What good is negative money, you ask? A good question, but first let's review what negative money is, and what it is not. It's not debt, because all proper debt (which excludes tax debt) cancels to zero when weighed against loaned principal. It's not tax, or theft, both of which rely on acts of aggression, the essential difference being that taxation is theft justified by a claim of sovereignty over the victim. The tax victim is treated as if morally obligated to endure the robbery without complaint, and to comply fully with all the demands of the thief.
Negative money is not a legal claim for money, because it is not enforceable. Negative money is a purely voluntary instrument; in this manner it resembles proper debt. It springs into existence without any preceding loan or adjudicated harm; in this manner it resembles taxation or theft. It may arise out of a perceived injustice or slight; in this manner it resembles a legal claim.
Negative money is not debt, it's not a legal claim, and it's not tax or theft. Negative money is a partly negotiable scarce instrument, publicly recordable, that is extinguishable by redemption. If given to a recipient (the "giftee"), it cannot be transferred from the giftee, except by purchase by another, or by redemption. To redeem a unit of negative money, a giftee can pay the equivalent amount of corresponding positive (regular) money to its giver in an automatic redemption, or convince the giver to otherwise voluntarily redeem the gift of negative money. The negative money then returns to the giver, or in some implementations, becomes the redeemed property of the giftee who after redemption holds the same title as the giver did prior to making the gift. A giftee can also sell the negative money to any willing purchaser without redemption, but only for a fixed amount of positive money, which may be less than or equal to the redemption amount (and may be zero), depending on system design. A person who receives negative money by purchase or redemption can spend it freely. In some implementations, all transactions are recorded publicly; in others, only current balances are publicly available; in others both balances and transactions are maintained in confidence. An heir to negative money stands in the position of the decedent. Likewise, a person who finds abandoned negative money stands in the position of the last recorded holder.
Negative money is not necessarily a digital thing. Any issuer or miner of money could place negative money into circulation. Imagine that an issuer of silver certificates issues negative money certificates in some fixed ratio to silver certificates issued. Perhaps, for example, the issuer provides ten silver certificates and five (or one, or ten, etc.) negative money certificates for each ten ounces of silver deposited. The issuer or some other party offers to maintain a registry of all negative money transactions for free, and pays for this service by charging a small fee for access to information in the registry. No transactions are recorded in the registry unless complying with the applicable negative money rule. Since a chain of title is needed to determine whether or not a negative money transaction is compliant, and recording is free, givers, purchasers, heirs and finders of abandoned negative money have an incentive to record every transaction.
Digital money just makes all of the functionality of negative money easier to implement. Digital money is, after all, programmable. Implemented as proof-of-work electronic coin, negative money is another variation of programmable money. It is a variation that is useful for incentivizing the orderly collection and publication of objective reputation information and settling claims without litigation. As a reputation information service, negative money ledgers are broader than credit reports, and capture negative feedback on a wide variety of non-neighborly behaviors beyond debt defaults or late payments.
Negative money may also serve as a litigation substitute. Instead of bringing suit over a claim, a claimant may make a gift of negative money to the defendant of the claim. This puts the defendant and the world on notice of the existence of a claim or complaint against the defendant, and provides an automatic mechanism for resolution of the dispute, without requiring any initiation of legal process. To publicly demonstrate that the claim is extinguished, the defendant must somehow cause the negative money to be redeemed, either automatically or otherwise. If the defendant sells the negative money to a third party, this provides a signal that the defendant considers the claim invalid, and is a strong invitation to sue. Especially if the third party is related to the defendant, and gifts the negative money back to the original giver. All of this can be accomplished via the programmable structure of the negative coin.
We should not fail to notice that the implications of programmable currency extend far beyond substituting for tradition money such as gold and silver, or fiat currencies. In general, programmable currencies may provide more versatile tools for incentivizing community-building behavior via voluntary exchange, if left free and unregulated. Negative money is just one small example.