Liquidity provider math

LPs sit on the same hypersphere as everyone else. They earn fees, take the AMM's directional risk, and cash out cleanly at resolution. Here's the math behind each LP operation.

Adding liquidity

A new LP deposits d units of collateral. The protocol scales all reserves up proportionally, which preserves the implied probability distribution:

h'ⱼ = hⱼ · (k + d) / k
k' = k + d

Because reserves scale uniformly, xⱼ = k − hⱼ also scales by the same factor, and so does xⱼ². The implied probabilities p̂ⱼ = xⱼ²/k² are unchanged - the market doesn't suddenly shift toward any outcome when liquidity is added.

In return, the LP receives s shares of the LP-share token, where:

s = d · L_total / k_before

(L_total is the total LP shares outstanding, k_before is the AMM norm immediately before the deposit.)

Removing liquidity (active market)

Before resolution, an LP can burn s shares to withdraw their share of the pool:

out = k · s / L_total

This is the LP's pro-rata share of the current AMM norm. Deposits before the trade activity earn fee accruals; deposits after earn from new trades onward.

Removing liquidity (resolved market)

Once the market has resolved with bin i_win as the outcome, the AMM is simpler. The vault holds h_{i_win} units of collateral that didn't go to winning trader claims, plus accumulated fees. An LP withdraws:

out = h_{i_win} · s / L_total + fee_share

The first term is the LP's slice of the leftover reserves on the winning outcome (winning-side tokens that no one claimed, weighted by LP share). The second term is the LP's accumulated portion of trade fees collected during the market's life.

Why baseline LP loss is 0%

A naive reading of the AMM math suggests LPs should lose by default - every trade pushes the AMM state away from neutral, and the AMM's holdings end up concentrated against the trader's position. In typical CFAMMs (e.g., Uniswap), this manifests as impermanent loss: a constant, unavoidable cost to LPs even at zero trade volume.

DekantPM avoids this. The reason:

LP collateral residual is computed from actual trader token holdings, not from AMM state alone.

Concretely, when the market resolves to bin i_win:

Each trader who holds tokens_{i_win} tokens of the winning bin redeems them at $1 each.
The vault then has h_{i_win} − Σ trader holdings of bin i_win left over.
That leftover plus accumulated fees is the LP pool.

If no one ever traded, no one holds any tokens at resolution, the vault has its full original collateral L, and LPs withdraw exactly what they deposited. Zero baseline loss.

The non-zero LP risk is adverse selection: informed traders identify when implied probabilities are stale and take the AMM's other side. This is the same risk every CFAMM faces. LPs are compensated by the 50% share of trade fees.

Break-even volume

For an LP to be net-profitable, fee revenue must exceed adverse-selection losses. As a rough order-of-magnitude:

break_even_volume_per_day ≈ adverse_selection_losses / (0.0015)

(0.3% trade fee × 50% LP share = 15 bps per trade goes to LPs.)

If the AMM is being arbed by L_total worth of informed flow per day, LPs need approximately L_total / 0.0015 ≈ 666·L_total daily trade volume to break even. In practice, healthy markets sit well above this - the math here is just for intuition.

What you don't get as an LP

	Why
No vote on resolution	Oracles are role-gated PDAs. LPs are not oracles.
No fee changes	Fee schedule is set once by superadmin, capped at 50%.
No directional position	Adding liquidity doesn't lean toward any outcome. To lean, place a separate trade.

If you want both fee income and a directional view, hold an LP position and a trader position separately. They live in different on-chain accounts and compose freely.

When to add or remove

You should add when	You should remove when
Healthy trade volume relative to AMM size	Approaching resolution and you'd rather lock in fees
You want passive exposure to fee flow	Implied probabilities have moved sharply away from your entry
You've LP'd before and accept the AS exposure	You need the capital elsewhere

The protocol enforces no minimum holding period. Adds and removes are atomic transactions.