kris paruch

mechanism design live publication

Economic Games as Estimators

Recasts token bonding curves as signal-processing systems: the spot price is an estimator, trades are noisy samples of agents' private valuations, and mechanism design becomes a question of estimation error bounds.

MARBLE 2020 (Springer) From Curved Bonding to Configuration Spaces — companion paper

What the paper argues

Every economic mechanism built on a shared ledger is, formally, a discrete event game: a system whose state changes only in response to discrete actions taken by agents, without requiring the designer to know the agents' objectives or decision rules. This paper's contribution is to notice that such a system can also be read as an estimator. Agents act on their own private preferences; those actions leave observable traces in the system state; and a well-designed mechanism turns the aggregate of those traces into a running estimate of something no single agent could report directly — most concretely, the market price of a token.

The paper builds the formal machinery for this claim, then applies it to the token bonding curve from From Curved Bonding to Configuration Spaces , showing that its spot price is exactly this kind of estimator — and testing it numerically against a set of standard control-theory input signals to characterize how quickly and accurately it tracks a moving target.

A game as a signal-processing system

The state transition is the same discrete-event structure as the companion paper — xt+1=f(xt,ut)x_{t+1} = f(x_t, u_t) — but this paper adds an explicit measurement layer. An agent's action produces an observable signal hatythat{y}_t, a noisy sample of that agent's private valuation:

y^t=h(xt,ut)\hat{y}_t = h(x_t, u_t)

The system-level conservation law VV forces the state trajectory onto a configuration space, and within that space a mapping GG produces an estimate yty_t of the same underlying quantity —

yt=G(xt)y_t = G(x_t)

with an estimation error ete_t defined as the gap between the two:

et=e(yt,y^t):=∥yt−y^t∥e_t = e(y_t, \hat{y}_t) := \| y_t - \hat{y}_t \|

This is the paper's central move: treat the conservation function not just as a constraint that keeps the system well-behaved (its role in the companion paper), but as the mechanism that produces a running estimate — the network itself becomes a distributed sensor.

The bonding curve as a price estimator

Applied to the same augmented-bonding-curve invariant as before,

V(Rt,St):=StκRt≡V0V(R_t, S_t) := \frac{S_t^\kappa}{R_t} \equiv V_0

the spot price is exactly the estimator function

Pt=G(xt)=κRtStP_t = G(x_t) = \kappa \frac{R_t}{S_t}

while any individual trade produces a realized price — the ratio of reserve moved to supply minted or burned,

P^t=ΔRΔS\hat{P}_t = \frac{\Delta R}{\Delta S}

which behaves like a noisy sample of the trading agent's private valuation. Because the curve's geometry guarantees that a realized price above the spot price always calls for bonding and a realized price below it always calls for burning, every trade nudges the spot price in the direction its own realized price implies — the estimator and the samples it is built from can never disagree in sign. That directional alignment is what lets the paper state a formal error bound between an agent's true private valuation and the system's running estimate of it:

∥ga(xt,εt)−G(xt)∥=∥pa,t−Pt∥≤ξ∥P^t−Pt∥    ∀a,∀t\| g_a(x_t,\varepsilon_t) - G(x_t) \| = \| p_{a,t} - P_t \| \le \xi \| \hat{P}_t - P_t \| \;\; \forall a, \forall t

with minimal assumptions about how any individual agent actually decides to trade.

Testing it as a signal-processing system

The paper then treats the bonding curve as a literal control-theory plant and drives it with four standard test signals — a square wave, a triangle wave, a sine wave, and a random walk — to see how the spot price tracks each one. The square-wave response looks like a high-gain proportional controller: fast convergence with overshoot. The triangle wave shows steady-state tracking error during the ramp. The sine wave tracks most closely at its peaks and troughs. The random walk is tracked with bounded error that scales with the walk's own volatility, rather than accumulating over time — evidence that the curve behaves like a stable estimator rather than one prone to drift.

Where this connects to what's on this site

The discrete-event-game and configuration-space machinery here is inherited directly from From Curved Bonding to Configuration Spaces : that paper proves the bonding curve's mechanism guarantees; this one asks what those guarantees are good for — treating the price they produce as a signal worth trusting, with a stated error bound rather than an assertion.

References