Replacing collateral with incentives
Algorithmic stablecoins take the most radical departure from the first two designs: rather than holding a reserve of value, whether off-chain dollars or locked crypto, that directly backs the token, they try to maintain a peg purely through rules that expand or contract the token's supply in response to price, combined with market incentives that are supposed to make those adjustments self-reinforcing. The basic logic mirrors a textbook idea from monetary economics: if a currency is trading above its target value, increase supply to bring the price back down, and if it's trading below target, reduce supply (or make reducing supply attractive to participants) to push the price back up. In principle, if the incentives are well-designed and participants behave as expected, the system never needs a pool of collateral sitting idle, it just needs the right feedback loop.
In practice, this design removes the one thing that made the first two categories robust: a hard, asset-backed floor. A fiat-collateralized token has a custodian's dollars behind it, and a crypto-collateralized token has over-collateralized crypto behind it. An algorithmic stablecoin's peg is only as strong as market participants' continued willingness to act on the incentives it offers, and that willingness is itself a function of confidence in the peg holding. This circularity, the mechanism depends on confidence, and confidence depends on the mechanism visibly working, is the defining structural feature of algorithmic designs, and it's what separates their risk profile from the other two categories entirely.