Abstract

Can a prediction market improve an automated market maker’s prices without creating an uncontrolled source of trading profit? The forecast can convey information about an event that affects asset value, but a trader may also manipulate that forecast. We study a coupling in which the forecast changes spot prices while the prediction reward remains independent of spot trading. For added prediction payments independent of the event outcome, we characterize exactly when the reward for truthful reporting survives. We then bound reporting distortion from finite outside positions and derive exact fees and trade limits for reversible trading cycles. The required protection depends on whether fees remain in the pool and whether traders can split their orders. A sponsor account can instead fund price adjustments. We bound its net spending when trades and forecast changes restore the initial inventory and pricing rule. These constructions retain a nonzero response to the forecast. Under independent Gaussian signal errors, a bounded correction strictly reduces mean squared log-price error. A synthetic study measures its trading effects. Market usefulness and endogenous participation remain separate empirical and equilibrium questions.

The author has a commercial interest in systems of the kind this paper describes.

1 Introduction

An automated market maker (AMM) quotes trades from a rule applied to its asset and cash reserves. Suppose an asset’s value depends on whether a binary event occurs. A prediction market trades claims on that event and supplies a forecast of its probability. If spot trading is too thin to update the asset price within the current trade window, that forecast may supply useful information. For an asset with two possible terminal values, a changed event probability changes the corresponding expected value. The question is how to make the AMM respond without giving traders an uncontrolled way to extract its reserves.

The difficulty starts when one person can trade in both markets. A trader can buy the asset, raise the forecast used in its quote, and sell the asset back at the changed price. The trader may then reverse the prediction trade as well. A charge recovered on reversal does not pay for the gain on the asset cycle. A reporter who already owns the asset can also benefit from an inflated forecast without completing such a cycle. These incentives arise even when the prediction market, considered alone, rewards truthful forecasts.

We first isolate the prediction reward. A strictly proper score gives a reporter the highest expected payment for stating the reporter’s actual belief. An added payment preserves that property, for every belief, exactly when it stays constant along the states that a report can move. This characterization concerns outcome-independent additions to the score. One-way coupling achieves it by allowing the forecast to change the AMM price while keeping spot state out of the prediction reward. Separate asset positions still create outside incentives, whose effect we bound for contracts held until the event resolves.

Trading protection requires a second calculation. We bound the forecast’s effect on price and compute the exact cash transfers when a trader changes that forecast between trades. The destination of the fees matters because fees retained in the pool change the reserves used for the next quote. Order splitting also matters: protection against two complete legs can fail across a longer sequence of partial fills. We give the corresponding fee thresholds and a finite-horizon alternative that limits total reserve movement.

A further construction assigns the cost of changing the pricing rule to a sponsor. At a fixed inventory, the sponsor transfers the cash required by the new rule into or out of the pool. Closed inventory and control sequences can then consume only the admitted sponsor balance. This construction keeps a nonzero response to the forecast, including coupling values below one. Its statistical value is conditional: a bounded correction reduces estimation error under the specified Gaussian signal law. The synthetic evaluation measures the resulting price error and trading outcomes under that law.

Proper scoring rules and cost-function prediction markets provide the base elicitation mechanism [1, 2, 3]. The failure caused by payoff feedback is related to decision markets and outside incentives [4, 5, 6]. The spot venue is a constant-function market maker of the type analysed by Angeris and co-authors [7, 8]. The present question is the composition of the two mechanisms.

This boundary follows the distinction between trade processing and settlement in the Principles for Financial Market Infrastructures [9]. Du’s payment-rail analysis also separates asset transfer from the liquidity, compliance, dispute, and service functions of a complete payment product [10, 11].

Event-Collect BFT defines the event-atomic ledger transition that fixes the trade window before plaintext is visible [12]. Parlay Identification of Ising Couplings in Correlated Binary Event Markets studies pairwise prediction inputs and the observations needed to identify them [13].

2 When the Prediction Reward Remains Truthful

A report can change both the prediction reward and the price of the underlying asset. We first hold the AMM reserves fixed and study only the prediction reward. The reporter may also be an asset trader; outside positions enter separately below.

Let Y\in\{0,1\} be an event, p\in(0,1) a trader’s belief, and r\in(0,1) the trader’s report. A scoring rule S(r,Y) is strictly proper when \Phi_p(r):=pS(r,1)+(1-p)S(r,0) has the unique maximum r=p for every p. We assume that \Phi_p is continuously differentiable in r.

Write R_{\!M} for the AMM’s cash reserve and R_{\!B} for its asset reserve. Cash uses a common unit of account, called the numeraire. The marginal price is the cash required per asset unit for an infinitesimal purchase. At fixed reserves (R_{\!M},R_{\!B}), let G be a positive continuously differentiable coupling factor. It sets the AMM marginal price P^c(r)=G(r)\frac{R_{\!M}}{R_{\!B}}. \tag{1} An outcome-independent, continuously differentiable transfer H(r,P) may also enter the prediction payoff. The report therefore traces the coupled path r\longmapsto \bigl(r,P^c(r)\bigr), \qquad \gamma(r):=H\bigl(r,P^c(r)\bigr). The scalar function \gamma is the added prediction payoff along that path. Only this added payment must be constant; the proper score still depends on the report and the eventual outcome. The theorem identifies when the report can affect price without changing the score’s preferred report.

The theorem concerns an outcome-independent transfer. Outcome-dependent changes to a scoring rule require the full proper-scoring-rule characterization and are not covered by this result.

This example explains why bidirectional dependence alone is not the relevant test. The relevant test is variation along the path that the report can move.

3 Choosing a Price Response from Noisy Signals

A truthful prediction reward still leaves the size of the AMM’s response to be chosen. The response should depend on how much information the forecast adds to the spot observation. A logit factor follows from a model in which both observations measure the same underlying log value with independent Gaussian errors. Log value means the natural logarithm of asset value.

Let v be latent log value. Suppose the spot venue and the prediction market provide independent unbiased Gaussian observations X_A=v+\varepsilon_A, \qquad X_P=a+s u=v+\varepsilon_P, \tag{4} with variances \sigma_A^2 and \sigma_P^2. Here u=\mathop{\mathrm{logit}}p. The constants a and s specify the intercept and slope that convert the logit observation into log-value units. The affine relation v=a+s u describes the noiseless calibration, not a universal valuation law. In the observation model, a+s u includes the error \varepsilon_P shown in (4). The variance of each error measures its uncertainty. The next result weights the observations to minimize the variance among unbiased linear estimates.

The full estimator is e^{\hat v}=e^{\kappa a}\,e^{(1-\kappa)X_A} \left(\frac{p}{1-p}\right)^{\kappa s}. The weight \kappa increases as the spot observation becomes less precise relative to the prediction observation. The full estimate changes both observation weights. Multiplying an unchanged spot price by the prediction-side factor is a separate overlay. It does not inherit the minimum-variance claim.

For a binary asset, the event may determine one of two values directly. The next proposition uses their probability-weighted value and measures its local sensitivity to the forecast. An operating probability is the point at which this linear approximation is fixed.

This is a fixed-operating-point extended-Kalman map in one dimension [14, 15]. Its interpretation is local. It does not establish global optimality for a nonlinear value function, a clipped factor, or an overlay on a different base price.

4 Bounded One-Way Coupling

The logit factor in (6) is unbounded as p approaches zero or one. An extreme report could therefore move the quote by an arbitrarily large factor. The following clipper keeps the central response and limits the price effect of extreme reports.

Write u for the logit of the signal supplied to the AMM. When that signal is the report r, the earlier factor is G(r)=g(\mathop{\mathrm{logit}}r). At truthful reporting it instead reads u=\mathop{\mathrm{logit}}p. A factor of one gives the reserve-ratio price; a factor below or above one lowers or raises that price at fixed reserves. The centre u_0 specifies the neutral signal, \delta_{\max} caps the logit adjustment, and w sets the width of its smooth transition. The parameter \alpha converts that bounded adjustment into a change in log price.

The product \alpha\delta_{\max} is the maximum absolute change in log price from the centred state. A comparison of opposite saturation states uses twice that amount. This distinction will determine the uniform fee threshold.

The bound applies at every later trade window because the current coupling depends on the current signal level, not on the sum of earlier changes. The reserves may change between windows. The factor itself does not compound.

Inside |u-u_0|\leq\delta_{\max}-w, the clipper is the identity. The statistical interpretation in Section 3 applies only when the same local signal model remains valid there. In the transition and saturated regions, clipping is a price bound. It is not an estimator optimality result. Without clipping, e^{\alpha u} has no finite uniform impact bound.

5 Finite Outside Positions

Clipping limits the price a report can induce, but an outside position can still reward a distorted report. The local calculation in Proposition 2.4 applies to small changes near a truthful report. We now allow finite changes and any included payoff with specified size and sensitivity bounds.

A reporting contract and a trading cycle have different economic endpoints. Here the prediction position remains outstanding until the event resolves. Reports leave the distribution of the event outcome fixed. Let b>0 be the logarithmic score scale, in units of numeraire. For belief p\in(0,1), its expected payment, up to a report-independent constant, is \Phi_p(r)=b\{p\log r+(1-p)\log(1-r)\}. Write U_p(r) for the included portfolio’s expected additional payoff at that belief. This quantity includes every report-dependent benefit charged to the modeled actor. Both terms use the same numeraire, horizon, and risk-neutral objective. Risk neutrality means that the reporter maximizes expected payment in that unit.

Two quantities describe the outside incentive. Its oscillation, \operatorname{osc}(U_p), is the least upper bound on payoff differences between any two reports. An L-Lipschitz payoff satisfies |U_p(r)-U_p(s)|\leq L|r-s|, so L bounds its change per unit change in report. The logarithmic score loss is expressed below through the binary relative entropy D(p\Vert r). It is zero at the truthful report and measures the expected logarithmic penalty for reporting r instead of p.

The proof compares the best report with truthful reporting and then bounds the slope available from the outside payoff. The first comparison controls total score loss; the second controls how far the optimum can move.

For a belief-specific bound, set \ell=L/b>0. Under the Lipschitz budget alone, the exact permitted interval is \frac{1+\ell-\sqrt{(1+\ell)^2-4\ell p}}{2\ell} \leq\widehat r\leq \frac{\ell-1+\sqrt{(1-\ell)^2+4\ell p}}{2\ell}. \tag{15} This follows by solving the two quadratic inequalities in (14). At L=0, the interval consists of p alone. An \eta-optimal report achieves an expected objective within \eta\geq0 of the maximum. For an \eta-optimal report, the same comparison gives bD(p\Vert r)\leq U_p(r)-U_p(p)+\eta. In particular, |r-p|\leq\sqrt{(M+\eta)/(2b)}. This last statement covers a solver with a certified objective error.

The two budgets can be imposed together. The next result characterizes every report attainable under some payoff that meets both budgets. Attainment permits ties: a permitted payoff may make several reports equally valuable.

To use these bounds, the contract needs one aggregate outside-payoff calculation for the reporter’s included positions. The following certificate obtains such a bound from signed asset quantities and clipped price factors. It permits long and short positions and includes every account assigned to the same beneficial owner.

The positive quantity m_j keeps r(1-r) away from zero wherever the corresponding price factor can change. It makes the probability-coordinate sensitivity bound finite.

The contract fixes the included beneficial owner, positions, payoff rules, quantities, and common valuation time before accepting the report. All accounts of that owner consume one aggregate exposure allowance. A position change requires a new bound before a later reporting decision. The certificate describes the included payoff. Realizable liquidation proceeds require their own execution model. A mark, an external derivative, and a reversible trading cycle can produce different functions U_p. The derivative bound can be tightened by maximizing the actual aggregate derivative, including genuine offsets, over the report interval.

For example, take one marked position of magnitude at most 100, with base price one. Set u_0=0, d=1, w=1/4, and \alpha=0.01. Then m>0.17, L<5.95, and score scale b=1000 gives probability distortion below 0.00149. The price ratio still ranges from e^{-0.01} to e^{0.01}. These quantities specify a synthetic contract, rather than a market calibration.

When the complete included payoff is observable and enforceable, the contract can settle its report-induced change separately. This requires funds on both sides because the adjustment can pay either party. The following proposition gives the exact cancellation and its funding requirement.

This contract preserves baseline ownership while settling the report-induced change separately. It requires a fixed observable payoff rule and enforceable transfers from both sides. The global distortion bound applies when the mechanism admits a bounded residual incentive instead. Neither construction identifies undisclosed positions or supplies external enforcement by itself. LMSR depth remains an elicitation parameter for the held-to-resolution contract. A completely reversed prediction-market cycle restores its cost-function state and requires the separate trading analysis below.

6 The Exact Round Trip

We now follow a trader who closes the asset position after changing the forecast. The relevant cost is the cash actually paid over the complete cycle. For the cost-function prediction market used here, restoring its state contributes no cost before fees. This holds even when an intermediate forecast was expensive to obtain.

Let the AMM hold numeraire reserve R_{\!M}>0 and asset reserve R_{\!B}>0. The following reserve relation specifies the executable price curve during each trade. Its exponent g is the bounded coupling factor already used in the marginal quote. At g=1, it is the constant-product reserve rule. During one leg, fix g>0 and preserve R_{\!M}R_{\!B}^{g}=k. \tag{17} Implicit differentiation gives the marginal asset price in numeraire units: P=-\frac{dR_{\!M}}{dR_{\!B}}=g\frac{R_{\!M}}{R_{\!B}}. \tag{18}

The constant k is fixed during a leg. In this construction, changing g holds the reserves fixed and selects the corresponding new k for the next leg. The funded construction in Section 8 uses a different update rule.

Suppose a trader buys \Delta_B=xR_{\!B} units, where 0<x<1, while the coupling is g_0. The trader then changes the prediction signal so that the coupling is g_1 and sells the same units back. Define \lambda=-\log(1-x)>0. The quantity \lambda measures the proportional reserve movement in logarithmic units. Logarithmic movements add when a trade is split into successive fills at the same coupling. The numeraire paid and received before fees are \begin{align*} N_{\mathrm{in}} &=R_{\!M}\left[(1-x)^{-g_0}-1\right], \tag{19} \\ N_{\mathrm{out}} &=R_{\!M}(1-x)^{-g_0}\left[1-(1-x)^{g_1}\right]. \tag{20} \end{align*}

Thus price impact alone does not charge a trader who restores the reserves at the same coupling level. The invariant returns the original numeraire reserve.

7 Fee Routing and Sequential Execution

The preceding cap concerns a complete two-leg trade under segregated fees. A venue also needs conservation across partial fills, signal changes, and repeated orders. Depositing each fee into the pool changes the cash reserve used to price the next fill. We compare that immediate retention with the segregated account above, then examine sequences with more than two trades.

The following ledger represents those events directly. Its M_t and B_t are the cash and asset reserves after step t; F_t is the separate fee balance. A tilde marks the gross reserves before the current fee is routed.

With immediate retention, the closing leg uses the fee enlarged cash reserve. This changes the sale-first result: the same threshold can now protect both complete two-leg directions at every size.

Retention changes the executable curve after every fee deposit. The segregated-fee sale cap therefore remains a separate theorem. An implementation records its routing choice before quoting either leg. Both mechanisms permit positive orders. Their two-leg statements assume that no other reserve-changing event occurs between the legs.

A trader may close one purchase through many smaller sales. Each fee deposit then changes the next execution curve, so the two-leg theorem does not cover the resulting sequence. The next theorem supplies a stronger threshold by finding a reserve quantity that never decreases at any trade.

The two-leg fee floor and the arbitrary-sequence floor solve different execution problems. For fee f>0, the latter still permits the positive signal range 0<a\leq-\log(1-f)/2. It supports repeated trading without a fixed cumulative turnover ceiling. The persistent-budget construction below offers a separate finite-horizon policy for larger signal ranges.

For the larger signal range, a venue can instead bound the total variation of log inventory over a fixed horizon. Total variation counts the absolute size of every movement, including movements that later reverse. It therefore records repeated trading even when the final inventory returns to its starting value. Pending orders must reserve that capacity before execution so that concurrent requests use the same allowance.

This bound permits arbitrary order interleaving and signal changes within the stated horizon. The reference is a constant-coupling reserve path, not a forecast of asset value. Unclosed asset inventory retains its market risk. Immediate retention uses its exact fee-deposit recurrence and the same funded reserve floors. A fee reinvestment, deposit, withdrawal, or rebalance has its own signed cash and asset entries. Such an event cannot silently restart the cumulative capacity counter.

Here v denotes a fixed external asset valuation in cash per unit, distinct from the earlier latent log value. The liquidity provider’s marked change is \Delta W_{\mathrm{LP}}=\Delta M+\Delta F+v\Delta B. Trader cash gain and this marked change are distinct measurements for an open inventory position. An empirical solvency assessment also needs the valuation path, external flows, and operational costs.

8 Funding the Control Change

The retained-fee and variation-budget constructions leave signal updates mechanically separate from the reserve curve. A different construction funds the reserve change needed to preserve a common state relation. Here a control is the pricing factor selected from the prediction signal. Changing the control changes the price available at the same asset inventory.

One might try to price every inventory and control movement by differences of a single function of the current state. Such a function is called a potential. Its charge over a sequence depends only on the endpoints, so every fully closed sequence has zero total charge. The next proposition explains why a price that responds to control conflicts with a control charge independent of inventory. Its generic coordinates x and z denote inventory and control; the funded AMM will then use its own explicit reserve coordinates.

A rectangular path changes inventory, then control, then reverses both movements. For P(x,z)=z and S(z)=0, the trader’s gain is (z_1-z_0)(x_1-x_0). This is a two-coordinate charging example with reversible transitions. Fees, restricted transitions, and explicit funding add economic state or change the charging rule. Each such addition requires its own accounting. The obstruction alone neither prices an executable AMM order nor evaluates an outside portfolio.

The sponsor account supplies the cash associated with a changed control at fixed inventory. The pool keeps one stated reserve relation across both kinds of movement. Asset trades exchange cash with traders; control updates exchange cash with the sponsor. In this construction, B_* is an anchor inventory, K is its cash reserve, and x is the logarithm of inventory relative to that anchor. The band parameter D limits the permitted distance from the anchor.

All pending control deposits reserve the same sponsor cash before execution. Concurrent requests share that account. Partial execution consumes its reservation cumulatively, and a retry retains the original identity. This is a funding rule for the mathematical mechanism. Actual deposit and withdrawal authority remains part of the account’s governing contract.

The complete economic state includes the sponsor balance. A trader can earn a positive amount while restoring (B,g) by consuming part of that balance. Restoring the entire state, including C, gives zero aggregate cash gain. For an open inventory position, cash gain must be combined with an explicit valuation of the remaining assets. The reserve band bounds each state. The cash admission rule bounds net sponsorship across repeated states. Returned cash can fund later updates, so gross transfers can exceed the initial balance.

Take B_*=K=100, g_\ell=0.8, g_h=1.2, and D=0.2. Starting from B=100 and g=0.8, buy down to B=100e^{-0.1}. An update to g=1.2 deposits 100(e^{0.12}-e^{0.08}), approximately 4.42098. Selling back to B=100 and restoring g=0.8 closes the inventory and control state. The trader receives exactly that deposit, and the sponsor account pays it once. A sponsor balance of 5 funds this cycle. It cannot fund a second identical cycle without replenishment. Every state in this example lies within the admitted band, including coupling values below one.

The sponsor purchases the defined control response. Its account, eligible updates, and cumulative budget are explicit parts of the mechanism. The reporter’s proper score can remain independent of that account. Any reporter ownership of the sponsor or trading positions enters the aggregate outside-payoff contract of Section 5. Thus funding a signal does not by itself establish truthful reporting. At the anchor, the price is the baseline reserve ratio multiplied by the selected control. Choosing K/B_*=e^{X_A} and g=e^{c(z_*)} realizes the bounded log-price estimator below. Here c(z_*) will be the clipped correction to the baseline log-price observation X_A. Its baseline X_A is fixed before the reporting window. Away from the anchor, the funded price law is the one in Theorem 8.3 and requires its own calibration.

9 Information Gain within the Trading Limits

Safety parameters also limit the information that the price can transmit. At f=0.003, the segregated-fee condition gives a<\operatorname{atanh}(0.003) and a full price ratio below (1.003)/(0.997)\approx1.006018. Whether this range is useful depends on the error scale, the trading depth, and the delay. The remaining question is whether a restricted price correction can still use information that the spot observation misses.

The next proposition returns to the two-observation model. Its D now denotes the difference between the prediction and spot observations, rather than the preceding section’s inventory-band parameter. The unrestricted correction z_* is the amount the calibrated estimate would add to the spot observation. The function c reduces that correction to the permitted range while preserving its direction.

The strict inequality states an information gain, not a minimum gain of practical size. A narrow permitted correction can yield a correspondingly small improvement.

For an affine prediction observation X_P=b+s u, the committed centre is u_0=(X_A-b)/s when s\ne0. Here b is the observation intercept, distinct from the earlier score scale. Apply the clip to \kappa s(u-u_0). This constructs the full estimator in its central region while bounding the overlay everywhere. The prediction payoff remains the proper score and reads neither the resulting spot price nor the fee account. Estimation error in \kappa, signal dependence, and delay require evaluation under their actual joint law.

10 Boundaries of the Result

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