One way to distribute funds without sybil attacks is... actual air drops, making manna rain from the sky. Print a million bio-degradable paper wallets, put EOS on every one of them, and just drop them over populated areas. Each wallet can access funds from a contract, and after a week, any funds that have not been claimed from the contract are returned to whoever organised it, and the wallets that are unused and undiscovered just decay.

It's quite an interesting way to distribute resources before there are identity systems and what-not in place, the sybil attack prevention is built-in with that people cannot be everywhere at once, and the airdrops are geographical.

Probability of discovering paper wallets in a person's daily life

People are mediated beings, that follow trails, and so every single day a person will walk along paths and follow trails that are completely unique, which means that just in a person's daily routine, they will walk by paper wallets from the air drop, and pick them up because, why not?

Benefits over airdrops with cash

Only funds that are discovered and used are withdrawn from the contract that manages the event, so there is no loss of money.