiative proper is whether the players understand what's going on well enough in the melee to coordinate their attacks. Better yet, initiative is the players having planned the melee before hand, lured the monsters to where they want them to be, and then to attack them with maximum force.
A die roll, you go first/they go first doesn't matter much when one side has overwhelming odds. Players who understand that, set the situation up to have overwhelming odds, and then engage--those are the players who actually have the initiative.
As referees, truthfully, we could always have the initiative. We have fuller knowledge of the environment, of the numbers the goblin (or whatever) tribe could bring to bear when fully alarmed, of the tricks, traps, and spells that can be brought to bear against the players. But the game is designed to allow the players to have the initiative (usually); they enter the goblin-caves of their own will; they are the ones sneaking around, choosing where to go; if they set off the alarm, so be it, but they have their own wits about them to try to silence any alarm they can.
Even when they do end up in melee combat, the player/characters can take the initiative. Stand in the hallway rather than the broad chamber so you can set up an un-flankable battle-line; lay down flaming oil to keep enemies off your flank when you can't withdraw in time; focus attacks on one enemy at a time to make sure they go down, rather than dividing attention among multiple foes; all of these and more are tactics that will see a party through, regardless of what they roll for initiative any particular round.
I don't know how many combats I've watched as a referee in which players just attack whatever, and nobody takes the initiative to direct their fellows as to the best course of action. Melees like that go fine against goblins ... but against well coordinated enemies, they tend to go poorly. I think initiative is already in the hands of the players ... the dice roll is just there to keep things from being stale (yeah, don't get me started on waiting an hour to get back around to my turn in 3e/Pathfinder/5e!!)
