Bounding the additional utility of double spends

[yondonfu]

Given the offline nature of probabilistic payments (parties do not wait for transactions to confirm on-chain to ensure global consensus on state), double spends cannot be completely prevented. However, double spends can be deterred using sender penalty escrows.

Chiesa et. al. have provided a formal economic analysis on double spending with probabilistic payments which the description below draws from.

In order for a penalty escrow to effectively deter double spends, its value must be greater than the additional utility a sender can gain from double spending across multiple recipients. If we consider an unlimited set of recipients without any limits on the value transacted for each recipient then a sender essentially has unbounded additional utility from double spending. As a result, a probabilistic payment system needs to implement restrictions on the following parameters in order to bound the additional utility gained from double spends:

1. T: The time required for a double spend to be detected.
2. N: The maximum number of recipients.
3. A: The cumulative value of probabilistic payments (expected values) sent before a double spend is detected.
4. W: The cumulative value of macropayments (face value of winning tickets) sent before a double spend is detected.

A affects the average case utility gain from double spends and W affects the worse case utility gain from double spends. For simplicity let's just consider W and the worst case utility gain from double spends.

All of these parameters should be considered on a per penalty escrow basis - bounds on the values of these parameters can be used to derive the additional utility gained from double spends which can be used to derive the required value for a particular sender's penalty escrow which must cover the maximum amount of financial activity that a sender can take part in during the time required to detect double spends.

Why do we need to bound these particular parameters?

• If T is unbounded, then recipients never detect double spends which means double spending senders have unbounded additional utility.
• If N is unbounded, then recipients can double spend across an unlimited number of recipients which means double spending senders have unbounded additional utility.
• If W is unbounded, then there is > 0 recipients that are accepting an unlimited number of payments so a double spending sender can work with those recipients to gain unbounded additional utility.

Related:

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[dob]

Clarifying that the practical execution of W would be something like:

• O is working with B within a given time window, and increments an accumulator with the value of all tickets that win.
• O would theoretically bound the maximum amount of this accumulator, and then stop doing business with B within the time window if it reached the bound.