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From Curved Bonding to Configuration Spaces

A framework generalizing token bonding curves to configuration spaces — proper subsets of a system's state space carved out by conservation laws — applied to the augmented bonding curve design for community funding.

IEEE ICBC 2020 (repec) Economic Games as Estimators — companion paper

What the paper argues

A bonding curve is usually described informally — a formula relating a token's price to its supply. This paper asks a more structural question: what does a bonding curve guarantee, independent of how any individual participant behaves? The answer is a configuration space — the proper subset of a system's full state space that remains reachable once a designed invariant is enforced on every state transition. Any property true everywhere in that subset is true for every possible sequence of agent actions, without the designer having to assume anything about what agents want or how they decide.

This reframes mechanism design as a dimensionality-reduction problem: pick the conservation function, and the reachable state space shrinks to a lower-dimensional manifold that guarantees the property by construction — never by hoping participants behave a certain way. The paper builds the general framework, then applies it in full to one concrete design: the augmented bonding curve, a funding mechanism for internet-native communities without a shared legal or geographic jurisdiction.

States, mechanisms, and conservation

The system is modeled as a discrete-time state machine: a global state xx evolves under a mechanism ff in response to an agent's action uu,

x0:=f(x,u)x_0 := f(x, u)

A conservation function V:XomathbbRV: X o mathbb{R} encodes a desired global property as a scalar. If VV is unchanged by every mechanism the system supports,

V(f(u,x))≡V(x)    ∀f∈F,∀u∈UV(f(u,x)) \equiv V(x) \;\; \forall f \in F, \forall u \in U

then VV is invariant along every reachable trajectory — the paper works this through first with a toy consumer-indifference example (an indifference curve is exactly this kind of restriction) before applying it to a real funding mechanism.

The augmented bonding curve

The augmented bonding curve binds a community's stakeholdership token to governance rights over a shared funding pool. Its state is a four-tuple — Reserve RR, Supply SS, spot Price PP, and Funding Pool FF — restricted to a configuration space by a single conservation function,

V(R,S):=SκR=V0V(R,S) := \frac{S^\kappa}{R} = V_0

where curvature kappakappa sets how sharply returns diminish. This single invariant pins down the spot price as the slope of the invariant's graph,

P=p(R,S):=−∂V/∂S∂V/∂R=κRSP = p(R,S) := -\frac{\partial V/\partial S}{\partial V/\partial R} = \frac{\kappa R}{S}

and the paper proves the two mechanisms that respect it. Bond-to-mint deposits reserve currency rr and issues new supply along the curve,

S+=V0(R+r)κ,P+=κ(R+r)(κ−1)/κV01/κS^+ = \sqrt[\kappa]{V_0(R+r)}, \qquad P^+ = \frac{\kappa (R+r)^{(\kappa-1)/\kappa}}{V_0^{1/\kappa}}

burn-to-withdraw destroys supply ss to recover reserve, symmetrically (net of an exit tax that funds the community pool),

R+=(S−s)κV0,P+=κ(S−s)κ−1V0R^+ = \frac{(S-s)^\kappa}{V_0}, \qquad P^+ = \frac{\kappa(S-s)^{\kappa-1}}{V_0}

and two lemmas — proved via L'Hôpital's rule on the realized-price expression — establish that the realized price of any bond or burn always sits on the correct side of the current spot price,

Pˉ(r)=rV0(R+r)κ−S\bar{P}(r) = \frac{r}{\sqrt[\kappa]{V_0(R+r)} - S}

converging to the spot price exactly as the transaction size shrinks to zero. That's the diminishing-returns guarantee stated formally: no participant can move the price in their favor for free, and the guarantee holds for every possible sequence of trades, not just well-behaved ones.

Hatching and vesting

Launching — "hatching" — an augmented bonding curve fixes its initial reserve, supply, and the fraction of raised funds that goes straight to the community pool rather than the reserve. The paper derives the resulting return multiple for early participants in closed form,

ρ=P0Phatch=κ⋅(1−θ)\rho = \frac{P_0}{P_{hatch}} = \kappa \cdot (1-\theta)

Because ho>1 ho > 1 for most reasonable parameter choices, early participants have a structural incentive to dump the moment their tokens are liquid — so the paper recommends a half-life vesting schedule rather than a hard unlock date, releasing tokens smoothly to avoid the vesting-cliff shock:

Svested=(1−2−γ(k−k0))ShatchS_{vested} = \left(1 - 2^{-\gamma(k-k_0)}\right) S_{hatch}

Numerical experiments sweep kappakappa and heta heta (the funding-pool share) to show how each parameter trades early-hatcher return against long-run community sustainability, and confirm the slippage and exit-tax dynamics predicted by the mechanism proofs.

Where this connects to what's on this site

This paper is the formal groundwork behind Economic Games as Estimators , which reinterprets the same bonding-curve mechanism as a price estimator rather than just a funding rule — using the configuration space defined here as the setting in which that estimation argument is made precise.

References