How PAMM Works

Deep dive into PNP’s Virtual Liquidity CPMM — virtual reserves, constant product math, buying and selling mechanics, price dynamics, and edge cases.

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How PAMM Works

PAMM runs a Uniswap V2-style constant product formula on virtual reserves — phantom pool balances derived from on-chain state. This page explains the mechanics from first principles.

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The Three Numbers

At any moment, a PNP market is fully described by three values:

Value	What it is
R (Reserve)	Total collateral locked in the contract. Real dollars.
S_YES (YES Supply)	Total YES tokens in circulation, held by traders.
S_NO (NO Supply)	Total NO tokens in circulation, held by traders.

From these three numbers, we derive everything: prices, trade outputs, slippage, liquidity depth.

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Virtual Reserves

The core insight. We define two virtual reserves that we use to run constant product math:

V_YES = R - S_YES
V_NO  = R - S_NO

Think of V_YES as “how much room is left before YES tokens saturate the reserve.” If nobody has bought YES yet (S_YES = 0), then V_YES = R — the full reserve is available as virtual liquidity for the YES side.

As traders buy YES tokens, S_YES grows and V_YES shrinks. The YES side gets “drained,” making YES more expensive. This is exactly how a Uniswap pool behaves when you drain one side.

Why “virtual”? No one deposited YES or NO tokens into a pool. The pool is a mathematical fiction derived from the reserve and circulating supplies — but it behaves identically to a real constant product AMM.

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Spot Prices

The price of the next infinitesimally small unit:

price_YES = V_NO / (V_YES + V_NO)
price_NO  = V_YES / (V_YES + V_NO)

These always sum to exactly 1.00. This is the fundamental property of a well-formed binary prediction market — probabilities are exhaustive.

Fresh market (no trades yet):

V_YES = R,  V_NO = R

price_YES = R / (R + R) = 0.50
price_NO  = R / (R + R) = 0.50

Every market starts at 50/50. The initial reserve determines liquidity depth, not starting price.

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Buying Tokens

When a trader pays collateral C to buy YES tokens, the mechanism uses a “complete set decomposition”:

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Step 1: Implicit Minting

In any binary market, 1 YES + 1 NO = 1 unit of collateral (a “complete set”). When the trader pays C:

• Conceptually, this mints C YES and C NO tokens
• The trader keeps the C YES tokens
• The C NO tokens are “sold” back to the virtual pool

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Step 2: The Swap

The implicit NO tokens get swapped for additional YES using constant product:

additional_YES = (C * V_YES) / (V_NO + C)

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Step 3: Total Output

total_YES_received = C + additional_YES
                   = C + (C * V_YES) / (V_NO + C)

The trader always receives more tokens than collateral paid (in units). How many extra depends on the current price.

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State Changes After a Buy

• R increases by C (money entered the system)
• S_YES increases by total_YES_received
• S_NO stays the same (implicit NO tokens absorbed into virtual pool)
• Prices adjust: YES becomes more expensive, NO becomes cheaper

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Selling Tokens

Selling is the reverse but mathematically trickier. A trader holds T YES tokens and wants collateral back.

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The Problem

To redeem collateral, you need a complete set (equal YES + NO). But the trader only has YES. So:

1. Sell x YES tokens to the pool for NO tokens
2. Pair remaining (T - x) YES with the NO received
3. Burn the complete sets for collateral

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The Quadratic

Combining the swap formula with the pairing constraint gives:

x^2 + (V_YES + V_NO - T) * x - T * V_YES = 0

The positive root gives x (how many to swap), and T - x gives the collateral released.

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Round-Trip Property

If a trader buys YES and immediately sells back the exact tokens (ignoring fees), they get back approximately what they paid. The curve is path-independent for small trades — it doesn’t “leak” value.

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Liquidity Depth

The initial reserve R_0 determines how much money it takes to move the price by a given amount.

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Small Reserve (R = 100 USDC)

10 USDC YES buy → price moves 50% → 54.75% (+4.75pp)

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Large Reserve (R = 1,000,000 USDC)

10 USDC YES buy → price moves 50% → ~50.0005% (negligible)

Market Size	Reserve	Suitable For
Micro	< 1,000	Casual bets, highly volatile
Small	1,000 - 10,000	Niche markets, moderate impact
Medium	10,000 - 100,000	Popular events, good trading experience
Large	100,000 - 1,000,000	High-profile markets, institutional depth

Deep liquidity = stable prices. The creator’s initial deposit directly controls this.

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Price Dynamics

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What Moves Prices

YES price increases when:

• Someone buys YES (increases S_YES, decreases V_YES)

YES price decreases when:

• Someone buys NO (increases S_NO, decreases V_NO)
• Someone sells YES (decreases both R and S_YES, net effect raises V_YES relative to V_NO)

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Slippage

The displayed price is the marginal price — what the next tiny unit costs. Larger trades experience slippage because each unit purchased moves the price before the next unit is priced.

Slippage increases with:

• Larger trade sizes
• Smaller reserves
• Prices near 0 or 1 (one virtual reserve is thin)

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Self-Correcting at Extremes

At 50/50 the market is equally liquid on both sides. At 90/10:

• Buying more YES is expensive and high-impact (thin V_YES)
• Buying NO is cheap and low-impact (thick V_NO)

This is desirable. Markets near consensus should be hard to push further and easy to push back toward center.

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Edge Cases

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Price Bounds

Prices can never reach exactly 0 or 1 while the market is active. As V_YES approaches zero, the price asymptotically approaches 1 but never reaches it. The constant product formula provides natural resistance at extremes.

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One-Sided Activity

If only YES is bought and NO supply stays at zero:

V_NO = R - 0 = R  (always equals full reserve)
V_YES = R - S_YES  (shrinks as YES is bought)

This works fine. If someone finally buys NO, they get excellent prices because V_NO is large. The market doesn’t need activity on both sides to function.

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Zero Trading Activity

If a market is created, nobody trades, and it expires — the creator calls claimRemainingReserve() and gets their full deposit back. No loss, no gain.

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First-Mover Advantage

The first trader always gets the best deal — buying at 50/50 when the full virtual reserve is available. This is inherent to all AMM-based markets and is desirable: it rewards early participants who take the risk of entering an untested market.