← Home

Bibliography

The literature behind each mechanism, 322 references, organized by topic. Titles link to their DOI or to the canonical source. A machine-readable version lives in research/bibliography.bib.

Proper scoring rules & distributional forecasting

• Bernardo, J. M. (1979). “Expected Information as Expected Utility.” The Annals of Statistics 7(3).
• Machete, R. L. (2013). “Contrasting Probabilistic Scoring Rules.” Journal of Statistical Planning and Inference 143(10).
• Yao, Y., Vehtari, A., Simpson, D. & Gelman, A. (2018). “Using Stacking to Average Bayesian Predictive Distributions (with Discussion).” Bayesian Analysis 13(3).
• Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S. & Lakshminarayanan, B. (2021). “Normalizing Flows for Probabilistic Modeling and Inference.” Journal of Machine Learning Research 22.
• Brier, G. W. (1950). “Verification of Forecasts Expressed in Terms of Probability.” Monthly Weather Review 78(1).
• Good, I. J. (1952). “Rational Decisions.” Journal of the Royal Statistical Society. Series B (Methodological) 14(1).
• McCarthy, J. (1956). “Measures of the Value of Information.” Proceedings of the National Academy of Sciences 42(9).
• Rosenblatt, M. (1952). “Remarks on a Multivariate Transformation.” The Annals of Mathematical Statistics 23(3).
• Bregman, L. M. (1967). “The Relaxation Method of Finding the Common Point of Convex Sets and Its Application to the Solution of Problems in Convex Programming.” USSR Computational Mathematics and Mathematical Physics 7(3).
• Savage, L. J. (1971). “Elicitation of Personal Probabilities and Expectations.” Journal of the American Statistical Association 66(336).
• Murphy, A. H. (1973). “A New Vector Partition of the Probability Score.” Journal of Applied Meteorology 12(4).
• Matheson, J. E. & Winkler, R. L. (1976). “Scoring Rules for Continuous Probability Distributions.” Management Science 22(10).
• DeGroot, M. H. & Fienberg, S. E. (1983). “The Comparison and Evaluation of Forecasters.” Journal of the Royal Statistical Society. Series D (The Statistician) 32(1--2).
• Dawid, A. P. (1984). “Present Position and Potential Developments: Some Personal Views. Statistical Theory: The Prequential Approach.” Journal of the Royal Statistical Society. Series A (General) 147(2).
• Schervish, M. J. (1989). “A General Method for Comparing Probability Assessors.” The Annals of Statistics 17(4).
• Diebold, F. X., Gunther, T. A. & Tay, A. S. (1998). “Evaluating Density Forecasts with Applications to Financial Risk Management.” International Economic Review 39(4).
• Hersbach, H. (2000). “Decomposition of the Continuous Ranked Probability Score for Ensemble Prediction Systems.” Weather and Forecasting 15(5).
• Banerjee, A., Guo, X. & Wang, H. (2005). “On the Optimality of Conditional Expectation as a Bregman Predictor.” IEEE Transactions on Information Theory 51(7).
• Banerjee, A., Merugu, S., Dhillon, I. S. & Ghosh, J. (2005). “Clustering with Bregman Divergences.” Journal of Machine Learning Research 6.
• Gneiting, T. & Raftery, A. E. (2007). “Strictly Proper Scoring Rules, Prediction, and Estimation.” Journal of the American Statistical Association 102(477).
• Gneiting, T., Balabdaoui, F. & Raftery, A. E. (2007). “Probabilistic Forecasts, Calibration and Sharpness.” Journal of the Royal Statistical Society. Series B (Statistical Methodology) 69(2).
• Bröcker, J. & Smith, L. A. (2007). “Increasing the Reliability of Reliability Diagrams.” Weather and Forecasting 22(3).
• Székely, G. J. & Rizzo, M. L. (2013). “Energy Statistics: A Class of Statistics Based on Distances.” Journal of Statistical Planning and Inference 143(8).
• Dawid, A. P. & Musio, M. (2014). “Theory and Applications of Proper Scoring Rules.” METRON 72(2).
• Gneiting, T. & Katzfuss, M. (2014). “Probabilistic Forecasting.” Annual Review of Statistics and Its Application 1.
• Gneiting, T. & Resin, J. (2023). “Regression Diagnostics Meets Forecast Evaluation: Conditional Calibration, Reliability Diagrams, and Coefficient of Determination.” Electronic Journal of Statistics 17(2).

Local scoring rules & score matching

• Hyvärinen, A. (2005). “Estimation of Non-Normalized Statistical Models by Score Matching.” Journal of Machine Learning Research 6.
• Hyvärinen, A. (2007). “Some Extensions of Score Matching.” Computational Statistics & Data Analysis 51(5).
• Hyvärinen, A. (2008). “Optimal Approximation of Signal Priors.” Neural Computation 20(12).
• Parry, M., Dawid, A. P. & Lauritzen, S. (2012). “Proper Local Scoring Rules.” The Annals of Statistics 40(1).
• Dawid, A. P., Lauritzen, S. & Parry, M. (2012). “Proper Local Scoring Rules on Discrete Sample Spaces.” The Annals of Statistics 40(1).
• Ehm, W. & Gneiting, T. (2012). “Local Proper Scoring Rules of Order Two.” The Annals of Statistics 40(1).
• Vincent, P. (2011). “A Connection Between Score Matching and Denoising Autoencoders.” Neural Computation 23(7).
• Lyu, S. (2009). “Interpretation and Generalization of Score Matching.” Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence (UAI).
• Song, Y., Garg, S., Shi, J. & Ermon, S. (2020). “Sliced Score Matching: A Scalable Approach to Density and Score Estimation.” Proceedings of the 35th Uncertainty in Artificial Intelligence Conference (UAI).
• Song, Y. & Ermon, S. (2019). “Generative Modeling by Estimating Gradients of the Data Distribution.” Advances in Neural Information Processing Systems (NeurIPS).
• Lin, L., Drton, M. & Shojaie, A. (2016). “Estimation of High-Dimensional Graphical Models Using Regularized Score Matching.” Electronic Journal of Statistics 10(1).
• Efron, B. (2011). “Tweedie's Formula and Selection Bias.” Journal of the American Statistical Association 106(496).
• Creal, D., Koopman, S. J. & Lucas, A. (2013). “Generalized Autoregressive Score Models with Applications.” Journal of Applied Econometrics 28(5).
• Hansen, P. R. & Tong, C. (2026). “Tweedie's Formula and Score-Driven Updating.” arXiv:2605.15902.

Property elicitation, elicitability & the geometry of scoring rules

• Hendrickson, A. D. & Buehler, R. J. (1971). “Proper Scores for Probability Forecasters.” The Annals of Mathematical Statistics 42(6).
• Dawid, A. P. (2007). “The Geometry of Proper Scoring Rules.” Annals of the Institute of Statistical Mathematics 59(1).
• Lambert, N. S., Pennock, D. M. & Shoham, Y. (2008). “Eliciting Properties of Probability Distributions.” Proceedings of the 9th ACM Conference on Electronic Commerce (EC).
• Schervish, M. J., Seidenfeld, T. & Kadane, J. B. (2009). “Proper Scoring Rules, Dominated Forecasts, and Coherence.” Decision Analysis 6(4).
• Reid, M. D. & Williamson, R. C. (2010). “Composite Binary Losses.” Journal of Machine Learning Research 11.
• Reid, M. D. & Williamson, R. C. (2011). “Information, Divergence and Risk for Binary Experiments.” Journal of Machine Learning Research 12.
• Gneiting, T. (2011). “Making and Evaluating Point Forecasts.” Journal of the American Statistical Association 106(494).
• Abernethy, J. D. & Frongillo, R. M. (2012). “A Characterization of Scoring Rules for Linear Properties.” Proceedings of the 25th Annual Conference on Learning Theory (COLT).
• Steinwart, I., Pasin, C., Williamson, R. C. & Zhang, S. (2014). “Elicitation and Identification of Properties.” Proceedings of the 27th Conference on Learning Theory (COLT).
• Frongillo, R. M. & Kash, I. A. (2015). “Vector-Valued Property Elicitation.” Proceedings of the 28th Conference on Learning Theory (COLT).
• Frongillo, R. M. & Kash, I. A. (2015). “On Elicitation Complexity.” Advances in Neural Information Processing Systems 28 (NeurIPS).
• Fissler, T. & Ziegel, J. F. (2016). “Higher Order Elicitability and Osband's Principle.” The Annals of Statistics 44(4).

Market scoring rules & cost-function makers

• Hanson, R. (2003). “Combinatorial Information Market Design.” Information Systems Frontiers 5(1).
• Ledyard, J., Hanson, R. & Ishikida, T. (2009). “An Experimental Test of Combinatorial Information Markets.” Journal of Economic Behavior & Organization 69(2).
• Hanson, R. (2007). “Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation.” The Journal of Prediction Markets 1(1).
• Chen, Y. & Pennock, D. M. (2007). “A Utility Framework for Bounded-Loss Market Makers.” Proceedings of the 23rd Conference on Uncertainty in Artificial Intelligence (UAI).
• Abernethy, J., Chen, Y. & Vaughan, J. W. (2013). “Efficient Market Making via Convex Optimization, and a Connection to Online Learning.” ACM Transactions on Economics and Computation 1(2).
• Abernethy, J., Chen, Y. & Vaughan, J. W. (2011). “An Optimization-Based Framework for Automated Market-Making.” Proceedings of the 12th ACM Conference on Electronic Commerce (EC).
• Chen, Y. & Vaughan, J. W. (2010). “A New Understanding of Prediction Markets via No-Regret Learning.” Proceedings of the 11th ACM Conference on Electronic Commerce (EC).
• Othman, A., Pennock, D. M., Reeves, D. M. & Sandholm, T. (2013). “A Practical Liquidity-Sensitive Automated Market Maker.” ACM Transactions on Economics and Computation 1(3).
• Othman, A. & Sandholm, T. (2011). “Liquidity-Sensitive Automated Market Makers via Homogeneous Risk Measures.” Internet and Network Economics (WINE 2011).
• Othman, A. & Sandholm, T. (2012). “Profit-Charging Market Makers with Bounded Loss, Vanishing Bid/Ask Spreads, and Unlimited Market Depth.” Proceedings of the 13th ACM Conference on Electronic Commerce (EC).
• Pennock, D. M. & Sami, R. (2007). “Computational Aspects of Prediction Markets.” Algorithmic Game Theory, Cambridge University Press.
• Chen, Y., Fortnow, L., Lambert, N., Pennock, D. M. & Wortman, J. (2008). “Complexity of Combinatorial Market Makers.” Proceedings of the 9th ACM Conference on Electronic Commerce (EC).
• Dudík, M., Lahaie, S. & Pennock, D. M. (2012). “A Tractable Combinatorial Market Maker Using Constraint Generation.” Proceedings of the 13th ACM Conference on Electronic Commerce (EC).
• Chen, Y., Ruberry, M. & Vaughan, J. W. (2013). “Cost Function Market Makers for Measurable Spaces.” Proceedings of the 14th ACM Conference on Electronic Commerce (EC).
• Chen, Y. & Pennock, D. M. (2010). “Designing Markets for Prediction.” AI Magazine 31(4).

Scoring rules ⇄ market makers: duality & online learning

• Abernethy, J., Bartlett, P. L. & Hazan, E. (2011). “Blackwell Approachability and No-Regret Learning are Equivalent.” Proceedings of the 24th Annual Conference on Learning Theory (COLT).
• Frongillo, R. M., Della Penna, N. & Reid, M. D. (2012). “Interpreting Prediction Markets: A Stochastic Approach.” Advances in Neural Information Processing Systems 25 (NeurIPS).
• Hu, J. & Storkey, A. (2014). “Multi-period Trading Prediction Markets with Connections to Machine Learning.” Proceedings of the 31st International Conference on Machine Learning (ICML).
• Frongillo, R. M. & Reid, M. D. (2015). “Convergence Analysis of Prediction Markets via Randomized Subspace Descent.” Advances in Neural Information Processing Systems 28 (NeurIPS).
• Abernethy, J. D., Frongillo, R. M. & Kutty, S. (2015). “On Risk Measures, Market Making, and Exponential Families.” ACM SIGecom Exchanges 13(2).
• Hazan, E. (2016). “Introduction to Online Convex Optimization.” Now Publishers.
• Frongillo, R. & Waggoner, B. (2018). “An Axiomatic Study of Scoring Rule Markets.” 9th Innovations in Theoretical Computer Science Conference (ITCS).
• Frongillo, R., Papireddygari, M. & Waggoner, B. (2024). “An Axiomatic Characterization of CFMMs and Equivalence to Prediction Markets.” 15th Innovations in Theoretical Computer Science Conference (ITCS).

Wagering mechanisms, decision markets & aggregation dynamics

• Lei, J., G'Sell, M., Rinaldo, A., Tibshirani, R. J. & Wasserman, L. (2018). “Distribution-Free Predictive Inference for Regression.” Journal of the American Statistical Association 113(523).
• Grünwald, P. & Roos, T. (2019). “Minimum Description Length Revisited.” International Journal of Mathematics for Industry 11(1).
• Feigenbaum, J., Fortnow, L., Pennock, D. M. & Sami, R. (2005). “Computation in a Distributed Information Market.” Theoretical Computer Science.
• Lambert, N. S., Langford, J., Wortman, J., Chen, Y., Reeves, D. M., Shoham, Y. & Pennock, D. M. (2008). “Self-Financed Wagering Mechanisms for Forecasting.” Proceedings of the 9th ACM Conference on Electronic Commerce (EC).
• Hanson, R. & Oprea, R. (2009). “A Manipulator Can Aid Prediction Market Accuracy.” Economica 76(302).
• Othman, A. & Sandholm, T. (2010). “Decision Rules and Decision Markets.” Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS).
• Chen, Y., Kash, I., Ruberry, M. & Shnayder, V. (2011). “Decision Markets with Good Incentives.” Internet and Network Economics (WINE 2011).
• Ostrovsky, M. (2012). “Information Aggregation in Dynamic Markets with Strategic Traders.” Econometrica 80(6).
• Harville, D. A. (1973). “Assigning Probabilities to the Outcomes of Multi-Entry Competitions.” Journal of the American Statistical Association 68(342).
• Hausch, D. B., Ziemba, W. T. & Rubinstein, M. (1981). “Efficiency of the Market for Racetrack Betting.” Management Science 27(12).
• Jha, A. & Wolak, F. A. (2023). “Can Forward Commodity Markets Improve Spot Market Performance? Evidence from Wholesale Electricity.” American Economic Journal: Economic Policy 15(2).
• Cramton, P. (2017). “Electricity Market Design.” Oxford Review of Economic Policy 33(4).
• Cotton, P. (2024). “Monteprediction: A Weekly Monte Carlo Competition in Eleven Dimensions.” Running weekly since January 2024
• Cotton, P. (2024). “A New Generative AI Competition in Eleven Dimensions.” Microprediction, Medium, 23 March 2024
• Craib, R., Bradway, G., Dunn, X. & Krug, J. (2017). “Numeraire: A Cryptographic Token for Coordinating Machine Intelligence.”
• Atanasov, P., Rescober, P., Stone, E., Swift, S. A., Servan-Schreiber, E., Tetlock, P., Ungar, L. & Mellers, B. (2017). “Distilling the Wisdom of Crowds: Prediction Markets vs. Prediction Polls.” Management Science 63(3).
• Cover, T. M. & Thomas, J. A. (2006). “Elements of Information Theory.” Wiley.
• Barron, A. R. & Cover, T. M. (1988). “A Bound on the Financial Value of Information.” IEEE Transactions on Information Theory 34(5).
• Kemp, J. T. & Bettencourt, L. M. A. (2022). “The Bayesian Origins of Growth Rates in Stochastic Environments.” arXiv:2209.09492.
• Arnold, S., Henzi, A. & Ziegel, J. F. (2023). “Sequentially Valid Tests for Forecast Calibration.” The Annals of Applied Statistics 17(3).
• Shaer, S., Maman, G. & Romano, Y. (2023). “Model-X Sequential Testing for Conditional Independence via Testing by Betting.” AISTATS.
• Grünwald, P., de Heide, R. & Koolen, W. M. (2024). “Safe Testing.” Journal of the Royal Statistical Society B 86(5).
• Vovk, V. & Wang, R. (2021). “E-values: Calibration, Combination and Applications.” The Annals of Statistics 49(3).
• Vovk, V., Nouretdinov, I. & Gammerman, A. (2003). “Testing Exchangeability On-Line.” International Conference on Machine Learning.
• Ville, J. (1939). “Étude critique de la notion de collectif.” Gauthier-Villars.
• Székely, G. J., Rizzo, M. L. & Bakirov, N. K. (2007). “Measuring and Testing Dependence by Correlation of Distances.” The Annals of Statistics 35(6).
• Lei, J. & Wasserman, L. (2014). “Distribution-Free Prediction Bands for Non-Parametric Regression.” Journal of the Royal Statistical Society B 76(1).
• Foygel Barber, R., Candès, E. J., Ramdas, A. & Tibshirani, R. J. (2021). “The Limits of Distribution-Free Conditional Predictive Inference.” Information and Inference 10(2).
• Vovk, V. (2012). “Conditional Validity of Inductive Conformal Predictors.” Asian Conference on Machine Learning.
• Simon-Gabriel, C. J., Barp, A., Schölkopf, B. & Mackey, L. (2023). “Metrizing Weak Convergence with Maximum Mean Discrepancies.” Journal of Machine Learning Research 24.
• Sklar, A. (1959). “Fonctions de répartition à $n$ dimensions et leurs marges.” Publications de l'Institut de Statistique de l'Université de Paris 8.
• Diks, C., Panchenko, V. & van Dijk, D. (2010). “Out-of-Sample Comparison of Copula Specifications in Multivariate Density Forecasts.” Journal of Economic Dynamics and Control 34(9).
• Joe, H. (1997). “Multivariate Models and Dependence Concepts.” Chapman & Hall.
• Csiszár, I. (1967). “Information-Type Measures of Difference of Probability Distributions and Indirect Observations.” Studia Scientiarum Mathematicarum Hungarica 2.
• Kumar, P. & Seppi, D. J. (1992). “Futures Manipulation with ``Cash Settlement''.” The Journal of Finance 47(4).
• Jarrow, R. A. (1994). “Derivative Security Markets, Market Manipulation, and Option Pricing Theory.” Journal of Financial and Quantitative Analysis 29(2).
• Lambert, N. S., Langford, J., Wortman Vaughan, J., Chen, Y., Reeves, D. M., Shoham, Y. & Pennock, D. M. (2015). “An Axiomatic Characterization of Wagering Mechanisms.” Journal of Economic Theory 156.
• Witkowski, J., Freeman, R., Wortman Vaughan, J., Pennock, D. M. & Krause, A. (2023). “Incentive-Compatible Forecasting Competitions.” Management Science 69(3).
• Wolfers, J. & Zitzewitz, E. (2006). “Interpreting Prediction Market Prices as Probabilities.” National Bureau of Economic Research NBER Working Paper 12200.

Parimutuel, dynamic & combinatorial prediction markets

• Gao, X. A., Chen, Y. & Pennock, D. M. (2009). “Betting on the Real Line.” Internet and Network Economics (WINE).
• Freeman, R., Lahaie, S. & Pennock, D. M. (2017). “Crowdsourced Outcome Determination in Prediction Markets.” Proceedings of the 31st AAAI Conference on Artificial Intelligence.
• Pennock, D. M. (2004). “A Dynamic Pari-Mutuel Market for Hedging, Wagering, and Information Aggregation.” Proceedings of the 5th ACM Conference on Electronic Commerce (EC).
• Conlon, B. (n.d.). “Totalisator History: A World's First.” http://tote-history-atl.net.au/.
• Conlon, B. (n.d.). “The First Automatic Totalisator.” The Rutherford Journal, https://rutherfordjournal.org/article020109.html.
• Ali, M. M. (1977). “Probability and Utility Estimates for Racetrack Bettors.” Journal of Political Economy 85(4).
• Thaler, R. H. & Ziemba, W. T. (1988). “Anomalies: Parimutuel Betting Markets: Racetracks and Lotteries.” Journal of Economic Perspectives 2(2).
• Baron, K. & Lange, J. (2007). “Parimutuel Applications in Finance: New Markets for New Risks.” Palgrave Macmillan. The bundle-constrained (self-hedging) parimutuel call auction behind Longitude's markets, where limit orders over bundled contingent claims let a single parimutuel pool replicate option-like payoffs.
• Snowberg, E. & Wolfers, J. (2010). “Explaining the Favorite-Long Shot Bias: Is It Risk-Love or Misperceptions?.” Journal of Political Economy 118(4).
• Ottaviani, M. & Sørensen, P. N. (2008). “The Favorite-Longshot Bias: An Overview of the Main Explanations.” Handbook of Sports and Lottery Markets, North-Holland.
• Ottaviani, M. & Sørensen, P. N. (2009). “Surprised by the Parimutuel Odds?.” American Economic Review 99(5).
• Chen, Y., Fortnow, L., Nikolova, E. & Pennock, D. M. (2007). “Betting on Permutations.” Proceedings of the 8th ACM Conference on Electronic Commerce (EC).
• Lange, J. & Economides, N. (2005). “A Parimutuel Market Microstructure for Contingent Claims.” European Financial Management 11(1).
• Peters, M., So, A. M. C. & Ye, Y. (2007). “Pari-Mutuel Markets: Mechanisms and Performance.” Internet and Network Economics (WINE 2007).
• Agrawal, S., Delage, E., Peters, M., Wang, Z. & Ye, Y. (2011). “A Unified Framework for Dynamic Prediction Market Design.” Operations Research 59(3).
• Laskey, K. B., Sun, W., Hanson, R., Twardy, C., Matsumoto, S. & Goldfedder, B. (2018). “Graphical Model Market Maker for Combinatorial Prediction Markets.” Journal of Artificial Intelligence Research 63.
• Plott, C. R., Wit, J. & Yang, W. C. (2003). “Parimutuel Betting Markets as Information Aggregation Devices: Experimental Results.” Economic Theory 22(2).

Automated market makers & DeFi

• Angeris, G., Kao, H. T., Chiang, R., Noyes, C. & Chitra, T. (2021). “An Analysis of Uniswap Markets.” Cryptoeconomic Systems 1(1).
• Angeris, G. & Chitra, T. (2020). “Improved Price Oracles: Constant Function Market Makers.” Proceedings of the 2nd ACM Conference on Advances in Financial Technologies (AFT).
• Angeris, G., Agrawal, A., Evans, A., Chitra, T. & Boyd, S. (2022). “Constant Function Market Makers: Multi-Asset Trades via Convex Optimization.” Handbook on Blockchain.
• Angeris, G., Evans, A. & Chitra, T. (2021). “Replicating Market Makers.” arXiv:2103.14769.
• Adams, H., Zinsmeister, N., Salem, M., Keefer, R. & Robinson, D. (2021). “Uniswap v3 Core.” Whitepaper, Uniswap.
• Martinelli, F. & Mushegian, N. (2019). “Balancer: A Non-Custodial Portfolio Manager, Liquidity Provider, and Price Sensor.” Whitepaper, Balancer Labs.
• Egorov, M. (2019). “StableSwap -- Efficient Mechanism for Stablecoin Liquidity.” Whitepaper, Curve Finance.
• Milionis, J., Moallemi, C. C., Roughgarden, T. & Zhang, A. L. (2022). “Automated Market Making and Loss-Versus-Rebalancing.” arXiv:2208.06046.
• Moallemi, C. C. & Robinson, D. (2024). “pm-AMM: A Uniform AMM for Prediction Markets.” Paradigm research report.
• Capponi, A. & Jia, R. (2021). “The Adoption of Blockchain-based Decentralized Exchanges.” arXiv:2103.08842.
• Lehar, A. & Parlour, C. A. (2025). “Decentralized Exchange: The Uniswap Automated Market Maker.” The Journal of Finance 80(1).
• Cartea, Á., Drissi, F. & Monga, M. (2024). “Decentralized Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision.” SIAM Journal on Financial Mathematics 15(3).
• Schlegel, J. C., Kwaśnicki, M. & Mamageishvili, A. (2023). “Axioms for Constant Function Market Makers.” Proceedings of the 24th ACM Conference on Economics and Computation (EC).
• Angeris, G., Chitra, T., Diamandis, T., Evans, A. & Kulkarni, K. (2024). “The Geometry of Constant Function Market Makers.” Proceedings of the 25th ACM Conference on Economics and Computation (EC).

Order books, double auctions & batch auctions

• Smith, V. L. (1962). “An Experimental Study of Competitive Market Behavior.” Journal of Political Economy 70(2).
• Gode, D. K. & Sunder, S. (1993). “Allocative Efficiency of Markets with Zero-Intelligence Traders: Market as a Partial Substitute for Individual Rationality.” Journal of Political Economy 101(1).
• Gjerstad, S. & Dickhaut, J. (1998). “Price Formation in Double Auctions.” Games and Economic Behavior 22(1).
• Budish, E., Cramton, P. & Shim, J. (2015). “The High-Frequency Trading Arms Race: Frequent Batch Auctions as a Market Design Response.” The Quarterly Journal of Economics 130(4).
• Vickrey, W. (1961). “Counterspeculation, Auctions, and Competitive Sealed Tenders.” The Journal of Finance 16(1).
• Kyle, A. S. (1985). “Continuous Auctions and Insider Trading.” Econometrica 53(6).
• Glosten, L. R. & Milgrom, P. R. (1985). “Bid, Ask and Transaction Prices in a Specialist Market with Heterogeneously Informed Traders.” Journal of Financial Economics 14(1).
• Daian, P., Goldfeder, S., Kell, T., Li, Y., Zhao, X., Bentov, I., Breidenbach, L. & Juels, A. (2020). “Flash Boys 2.0: Frontrunning in Decentralized Exchanges, Miner Extractable Value, and Consensus Instability.” 2020 IEEE Symposium on Security and Privacy (SP).
• Roughgarden, T. (2021). “Transaction Fee Mechanism Design.” Proceedings of the 22nd ACM Conference on Economics and Computation (EC).
• Friedman, D. & Rust, J. (1993). “The Double Auction Market: Institutions, Theories, and Evidence.” Addison-Wesley.
• Satterthwaite, M. A. & Williams, S. R. (1989). “Bilateral Trade with the Sealed Bid k-Double Auction: Existence and Efficiency.” Journal of Economic Theory 48(1).
• Smith, V. L., Williams, A. W., Bratton, W. K. & Vannoni, M. G. (1982). “Competitive Market Institutions: Double Auctions vs. Sealed Bid-Offer Auctions.” The American Economic Review 72(1).
• Cotton, P. & Papanicolaou, A. (2022). “Trading Illiquid Goods: Market Making as a Sequence of Sealed-Bid Auctions, with Analytic Results.” https://github.com/microprediction/home/blob/main/workingpapers/trading_illiquid_goods.pdf.

Perpetual futures & demand lending pools

• Shiller, R. J. (1993). “Measuring Asset Values for Cash Settlement in Derivative Markets: Hedonic Repeated Measures Indices and Perpetual Futures.” The Journal of Finance 48(3).
• Makarov, I. & Schoar, A. (2020). “Trading and Arbitrage in Cryptocurrency Markets.” Journal of Financial Economics 135(2).
• Angeris, G., Chitra, T., Evans, A. & Lorig, M. (2023). “Short Communication: A Primer on Perpetuals.” SIAM Journal on Financial Mathematics 14(1).
• Ackerer, D., Hugonnier, J. & Jermann, U. (2024). “Perpetual Futures Pricing.” NBER Working Paper No. 32936.
• He, S., Manela, A., Ross, O. & von Wachter, V. (2022). “Fundamentals of Perpetual Futures.” arXiv:2212.06888.
• Chitra, T., Diamandis, T., Sheng, N., Sterle, L. & Yusubov, K. (2025). “Perpetual Demand Lending Pools.” arXiv:2502.06028.
• BitMEX (n.d.). “Perpetual Contracts Guide.” BitMEX documentation.
• GMX (n.d.). “GMX Documentation: Decentralized Perpetual Exchange.” Online documentation.
• Hyperliquid (n.d.). “Protocol Vaults (HLP).” Hyperliquid documentation.

Peer prediction (no ground truth)

• Miller, N., Resnick, P. & Zeckhauser, R. (2005). “Eliciting Informative Feedback: The Peer-Prediction Method.” Management Science 51(9).
• Prelec, D. (2004). “A Bayesian Truth Serum for Subjective Data.” Science 306(5695).
• Jurca, R. & Faltings, B. (2009). “Mechanisms for Making Crowds Truthful.” Journal of Artificial Intelligence Research 34.
• Witkowski, J. & Parkes, D. C. (2012). “A Robust Bayesian Truth Serum for Small Populations.” Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI).
• Witkowski, J. & Parkes, D. C. (2012). “Peer Prediction without a Common Prior.” Proceedings of the 13th ACM Conference on Electronic Commerce (EC).
• Dasgupta, A. & Ghosh, A. (2013). “Crowdsourced Judgement Elicitation with Endogenous Proficiency.” Proceedings of the 22nd International Conference on World Wide Web (WWW).
• Radanovic, G., Faltings, B. & Jurca, R. (2016). “Incentives for Effort in Crowdsourcing Using the Peer Truth Serum.” ACM Transactions on Intelligent Systems and Technology 7(4).
• Shnayder, V., Agarwal, A., Frongillo, R. & Parkes, D. C. (2016). “Informed Truthfulness in Multi-Task Peer Prediction.” Proceedings of the 2016 ACM Conference on Economics and Computation (EC).
• Frongillo, R. M. & Witkowski, J. (2017). “A Geometric Perspective on Minimal Peer Prediction.” ACM Transactions on Economics and Computation 5(3).
• Faltings, B. & Radanovic, G. (2017). “Game Theory for Data Science: Eliciting Truthful Information.” Morgan & Claypool.
• Kong, Y. & Schoenebeck, G. (2018). “Water from Two Rocks: Maximizing the Mutual Information.” Proceedings of the 2018 ACM Conference on Economics and Computation (EC).
• Kong, Y. & Schoenebeck, G. (2019). “An Information Theoretic Framework for Designing Information Elicitation Mechanisms That Reward Truth-Telling.” ACM Transactions on Economics and Computation 7(1).
• Kong, Y. (2024). “Dominantly Truthful Peer Prediction Mechanisms with a Finite Number of Tasks.” Journal of the ACM 71(2).
• Feng, S., Yu, F. Y. & Chen, Y. (2022). “Peer Prediction for Learning Agents.” Advances in Neural Information Processing Systems (NeurIPS).
• Zhang, Y., Xu, S., Pennock, D. & Schoenebeck, G. (2025). “Stochastically Dominant Peer Prediction.” arXiv:2506.02259.
• Kong, Y., Schoenebeck, G., Tao, B. & Yu, F. Y. (2020). “Information Elicitation Mechanisms for Statistical Estimation.” Proceedings of the AAAI Conference on Artificial Intelligence.
• Schoenebeck, G. & Yu, F. Y. (2020). “Two Strongly Truthful Mechanisms for Three Heterogeneous Agents Answering One Question.” Web and Internet Economics (WINE 2020).
• Liu, Y., Wang, J. & Chen, Y. (2022). “Surrogate Scoring Rules.” ACM Transactions on Economics and Computation 10(3).

Forecast aggregation, prediction-market economics & info finance

• Cochrane, J. H. (2005). “Asset Pricing.” Princeton University Press.
• Galton, F. (1907). “Vox Populi.” Nature 75(1949).
• Stone, M. (1961). “The Opinion Pool.” The Annals of Mathematical Statistics 32(4).
• Prelec, D., Seung, H. S. & McCoy, J. (2017). “A Solution to the Single-Question Crowd Wisdom Problem.” Nature 541(7638).
• Palley, A. B. & Soll, J. B. (2019). “Extracting the Wisdom of Crowds When Information Is Shared.” Management Science 65(5).
• Genest, C. & Zidek, J. V. (1986). “Combining Probability Distributions: A Critique and an Annotated Bibliography.” Statistical Science 1(1).
• Bates, J. M. & Granger, C. W. J. (1969). “The Combination of Forecasts.” Operational Research Quarterly 20(4).
• Timmermann, A. (2006). “Forecast Combinations.” Handbook of Economic Forecasting, Elsevier.
• Smith, J. & Wallis, K. F. (2009). “A Simple Explanation of the Forecast Combination Puzzle.” Oxford Bulletin of Economics and Statistics 71(3).
• Ranjan, R. & Gneiting, T. (2010). “Combining Probability Forecasts.” Journal of the Royal Statistical Society. Series B (Statistical Methodology) 72(1).
• Satopää, V. A., Baron, J., Foster, D. P., Mellers, B. A., Tetlock, P. E. & Ungar, L. H. (2014). “Combining Multiple Probability Predictions Using a Simple Logit Model.” International Journal of Forecasting 30(2).
• Baron, J., Mellers, B. A., Tetlock, P. E., Stone, E. & Ungar, L. H. (2014). “Two Reasons to Make Aggregated Probability Forecasts More Extreme.” Decision Analysis 11(2).
• Surowiecki, J. (2004). “The Wisdom of Crowds.” Doubleday.
• Tetlock, P. E. & Gardner, D. (2015). “Superforecasting: The Art and Science of Prediction.” Crown.
• Wolfers, J. & Zitzewitz, E. (2004). “Prediction Markets.” Journal of Economic Perspectives 18(2).
• Rhode, P. W. & Strumpf, K. S. (2004). “Historical Presidential Betting Markets.” Journal of Economic Perspectives 18(2).
• Arrow, K. J., Forsythe, R., Gorham, M., Hahn, R., Hanson, R., Ledyard, J. O., Levmore, S., Litan, R., Milgrom, P., Nelson, F. D., Neumann, G. R., Ottaviani, M., Schelling, T. C., Shiller, R. J., Smith, V. L., Snowberg, E., Sunstein, C. R., Tetlock, P. C., Tetlock, P. E., Varian, H. R., Wolfers, J. & Zitzewitz, E. (2008). “The Promise of Prediction Markets.” Science 320(5878).
• Berg, J. E., Nelson, F. D. & Rietz, T. A. (2008). “Prediction Market Accuracy in the Long Run.” International Journal of Forecasting 24(2).
• Hanson, R. (2013). “Shall We Vote on Values, But Bet on Beliefs?.” Journal of Political Philosophy 21(2).
• Kelly, J. L. (1956). “A New Interpretation of Information Rate.” The Bell System Technical Journal 35(4).
• Thorp, E. O. (2006). “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market.” Handbook of Asset and Liability Management, North-Holland.
• MacLean, L. C., Thorp, E. O. & Ziemba, W. T. (2011). “The Kelly Capital Growth Investment Criterion: Theory and Practice.” World Scientific.
• Fernholz, E. R. (2002). “Stochastic Portfolio Theory.” Springer.
• Karatzas, I. & Fernholz, E. R. (2009). “Stochastic Portfolio Theory: An Overview.” Handbook of Numerical Analysis: Mathematical Modeling and Numerical Methods in Finance, Elsevier.
• Cotton, P. (n.d.). “A Gentle Introduction to Stochastic Portfolio Theory (and Its Inverse).” https://microprediction.medium.com/a-gentle-introduction-to-stochastic-portfolio-theory-and-its-inverse-241d3bc72917.
• Buterin, V. (2024). “From Prediction Markets to Info Finance.” https://vitalik.eth.limo/general/2024/11/09/infofinance.html.

Microprediction & composition

• Cotton, P. (2022). “Microprediction: Building an Open AI Network.” MIT Press.
• Cotton, P. (2019). “Self-Organizing Supply Chains for Micro-Prediction: Present and Future Uses of the ROAR Protocol.” arXiv:1907.07514
• Abernethy, J. D. & Frongillo, R. M. (2011). “A Collaborative Mechanism for Crowdsourcing Prediction Problems.” Advances in Neural Information Processing Systems (NeurIPS).
• Falconer, T., Kazempour, J. & Pinson, P. (2024). “Bayesian Regression Markets.” Journal of Machine Learning Research 25.
• Smith, A. (1776). “An Inquiry into the Nature and Causes of the Wealth of Nations.” W. Strahan and T. Cadell.
• Cotton, P. (n.d.). “Monteprediction: A Monte Carlo Distributional Forecasting Contest.” https://monteprediction.com.
• Cotton, P. (2026). “skaters: Fast online time-series models with composable transforms.”
• Cotton, P. (2021). “timemachines: Time-series prediction with a uniform skater interface.”

Convex duality, risk sharing & composition

• Rockafellar, R. T. (1970). “Convex Analysis.” Princeton University Press.
• Moreau, J. J. (1965). “Proximité et dualité dans un espace hilbertien.” Bulletin de la Société Mathématique de France 93.
• Barrieu, P. & El Karoui, N. (2005). “Inf-convolution of Risk Measures and Optimal Risk Transfer.” Finance and Stochastics 9(2).
• Jouini, E., Schachermayer, W. & Touzi, N. (2008). “Optimal Risk Sharing for Law Invariant Monetary Utility Functions.” Mathematical Finance 18(2).
• Sejdinovic, D., Sriperumbudur, B., Gretton, A. & Fukumizu, K. (2013). “Equivalence of Distance-Based and RKHS-Based Statistics in Hypothesis Testing.” The Annals of Statistics 41(5).
• Bhaskara, A., Frongillo, R. & Papireddygari, M. (2023). “A General Theory of Liquidity Provisioning for Prediction Markets.” arXiv:2311.08725.

The sections below collect the prior art for the three candidate-original generalizations in research/. Exhaustive references are the point.

Risk sharing & syndicates

• Borch, K. (1962). “Equilibrium in a Reinsurance Market.” Econometrica 30(3).
• Wilson, R. (1968). “The Theory of Syndicates.” Econometrica 36(1).

Parimutuel theory & the Fisher-market connection

• Eisenberg, E. & Gale, D. (1959). “Consensus of Subjective Probabilities: The Pari-Mutuel Method.” The Annals of Mathematical Statistics 30(1).
• Zhang, L. (2011). “Proportional Response Dynamics in the Fisher Market.” Theoretical Computer Science 412(24).

Competitive / parimutuel scoring rules

• Kilgour, D. M. & Gerchak, Y. (2004). “Elicitation of Probabilities Using Competitive Scoring Rules.” Decision Analysis 1(2).
• Johnstone, D. J. (2007). “The Parimutuel Kelly Probability Scoring Rule.” Decision Analysis 4(2).

Wagering mechanisms (continued)

• Freeman, R. & Pennock, D. M. (2018). “An Axiomatic View of the Parimutuel Consensus Mechanism.” Proceedings of the 27th International Joint Conference on Artificial Intelligence (IJCAI).
• Raja, A. A., Pinson, P., Kazempour, J. & Grammatico, S. (2022). “A Market for Trading Forecasts: A Wagering Mechanism.” arXiv:2205.02668.

Local scoring / score-matching bridge

• Dawid, A. P. & Musio, M. (2015). “Bayesian Model Selection Based on Proper Scoring Rules.” Bayesian Analysis 10(2).
• Waghmare, K. & Ziegel, J. (2026). “Proper Scoring Rules for Estimation and Forecast Evaluation.” Annual Review of Statistics and Its Application 13.

Score-based generative models & the mode-mass blindness of score matching

• Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S. & Poole, B. (2021). “Score-Based Generative Modeling through Stochastic Differential Equations.” International Conference on Learning Representations (ICLR).
• Wenliang, L. K. & Kanagawa, H. (2020). “Blindness of Score-Based Methods to Isolated Components and Mixing Proportions.” arXiv:2008.10087.
• Zhang, M., Key, O., Hayes, P., Barber, D., Paige, B. & Briol, F. X. (2022). “Towards Healing the Blindness of Score Matching.” arXiv:2209.07396.
• Koehler, F., Heckett, A. & Risteski, A. (2023). “Statistical Efficiency of Score Matching: The View from Isoperimetry.” International Conference on Learning Representations (ICLR).

Continuous / measurable-space prediction markets

• Dudík, M., Wang, X., Pennock, D. M. & Rothschild, D. M. (2021). “Log-time Prediction Markets for Interval Securities.” Proceedings of the 20th International Conference on Autonomous Agents and Multiagent Systems (AAMAS).

Energy / kernel scores

• Gneiting, T., Stanberry, L. I., Grimit, E. P., Held, L. & Johnson, N. A. (2008). “Assessing Probabilistic Forecasts of Multivariate Quantities.” TEST 17(2).
• Gretton, A., Borgwardt, K. M., Rasch, M. J., Schölkopf, B. & Smola, A. (2012). “A Kernel Two-Sample Test.” Journal of Machine Learning Research 13.
• Steinwart, I. & Ziegel, J. F. (2021). “Strictly Proper Kernel Scores and Characteristic Kernels on Compact Spaces.” Applied and Computational Harmonic Analysis 51.

Weighted / localized proper scoring rules (the properness boundary)

• Bolin, D. & Wallin, J. (2023). “Local Scale Invariance and Robustness of Proper Scoring Rules.” Statistical Science 38(1).
• Gneiting, T. & Ranjan, R. (2011). “Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules.” Journal of Business & Economic Statistics 29(3).
• Diks, C., Panchenko, V. & van Dijk, D. (2011). “Likelihood-Based Scoring Rules for Comparing Density Forecasts in Tails.” Journal of Econometrics 163(2).
• Holzmann, H. & Klar, B. (2017). “Focusing on Regions of Interest in Forecast Evaluation.” The Annals of Applied Statistics 11(4).
• Allen, S., Ginsbourger, D. & Ziegel, J. (2023). “Evaluating Forecasts for High-Impact Events Using Transformed Kernel Scores.” SIAM/ASA Journal on Uncertainty Quantification 11(3).
• de Punder, R., Diks, C., Laeven, R. J. A. & van Dijk, D. (2026). “Localizing Strictly Proper Scoring Rules.” Journal of the American Statistical Association. Published online 5 Jan 2026, pp. 1--13
• Pic, R., Dombry, C., Naveau, P. & Taillardat, M. (2025). “Proper Scoring Rules for Multivariate Probabilistic Forecasts based on Aggregation and Transformation.” Advances in Statistical Climatology, Meteorology and Oceanography 11(1).
• Ehm, W., Gneiting, T., Jordan, A. & Krüger, F. (2016). “Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations and Forecast Rankings.” Journal of the Royal Statistical Society: Series B 78(3).

Sliced Wasserstein & direction-adaptation variants

• Rabin, J., Peyré, G., Delon, J. & Bernot, M. (2012). “Wasserstein Barycenter and Its Application to Texture Mixing.” Scale Space and Variational Methods (SSVM).
• Bonneel, N., Rabin, J., Peyré, G. & Pfister, H. (2015). “Sliced and Radon Wasserstein Barycenters of Measures.” Journal of Mathematical Imaging and Vision 51(1).
• Deshpande, I., Hu, Y. T., Sun, R., Pyrros, A., Siddiqui, N., Koyejo, S., Zhao, Z., Forsyth, D. & Schwing, A. G. (2019). “Max-Sliced Wasserstein Distance and Its Use for GANs.” IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR).
• Kolouri, S., Nadjahi, K., Şimşekli, U., Badeau, R. & Rohde, G. K. (2019). “Generalized Sliced Wasserstein Distances.” Advances in Neural Information Processing Systems (NeurIPS).
• Nguyen, K., Ho, N., Pham, T. & Bui, H. (2021). “Distributional Sliced-Wasserstein and Applications to Generative Modeling.” International Conference on Learning Representations (ICLR).
• Chen, X., Yang, Y. & Li, Y. (2022). “Augmented Sliced Wasserstein Distances.” International Conference on Learning Representations (ICLR).
• Nguyen, K. & Ho, N. (2023). “Energy-Based Sliced Wasserstein Distance.” Advances in Neural Information Processing Systems (NeurIPS).
• Nguyen, K., Ren, T. & Ho, N. (2023). “Markovian Sliced Wasserstein Distances: Beyond Independent Projections.” Advances in Neural Information Processing Systems (NeurIPS).
• Nguyen, K., Zhang, S., Le, T. & Ho, N. (2024). “Sliced Wasserstein with Random-Path Projecting Directions.” International Conference on Machine Learning (ICML).
• Paty, F. P. & Cuturi, M. (2019). “Subspace Robust Wasserstein Distances.” International Conference on Machine Learning (ICML).
• Lin, T., Fan, C., Ho, N., Cuturi, M. & Jordan, M. I. (2020). “Projection Robust Wasserstein Distance and Riemannian Optimization.” Advances in Neural Information Processing Systems (NeurIPS).
• Bonet, C., Drumetz, L. & Courty, N. (2025). “Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds.” Journal of Machine Learning Research 26(32).
• Nadjahi, K., Durmus, A., Chizat, L., Kolouri, S., Shahrampour, S. & Şimşekli, U. (2020). “Statistical and Topological Properties of Sliced Probability Divergences.” Advances in Neural Information Processing Systems (NeurIPS).
• Gong, W., Li, Y. & Hernández-Lobato, J. M. (2021). “Sliced Kernelized Stein Discrepancy.” International Conference on Learning Representations (ICLR).

Covariance shrinkage & closed-form intensity

• Stein, C. (1956). “Inadmissibility of the Usual Estimator for the Mean of a Multivariate Normal Distribution.” Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability.
• James, W. & Stein, C. (1961). “Estimation with Quadratic Loss.” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability.
• Ledoit, O. & Wolf, M. (2004). “A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices.” Journal of Multivariate Analysis 88(2).
• Ledoit, O. & Wolf, M. (2004). “Honey, I Shrunk the Sample Covariance Matrix.” The Journal of Portfolio Management 30(4).
• Ledoit, O. & Wolf, M. (2012). “Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices.” The Annals of Statistics 40(2).
• Ledoit, O. & Wolf, M. (2020). “Analytical Nonlinear Shrinkage of Large-Dimensional Covariance Matrices.” The Annals of Statistics 48(5).
• Dobriban, E. & Wager, S. (2018). “High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification.” The Annals of Statistics 46(1).
• Schäfer, J. & Strimmer, K. (2005). “A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics.” Statistical Applications in Genetics and Molecular Biology 4(1).

High-dimensional two-sample tests via random projection

• Bai, Z. & Saranadasa, H. (1996). “Effect of High Dimension: By an Example of a Two Sample Problem.” Statistica Sinica 6(2).
• Srivastava, M. S. & Du, M. (2008). “A Test for the Mean Vector with Fewer Observations than the Dimension.” Journal of Multivariate Analysis 99(3).
• Chen, S. X. & Qin, Y. L. (2010). “A Two-Sample Test for High-Dimensional Data with Applications to Gene-Set Testing.” The Annals of Statistics 38(2).
• Lopes, M. E., Jacob, L. J. & Wainwright, M. J. (2011). “A More Powerful Two-Sample Test in High Dimensions Using Random Projection.” Advances in Neural Information Processing Systems (NeurIPS).

Markets as online learning; machine-learning markets

• Nueve, E. & Waggoner, B. (2025). “Smooth Quadratic Prediction Markets.” arXiv:2505.02959.
• Storkey, A. J. (2011). “Machine Learning Markets.” Proceedings of the 14th International Conference on Artificial Intelligence and Statistics (AISTATS).
• Storkey, A. J., Millin, J. & Geras, K. (2012). “Isoelastic Agents and Wealth Updates in Machine Learning Markets.” International Conference on Machine Learning (ICML).
• Hu, J. (2012). “Combinatorial Modelling and Learning with Prediction Markets.” arXiv:1201.3851.
• Barbu, A. & Lay, N. (2012). “An Introduction to Artificial Prediction Markets for Classification.” Journal of Machine Learning Research 13.
• Lay, N. & Barbu, A. (2012). “The Artificial Regression Market.” arXiv:1204.4154.

Boosting as functional gradient descent

• Rosset, S. & Segal, E. (2002). “Boosting Density Estimation.” Advances in Neural Information Processing Systems 15 (NeurIPS).
• Freund, Y. & Schapire, R. E. (1997). “A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting.” Journal of Computer and System Sciences 55(1).
• Mason, L., Baxter, J., Bartlett, P. L. & Frean, M. (1999). “Boosting Algorithms as Gradient Descent.” Advances in Neural Information Processing Systems (NeurIPS).
• Friedman, J., Hastie, T. & Tibshirani, R. (2000). “Additive Logistic Regression: A Statistical View of Boosting.” The Annals of Statistics 28(2).
• Friedman, J. H. (2001). “Greedy Function Approximation: A Gradient Boosting Machine.” The Annals of Statistics 29(5).

The reliability-dial machinery (Schur damping)

• Cotton, P. (2024). “Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios.” arXiv:2411.05807.
• Cotton, P. (2026). “Two Sides of Schur Damping: High-Dimensional Pseudo-Likelihoods and Portfolio Allocation.” arXiv:2606.14798.

Mollified scoring, fair scores & the heat ladder

• Theis, L., van den Oord, A. & Bethge, M. (2016). “A Note on the Evaluation of Generative Models.” International Conference on Learning Representations (ICLR).
• Fricker, T. E., Ferro, C. A. T. & Stephenson, D. B. (2013). “Three Recommendations for Evaluating Climate Predictions.” Meteorological Applications 20(2).
• Ferro, C. A. T. (2014). “Fair Scores for Ensemble Forecasts.” Quarterly Journal of the Royal Meteorological Society 140(683).
• Ferro, C. A. T. (2017). “Measuring Forecast Performance in the Presence of Observation Error.” Quarterly Journal of the Royal Meteorological Society 143(708).
• Bröcker, J. & Smith, L. A. (2007). “Scoring Probabilistic Forecasts: The Importance of Being Proper.” Weather and Forecasting 22(2).
• Bröcker, J. (2012). “Evaluating Raw Ensembles with the Continuous Ranked Probability Score.” Quarterly Journal of the Royal Meteorological Society 138(667).
• Bröcker, J. & Smith, L. A. (2008). “From Ensemble Forecasts to Predictive Distribution Functions.” Tellus A 60(4).
• Kimpara, D., Frongillo, R. & Waggoner, B. (2023). “Proper Losses for Discrete Generative Models.” International Conference on Machine Learning (ICML).
• Krüger, F., Lerch, S., Thorarinsdottir, T. & Gneiting, T. (2021). “Predictive Inference Based on Markov Chain Monte Carlo Output.” International Statistical Review 89(2).
• Siegert, S., Ferro, C. A. T., Stephenson, D. B. & Leutbecher, M. (2019). “The Ensemble-Adjusted Ignorance Score.” Quarterly Journal of the Royal Meteorological Society 145(S1).
• Saetra, Ø., Hersbach, H., Bidlot, J. R. & Richardson, D. S. (2004). “Effects of Observation Errors on the Statistics for Ensemble Spread and Reliability.” Monthly Weather Review 132(6).
• Candille, G. & Talagrand, O. (2008). “Impact of Observational Error on the Validation of Ensemble Prediction Systems.” Quarterly Journal of the Royal Meteorological Society 134(633).
• Bessac, J. & Naveau, P. (2021). “Forecast Score Distributions with Imperfect Observations.” Advances in Statistical Climatology, Meteorology and Oceanography 7.
• Patrini, G., Rozza, A., Menon, A. K., Nock, R. & Qu, L. (2017). “Making Deep Neural Networks Robust to Label Noise: A Loss Correction Approach.” IEEE Conference on Computer Vision and Pattern Recognition (CVPR).
• van Rooyen, B. & Williamson, R. C. (2018). “A Theory of Learning with Corrupted Labels.” Journal of Machine Learning Research 18(228).
• Stam, A. J. (1959). “Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon.” Information and Control 2(2).
• Barron, A. R. (1986). “Entropy and the Central Limit Theorem.” The Annals of Probability 14(1).
• Song, Y., Durkan, C., Murray, I. & Ermon, S. (2021). “Maximum Likelihood Training of Score-Based Diffusion Models.” Advances in Neural Information Processing Systems (NeurIPS).
• Lang, S., Leutbecher, M. & Maciel, P. (2025). “A Multi-Scale Loss Formulation for Learning a Probabilistic Model with Proper Score Optimisation.” arXiv:2506.10868.
• Wei, J., Fu, Z., Liu, Y., Li, X., Yang, Z. & Wang, Z. (2021). “Sample Elicitation.” International Conference on Artificial Intelligence and Statistics (AISTATS).

Cited in the papers: gambling systems, random projections, spatial processes

• Breiman, L. (1961). “Optimal Gambling Systems for Favorable Games.” Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability.
• Johnson, W. B. & Lindenstrauss, J. (1984). “Extensions of Lipschitz Mappings into a Hilbert Space.” Conference on Modern Analysis and Probability, American Mathematical Society.
• Vecchia, A. V. (1988). “Estimation and Model Identification for Continuous Spatial Processes.” Journal of the Royal Statistical Society: Series B 50(2).
• CrunchDAO (2024). “The MidOne Contest.”
• Cotton, P. (2024). “density: Conventions for Densities Used in Various Games.”
• Cotton, P. (2020). “The Lottery Paradox: A New Use.” Talk at MIT CSAIL, December 1, 2020
• Cotton, P. (2026). “Scoring Point-Cloud Distributional Submissions.” Working draft
• Cotton, P. (2026). “Betting Against a Conformal Predictor: a Parimutuel Account of the Information Gap.” Companion note to emphMarginally Useful
• Cotton, P. (2026). “An Algebra of Prediction-Rewarding Mechanisms.” Working draft
• Cotton, P. (2026). “Multi-Stage Solicitation of Probability Distributions: Experiments, Theory and Perspective on Conformal Prediction.” Working draft
• Cotton, P. (2026). “Likelihood versus CRPS: A New Perspective.” Working draft
• Cotton, P. (2020). “A Call for Contributions to a Copula Contest.” LinkedIn, 11 July 2020; z-curve copula streams on cryptocurrency comovements

Working on a related mechanism or implementation? Open an issue on the mechanisms repo and it will be added.