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Many games like Savage Worlds or some of the Fria Ligan games (like Alien) use a random card draw to decide initiative order. Fun fun fun. You draw a card, no-one gets a duplicate so there are no ties, and then when you do your action you flip your card over to indicate that you're done.
Until now, It hasn't been entirely clear to me how to implement this in 5e, where you have a bonus like +3 or +5 or -2 or whatever. In Savage Worlds you can have "edges" (special abilitities) that let's you draw more than one card and keep the best. But that's a lot stronger than a simple +1.
Ideally you'd have a communal (everyone draws from it) 20 card deck with all the backs the same. On the front there is a number and an upside down number. For example 3 and 23. There's always a difference of 20 between the two numbers.
Instead of a bonus, note your bonus plus one. That's the lowest number you can use. Write it as a range of 1-20 offset by your bonus. So with +3 you'd note "4-23" and if you draw the 2/22 card, you use the 22, rotating the card 180° to show it. If you draw a 4/24 you use the 4. Each card will always have one number within your range and one outside of it.
Here is a suggestion for which twenty cards to use.
┌─────┬──────┐ │ Low │ High │ ╞═════╪══════╡ │ -7 │ 13 │ ├─────┼──────┤ │ -6 │ 14 │ ├─────┼──────┤ │ -5 │ 15 │ ├─────┼──────┤ │ -4 │ 16 │ ├─────┼──────┤ │ -3 │ 17 │ ├─────┼──────┤ │ -2 │ 18 │ ├─────┼──────┤ │ -1 │ 19 │ ├─────┼──────┤ │ 0 │ 20 │ ├─────┼──────┤ │ 1 │ 21 │ ├─────┼──────┤ │ 2 │ 22 │ ├─────┼──────┤ │ 3 │ 23 │ ├─────┼──────┤ │ 4 │ 24 │ ├─────┼──────┤ │ 5 │ 25 │ ├─────┼──────┤ │ 6 │ 26 │ ├─────┼──────┤ │ 7 │ 27 │ ├─────┼──────┤ │ 8 │ 28 │ ├─────┼──────┤ │ 9 │ 29 │ ├─────┼──────┤ │ 10 │ 30 │ ├─────┼──────┤ │ 11 │ 31 │ ├─────┼──────┤ │ 12 │ 32 │ └─────┴──────┘
This way, it works for a malus (negative "bonus") too. For example a -2 to initiative means a range of -1 to 18.
This is great, you have the same probabilitites as the RAW but the fun fun fun of cards. No duplicates, no ties, and you can flip your card over when you've acted. Obv for advantage or disadvantage, draw two cards and use the most appropriate one.
You can make the cards with a marker and some blank index cards.
What if you don't have any fancy custom cards? You can just use plain numbered one through twenty cards such as the ones found in a tarot deck or some other card games like The Mind. Note your range as above. If you draw a card within your range, use it straight up. Under your range you add twenty to your cards result (remember it somehow --- maybe the 180° idea still works) and over your range subtract twenty from it.
Maybe you can't easily find cards that go up to twenty but you can easily scavenge 1 through 10 from an Uno deck or from ordinary playing cards. Or you want to use the cozy 1 through 10 cards in the Essentials Kit just for the sake of using them. A ten card deck can sometimes be quicker and easier to shuffle and the math will be easier too.
The heavy lifting comes upfront, when calculating your range. Divide it in two and round down. This makes the initiative bonuses more coarse-grained. A +5 is no better than a +4, since both would have a 3-12 range. But that only matters a tenth of the time. Anyway, same idea as before. If you draw below your range, treat it as ten higher. If you draw over your range, treat it as ten lower.
A higher number can interrupt and act before a lower number. When everyone has flipped over all their cards (indicating that they have acted), shuffle them all up and draw new ones. Or use the same numbers round after round, only drawing new ones after the fight. You can draw you cards after fights too, so you have a number prepared for the next fight. Maybe the monsters can hold off om drawing theirs so there's still some uncertainty.