# ArbTaxDecay

> Prices the staleness of a pool: the longer a pool goes untraded, the more the next swap pays.

A production Uniswap v4 hook. Source: https://github.com/nirholas/arb-tax-decay. Part of the HookForge catalogue: https://hookforge.pages.dev

## How it works
Loss-versus-rebalancing is the dominant cost of providing liquidity to a constant-function AMM. It is paid when an arbitrageur brings a stale pool price back to the market price, and the size of that arbitrage grows with how long the pool sat unpriced. Ordinary flow does not have this property: a swap that lands one second after another swap is almost certainly not an arbitrage, because there was no time for the reference price to drift. This hook turns that observation into a fee. It measures the time since the pool last traded and adds a surcharge that grows with it along a saturating curve, capped at `maxSurcharge`:   surcharge(elapsed) = maxSurcharge * elapsed / (elapsed + halfLife) At `elapsed == halfLife` the arbitrageur pays half the cap; a swap in the same second as the previous one pays only `baseFee`. The surcharge is an LP fee, so the value it captures is paid to in-range liquidity providers. The hook never custodies funds and holds no privileged role. Two properties make this cheap to reason about. It needs no oracle, so there is nothing to manipulate and no liveness dependency. And it is monotone in a quantity the arbitrageur cannot control: waiting longer to arbitrage a pool only raises the toll, so the strategy that minimizes the tax is to trade the pool more often, which is exactly the behaviour that keeps the price fresh for everyone else. Prior art: dynamic-fee hooks keyed on realized volatility or on price movement are common, and the LVR literature (Milionis, Moallemi, Roughgarden, Zhang) motivates charging arbitrageurs more. Keying the fee on time-since-last- trade rather than on a price signal is the part that is new here, and it is what removes the oracle. Limitation, stated plainly: on a pool that trades continuously the surcharge is near zero, so this hook does nothing for a busy major pair. It is aimed at the long tail, where pools are quiet for minutes or hours at a time and the arbitrage on the first trade back is the whole of the LP's loss.

## Prior art
Dynamic-fee hooks keyed on realized volatility or on price movement are common, and the loss-versus-rebalancing literature (Milionis, Moallemi, Roughgarden, Zhang) motivates charging arbitrageurs more. Keying the fee on time since the last trade rather than on a price signal is what is new here, and it is what removes the oracle.

## Where it does not help
On a pool that trades continuously the surcharge is near zero, so this does nothing for a busy major pair. It is aimed at the long tail, where pools sit quiet for minutes or hours and the arbitrage on the first trade back is the whole of the provider loss.

## Facts
Slug: arb-tax-decay
Contract: ArbTaxDecayHook
Callbacks: beforeSwap, afterInitialize
Parameters: baseFee (uint24), maxSurcharge (uint24), halfLife (uint32)
Dynamic fee required: yes

## Caveats
- Unaudited.
- A deployment with status "deterministic" is a mined CREATE2 address with no code at it yet. Never present one as live.