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Your standard (or stereotypical) initiative rules do far more damage than good. The good they do is represent the characters' abilities and skills - some people get the drop on others faster. It makes sense that a skilled warrior can smack a lumbering zombie before the zombies have a chance to strike him; attacking first in the round will always (at some point) mean killing someone, and therefore avoiding the attack an enemy would have otherwise made.
In your basic D&D-like setting, each player may roll Initiative, record it, then attack, and miss. You've just made an additional roll, and kept a record, rather than just make a roll, so the Initiative rule almost doubles the motions players go through to resolve combat.
Adding movement, special abilities, and secondary attacks will dilute the problem, but none of them really solve the problem.
A lot of people discard the Initiative system and allow people to attack whenever, as narrative or volume dictates. I dislike this approach to combat, but I can see exactly why people do it.
A related method from A,D&D involves each side rolling 1D10, and the side with the highest roll goes first - either all players, or all monsters.
I've never seen this approach, but for a slight improvement on the existing system, we could simply read the attack result twice.
In D&D-style Initiative, this would mean rolling 1D20, and using it for both attack and Initiative. Add any Initiative bonus to this roll, then whoever goes first also finds out if they have hit at the same time by adding their attack score. We don't need to stop to record Initiative, because the dice on the table keep the record.
Players could let themselves mix turn order between themselves - all that matters is the initiative score in relation to the monsters.
Returning to the A,D&D rule of 'both sides roll 1D10', we could let each individual character add their own Initiative Bonus. If the group rolls an '8', and your Initiative Bonus is '+2', you get a '10'.
This significantly cuts down on the rolls, and lets everyone know who's going first immediately - it's just whoever has the highest Initiative.
Take five players rolling 1D20 for Initiative, vs one monster, and the monster will almost certainly never go first organically. But here, with a single D10 rolling for everyone, we retain the possibility of the monster striking first.
All previous alternatives relied on reducing Initiative rolls, but we might tip the scales another way - instead of reducing the initiative mechanics, we could increase the value, until we can justify the rolls.
For BIND, instead of removing Initiative, I decided to use it as a resource to spend. Attacking costs 6, moving costs 2, and so on.
This provides a nice stack of presumed actions - a fighter on Initiative 15 can expect to attack again at 9, then again at 3. We can't demand that people have enough Initiative points to attack (everyone should be able to attack once), so we just let him spend 6 and go down to -3 (otherwise, those with low Initiative may wait several rounds to attack once).
Knowing he will end on 3 Initiative, our fighter knows he can spend 2 Initiative on movement, or some other small action, and retain a full attack on Initiative 1. This adds a tactical element, as someone spending these Initiative points must keep in mind where they will end, and what their expenditures might lose them.