Papers
Original write-ups developed in this repository. The literature surveys live under bibliography; these are new work.
Scoring Point-Cloud Distributional Submissions
The Nearest-the-Pin Parimutuel, Jittered Outcomes, and the Heat Ladder
A parimutuel over a continuum splits the pot in proportion to the density each participant placed at the realised outcome: the nearest-the-pin reward behind monteprediction.com. When the submission is a cloud of samples smoothed into a KDE, scoring at the raw outcome pays the deconvolution of the belief; jittering the pin with the same kernel restores strict propriety, and repeating the repaired score across smoothing scales splits the edge into Fisher-divergence payments.
Peter Cotton · Microprediction · July 2026 · full draft · pdf · SSRN · interactive demo · implementation · prior-art audit
An Algebra of Prediction-Rewarding Mechanisms
Scoring Rules, Market Makers, and Opinion Pools Generated by One Convex Function
A primer. Scoring rules, cost-function market makers, and opinion pools are taught as separate subjects, and all three are generated by one convex function of the forecast: a proper scoring rule is that function read at the report, a cost-function market maker is its Fenchel conjugate, and the linear and logarithmic opinion pools are the two Kullback-Leibler barycenters. Merging two market makers is infimal convolution, so liquidity adds. The mechanisms are given one input-output signature so that they can be chained, and Bayesian model averaging, least squares, the Kalman filter and split-conformal prediction are then chains with one degree of freedom frozen. Every statement is worked on a single three-outcome example with a liquidity of ten.
Peter Cotton · working draft v0.5 · full draft · pdf
Multi-Stage Solicitation of Probability Distributions
Experiments, Theory and Perspective on Conformal Prediction
The microprediction platform ran several chained-elicitation games at once: a pool on a live number, a stream predicting that pool's own calibration, dependence streams that priced copulas through a space-filling curve, and a stacked lottery of calibration maps. This note describes those experiments and then asks, of each, whether it was a valid game: whether truthful reporting was really the best move. A chain is proper where an exogenous outcome anchors every stage, and a downstream contribution can then be scored in its own market or, for the log score equivalently, folded up to the top level. The base pool scored smoothed samples at the raw outcome and would have been improper, but an accidental jitter for discrete outcomes stood in for the fix the theory prescribes; the copula streams settled through a curve whose nearness is not the joint's. A chain is as valid as its weakest anchor and its settlement transform.
Peter Cotton · working draft v0.3 · full draft · pdf
Likelihood versus CRPS: A Compositional Perspective
Composition, Chained Elicitation, and the Score under Which Density Refinements Add Up
The choice between the logarithmic score and CRPS is usually argued on locality, propriety, robustness, and interpretability. This note adds a fifth consideration the debate rarely weighs. When the elicited object is a reusable density and forecasts are chained, each stage refining the residual of the last, only the log score's per-stage increment is itself the proper score of what that stage added, the prequential and logarithmic-market-scoring structure. Composability becomes the organizing axis of the choice, tied to chained elicitation. A worked example shows the two scores preferring measurably different Gaussians on heavy-tailed data, and conformal prediction appears as the limiting case: a method that need not produce a density, scored on the one metric that does not notice.
Peter Cotton · working draft v0.1 · full draft · pdf
Non-Convex Market Makers
Coherence without Convexity, Friction, and Parallel Composition
Cost-function market makers are always assumed convex; convexity is not what makes them coherent. The exact no-arbitrage condition is a chord condition — every chord slope of the cost lies in the convex hull of payoffs — and convexity enters the standard axiomatics through a separate requirement, monotone information incorporation. Rational flow against a non-convex maker sees the convex envelope, so non-convexity costs expressiveness rather than soundness, and bounded incoherence is priced: a proportional fee of at least the arbitrage depth restores no-arbitrage and, conjugated, is exactly a bid-ask spread. Combining fee-bearing makers is an infimal convolution solved by one clearing-price root-find, the fees act as an L1 penalty so participation is sparse, and the aggregate supply curve is a consolidated limit order book. Separation makes model inconsistency financially discoverable, with coherence relative to friction and to the cost of finding certificates.
Peter Cotton · working draft v0.1 · full draft · pdf
Bounded Market Makers from Convex Duality
Self-Set Fees, Dead Bands, and the Order Book as a Theorem
Automated market makers for contracts that settle to bounded payoffs, built from the cost function $C(q)=b\sum_d\log\cosh(\langle q,v_d\rangle/b)$ over residual directions $v_d$ supplied by the maker and capital $b$. The conjugate is finite exactly on a zonotope, which gives an exact worst-case loss of $b\,D\log 2$, the price set as the zonotope's interior, and the Hessian as a quoted covariance that depletes as inventory accumulates. A per-unit fee is a dead band in price space and the trader's problem is a lasso. Aggregating fee-bearing makers is an inf-convolution whose supply correspondence is a consolidated limit order book, recovered as a theorem rather than imposed. Every statement is checked numerically by a script distributed with the source.
Peter Cotton · working draft v0.1 · LaTeX source in papers/bounded-market-makers · pdf