When the Market Knows the Controller
A furnace has no opinion about the temperature its thermostat is set to. It does not know the rule the thermostat uses to decide when to switch it on, and it could not act differently even if it did. It only knows what the thermostat is telling it right now, not what the thermostat will tell it later as the room warms up. The classical stability toolkit, gain and phase margins, Bode and Nyquist plots, assumes exactly this: the thing being controlled, the "plant," is unaware of the controller.
Onchain DeFi controllers, such as RAI Dollar's peg controller, operate in a different environment. They are public bytecode: every parameter is visible, every response is predictable given its inputs, and anyone can execute them in advance. The "plant" being controlled is a market full of profit-seeking actors who can use this to trade against the controller's future. The moment they do, the plant's dynamics change, and the plant can no longer be modeled on its own: the controller is now part of it.
This sounds like a flaw, even a paradox, but it is closer to the opposite. The phenomenon has a name in several fields, and two RAI Dollar scenarios show how it plays out.
The controller is part of the market process
Classical control analysis assumes the process only reacts to what the controller does. But the market also reacts to what the controller will do. A published control mechanism turns every observation of the state into a forecast of the actuation.
The same idea in other fields
Economists call this the Lucas critique (1976): behaviour measured under one policy rule does not survive a change of rule, so you cannot evaluate a new policy with dynamics estimated under the old one. Kydland and Prescott (1977) reinforced it: policy-making is "not a game against nature but a game against rational economic agents." Machine learning discovered it as performative prediction (Perdomo et al., 2020): a deployed model changes the data it will be judged on, and the stable solutions are fixed points where the policy is optimal against the world it induces. Informally, it is Soros's reflexivity.
Commitment
Each of those framings hinges on the same ingredient: commitment. Actors only price a rule they believe will be followed, and onchain controllers take that belief further than any institution can. A central bank can abandon a published rule, and markets price that option continuously. Immutable code like RAI Dollar's has no discretion to abandon. There is no governance, no pause switch, no parameter update. The rule is not a promise about future behavior; it is the future behavior. The rule is committed; its inputs, the oracles and the collateral tokens behind them, are not, and Case 1 below is a failure of an input.
Krugman's S-curve
Krugman's target zone model (1991) shows what commitment buys. A central bank credibly commits to defend an exchange-rate band, and the anticipation of the defense stabilizes the rate inside the band before any intervention occurs. The equilibrium price traces an S-curve (the "honeymoon effect"), flattening as it approaches the edges ("smooth pasting"). The fundamental does reach the edges and the bank does intervene there, but only in the smallest amounts needed to turn it back; inside the band the commitment does all the work. The model fit the European Monetary System poorly for a telling reason: central-bank commitment was never fully credible, and devaluation risk reverses the honeymoon (Bertola and Svensson 1993). An immutable rule has no devaluation option to price.
So what does the market do with a controller it can read perfectly? Two scenarios from RAI Dollar's Peg Control simulator put the question sharply: does anticipation eat itself, or does Krugman's S-curve hold?
Case 1: buying when collateral impairment causes under-peg
Suppose an asset worth 5% of the redemption basket fails silently: the oracle still reports a price, but the market will not clear at it. A loss of redemption backing can cause a pronounced stablecoin depeg. RD redeems for collateral at the oracle's price, so the hole lands in the redemption floor, and because the oracle is still reporting, the oracle-failure shutdown never triggers and the impairment persists (the simulator's "silent failure" scenario).
The stress state
The redemption price floor gaps down, and coincident selling can drive the price onto it, which maxes out the rate, possibly to no effect (yield paid in an impaired coin cannot restore its backing). RAI Dollar has a lever no other $1 stablecoin has: moving par. The control baton passes from the rate (the interest rate all borrowers pay, the controller's everyday lever) to par, which begins its slew-limited climb (at most $0.001 per hour) toward ≈ $1/(non-impaired basket %), re-denominating RD to move the market price back towards $1.
The market anticipates and responds
Every step of that sequence is legible in advance. The one thing the controller cannot see, the size of the hole, it does not need to: the market price carries it. Once redemption arbitrage has pinned RD to its new floor, the discount is the hole, and par's path follows from the contract's constants and that price. A rational actor reads the path and buys the under-peg before par has moved: RD at $0.94 that will trade near $1.00 within weeks is a yield anyone can compute.
Here is the apparent paradox: the buying itself lifts the price, which shrinks the controller's error, which slows the very par climb the actor was front-running. If everyone anticipates, does the trade still work? It seems to defeat itself.
Case 2: selling ahead of the over-peg arbitrage that never comes
Now the mirror image. A sustained demand surge pushes RD above the peg. RD has a similar price ceiling to Liquity's, par × MCR (as low as 110%), where a riskless trade opens: lock collateral at the minimum ratio, mint RD, sell it above the collateral's value, and walk away from the trove. That is a hard ceiling; moving par handles the milder over-pegs below it. The controller's predictable response: rate cuts to its floor, the baton passes, and par steps down, dragging the walk-away ceiling down with it until the price comes home.
An actor holding RD reads this and sells early: why ride par down when you can exit at the premium? But again the paradox: if everyone sells early, the price never reaches par × MCR, par never needs to fall that far, and the "guaranteed" walk-away arbitrage never materializes. The would-be minter who waited for the riskless trade finds it was competed out of existence by the holders who didn't wait.
The market-controller relationship is a fixed point
The anticipation is not defeating the controller. It is performing the controller's work ahead of schedule, and the equilibrium is the point where the remaining profit exactly compensates the marginal actor.
Notice the structure both cases share. Buying the under-peg makes the next unit of under-peg buying less profitable (the price is higher, the remaining par climb smaller). Selling the over-peg makes the next unit of selling less profitable. In game-theoretic terms, trades against RAI Dollar's moving par are strategic substitutes: each actor's move shrinks the others' payoff from the same move.
Strategic substitutes vs complements
If an action taken by others decreases your incentive to take it, that action is a strategic substitute (Bulow, Geanakoplos and Klemperer 1985).
If an action taken by others increases your incentive to take it, that action is a strategic complement.
Games of strategic substitutes
Games of strategic substitutes have self-limiting equilibria: activity ramps until the marginal action has no payoff, then stops. No strategy dominates, since your payoff depends on how many others act, but there is a clean threshold rule: act while the expected drift at the current price still clears your required return over the holding period.
In the price-floor case: say your profit hurdle is 20% a year, and the controller's preview says par's climb brings RD up to $1.00 at market in about three weeks. At $0.94, the six cents of drift left on the table is over 100% a year: buy. Everyone reading the same preview buys too, and the price lifts. At $0.97, three cents is still about 50% a year: keep buying. Near $0.99, one cent over three weeks is roughly 20% a year, your hurdle: stop. Buyers with a lower hurdle or a longer horizon carry it a little further, and the price settles where the last willing buyer's hurdle is just met. The three weeks is itself part of the fixed point: buying slows par, which stretches the horizon, which lowers the return.
Games of strategic complements
Games of strategic complements have self-reinforcing, all-or-nothing equilibria: each actor's move raises the next actor's incentive to make the same move, so activity either fizzles out or cascades to a corner where everyone has acted. There is no interior resting point and no computable margin to stand on; which corner the game lands in is chosen by beliefs about what others will do.
Fiat stablecoin runs are games of strategic complements
Contrast the game feared in fiat stablecoins: the run. A run worsens the position of everyone who stays, in one of two ways. If the reserve is fractional or must be fire-sold, exits consume the backing behind the remaining claims, and whoever moves last recovers least. If the reserve is whole but opaque, the harm is informational: most holders cannot redeem, the market price is the best public evidence about the backing, and the discount your exit deepens pushes the next holder toward the exit. Either way, every exit moves the next holder closer to selling, and enough selling tips it.
Complementarity does not mean a holder should always sell into a depeg; if few others run and the reserves are whole, staying or buying the discount is the better response, which is why most fiat depegs snap back. It means the incentive to sell grows with the selling, so the staying equilibrium rests on beliefs about the reserves and about other holders, and enough selling can tip it with no change in fundamentals. That is what produces two equilibria over the same fundamentals, everyone stays or everyone runs: the bank runs of Diamond and Dybvig (1983) and the currency attacks of Obstfeld (1996). Substitutes do the opposite: each arbitrageur's trade eats the next one's profit, so activity is self-limiting and, inside the regime described below, the equilibrium is unique. One structure amplifies crowds; the other absorbs them.
RAI Dollar changes the game to strategic substitutes
A feedback controller of the redemption price, which reads backing value from a continuous public market, converts a stablecoin peg from a coordination game into an arbitrage game. Holding through a dip is backed by a computable par trajectory rather than by hoping others don't run, and a holder who cannot price the backing can free-ride on the arbitrageurs who can. That, not crypto versus fiat, is the dividing line: USDC's recovery in March 2023 was arbitrage too, once the exposure was known. A retail holder of an opaque coin can neither price the reserves nor redeem against them, so holding through its depeg is a bet that the reserves are whole and that others will not test them first. The conversion from complements to substitutes is the deep stability property.
Both RAI Dollar cases mirror Krugman's target zone. An over-peg price that moves par down is an intervention threat that, in equilibrium, disciplines the price without firing. The redemption floor, with the par climb behind it, commits to a similar threat. These committed rules (market versus controller) make the traders' game (actor versus actor) a substitutes game, because they give every deviation a computable expiry date.
Where the substitutes regime ends
The substitutes property holds while three conditions do.
Redemption throughput. The floor is enforced only as fast as redemptions clear, about 1% of debt at once and 4% per day. A larger run queues, and while it queues the price can sit below the floor. This is the binding constraint in Case 1.
Par capacity. Par is capped at $1.30, so re-denomination can cover an impairment of at most about 23%. Past that the defense is known to fall short, and foresight brings the attack forward instead of deferring it (Krugman 1979).
DEX depth. The DEX quotes the impaired asset's marginal price, not the average recovery, so each redeemer's sale lowers it for the next. The effect is bounded by the slice's weight, 5% here. Where an impaired asset has no usable DEX price at all, holders are back to noisy private signals (Morris and Shin 1998).
Who takes the loss
Inside the regime, someone still pays. Re-denomination moves the hole rather than filling it: raising par by Δ raises every trove's dollar debt by Δ, so borrowers carry the loss, and the thinnest troves, those under (1 + Δ) times their minimum ratio, are liquidated. Because a slow par climb liquidates them just under the minimum, their collateral still exceeds their debt, so the Stability Pool or the surviving troves that absorb them gain rather than lose. There is no cascade without a collateral crash, which is a separate risk.
Fixed point formulation
Let p(t) be the market price, Par(p(·)) the par path the published controller will generate given that price path (computable from the on-chain constants and the market price), and r a trader's hurdle rate.
A holder earns the price's drift, and par drives it: on the floor the price moves with par, scaled by the recovered fraction and the redemption fees; above it, toward where par is taking the market. Buying under the peg pays while that expected drift, as a return on the price paid, beats the hurdle:
E[ dp/dt ] / p(t) > r
Each purchase raises p(t), which shrinks the left side twice: the smaller error shrinks the par drift the controller will generate, and whatever drift remains is a smaller return on a higher price. Buying stops where the marginal trader is indifferent:
E[ dp/dt ] / p(t) = r
with the price ending where par takes it: about $1.00 at market, a percent less at redemption after fees. Traders differ in r, so the hurdle that clears rises with the capital already deployed.
The over-peg case is the mirror: the drift is negative, and holders sell until the remaining step-down is not worth front-running.
It is a fixed point rather than a formula because the drift on the left depends on the price path the traders' own buying produces: the equilibrium p(t) is the path that, run through the controller, yields exactly the drift that makes the marginal trader indifferent. Economists call this a rational-expectations equilibrium. Substitutes alone do not make it unique; that the payoff to buying falls strictly as total buying rises does, inside the regime described above and within the envelope simulated. The more anticipating capital there is, the less the controller actually actuates. Anticipation substitutes for actuation.
Carry is omitted. RD deployed in a Stability Pool or LP position earns yield while waiting (idle RD earns nothing), which adds to the left side. It is left out to keep the focus on par.
Watch it happen
The simulator below runs the exact RateParControl port from the Baton Pass against the same agent-based market, except that a share α of the market's capital now does what a profit actor would do: every few hours it re-runs the controller's own preview over a lookahead window, ten days by default (onchain, nextParWithPrice() in a loop), with the price held at its current value, and trades against where par and the arbitrage boundaries are headed. These agents are not solving the fixed point; they re-project every four hours and let their own trades move the price, and the chart shows that adaptive loop converging. It does converge: across sweeps of α and of the anticipators' gains, the late-run price settles in every case (the market's anchoring strength was not varied). Drag α and the lookahead and watch both paradoxes resolve in real time. The full agent specification is on the standalone simulator.
Things to notice:
In the impairment scenario, the crash gets shallower as α rises, but par's climb barely changes. Anticipators buying the under-peg cushion the panic (the crisis low lifts by nearly three cents, and time spent deep under peg collapses), yet par ends up within about half a cent of the same place: it starts about half a day later and climbs a little more gently, because the buying shrank the error that feeds it, but the destination is set by the hole, not the panic. A 5% hole in the redemption basket is a real loss, and no amount of anticipation can fake solvency. Anticipation smooths the path; it cannot move the destination. With a permanent hole that destination is a band rather than a point: once the price is back near $1 the controller hands back to the rate, par leaks toward $1.00, the error reappears and par re-engages, a slow cycle that holds the price within about a cent of the peg.
In the over-peg scenario, the ceiling recedes as the market reaches for it. At α = 0 (no anticipation) the price runs up toward par × MCR, par's step-down drags the ceiling down onto it, and the price pins there until par brings the ceiling home: the walk-away arbitrage actually fires. At high α the price never gets within two cents of the ceiling, so the walk-away trade never executes; par still steps down against the surge, just less. A crawling band, live: anticipatory selling holds the price well inside a ceiling that is itself coming down. The most important trade in the scenario is the one that never happens.
What this means for designing open DeFi controllers
If the market is going to run your controller forward and trade against it, design for it. RAI Dollar design choices are what the game-theoretic frame prescribes.
Immutability is the commitment device that makes anticipation rational. The honeymoon effect rests on credibility: Krugman's band only stabilizes if the defense is believed. A governance vote that could soften the controller is an option the market must price, and every basis point of it weakens the pre-emptive flows. Removing discretion converts "the controller will probably do X" into "X is a property of the chain"; pre-emptive stabilization is the dividend.
Slew limits make the trajectory legible, and legibility is stabilizing. Par moves at most $0.001 per hour. In a classical frame that is damping; in the strategic frame it is information: a bounded trajectory is one the market can price with confidence, which is what lets anticipation do the controller's work. Stabilization then arrives at the speed of expectation rather than actuation: the market moves in hours what par takes weeks to do (its three-week pace in Case 1 comes from the controller's gain, not the cap, which would allow it in two days).
The metric of success is less clear. In the classical frame, a controller that barely actuates looks lazy. In the strategic frame it is working perfectly: the rate that never pins, the par that never moves, the walk-away arbitrage that never fires are the successes. The measurable version is counterfactual: cumulative deviation and time under peg against the α = 0 baseline, which the simulator reports.
RAI Dollar is built for an onchain market
How a published controller behaves when the market can read it is an old question that has never had a clean setting. Immutable mechanisms and onchain markets now provide one: total commitment, perfect observability, permissionless response, and straightforward incentives. RAI Dollar continues the tradition of the original RAI: permissionless onchain control of par as the monetary policy of a stablecoin.
References
- Lucas, R.E. (1976). "Econometric Policy Evaluation: A Critique." Carnegie-Rochester Conference Series on Public Policy 1, 19–46.
- Kydland, F.E. & Prescott, E.C. (1977). "Rules Rather than Discretion: The Inconsistency of Optimal Plans." Journal of Political Economy 85(3), 473–491.
- Krugman, P.R. (1991). "Target Zones and Exchange Rate Dynamics." Quarterly Journal of Economics 106(3), 669–682.
- Morris, S. & Shin, H.S. (1998). "Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks." American Economic Review 88(3), 587–597.
- Perdomo, J.C., Zrnic, T., Mendler-Dünner, C. & Hardt, M. (2020). "Performative Prediction." ICML 2020.
- Krugman, P. (1979). "A Model of Balance-of-Payments Crises." Journal of Money, Credit and Banking 11(3), 311–325.
- Diamond, D.W. & Dybvig, P.H. (1983). "Bank Runs, Deposit Insurance, and Liquidity." Journal of Political Economy 91(3), 401–419.
- Bulow, J.I., Geanakoplos, J.D. & Klemperer, P.D. (1985). "Multimarket Oligopoly: Strategic Substitutes and Complements." Journal of Political Economy 93(3), 488–511.
- Bertola, G. & Svensson, L.E.O. (1993). "Stochastic Devaluation Risk and the Empirical Fit of Target-Zone Models." Review of Economic Studies 60(3), 689–712.
- Obstfeld, M. (1996). "Models of Currency Crises with Self-Fulfilling Features." European Economic Review 40(3–5), 1037–1047.
This post was constructed with the help of Anthropic's Claude Fable 5: the literature survey, the agent-based simulator embedded above, and the traced case figures were built collaboratively with it. The mechanism claims were checked against RateParControl.sol, and the citations verified, before publication.