Chapter 8: Further Reading

Foundational Papers

Hanson's Original LMSR Paper

Hanson, R. (2003). "Combinatorial Information Market Design." Information Systems Frontiers, 5(1), 107-119.

The paper that started it all. Hanson introduces the concept of market scoring rules as automated market makers and presents the logarithmic market scoring rule (LMSR) as the most practical variant. The paper covers the general framework, the combinatorial extension, and the bounded loss property. Essential reading for anyone serious about prediction market design.

Hanson's Market Scoring Rules

Hanson, R. (2007). "Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation." Journal of Prediction Markets, 1(1), 3-15.

A more detailed treatment of market scoring rules, with particular emphasis on the logarithmic variant. Hanson explains how scoring rules can be chained together to create markets and provides the formal proof of bounded loss.

Othman's LS-LMSR

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), Article 14.

The definitive paper on the Liquidity-Sensitive LMSR. Othman shows that making the liquidity parameter depend on trading volume solves the fundamental tension between price sensitivity and subsidy cost. Includes theoretical analysis and experimental results from a real prediction market.

Chen and Pennock on Cost Function Properties

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).

A rigorous treatment of the mathematical properties required of cost-function-based market makers. Establishes the connection between cost functions, convexity, path independence, and no-arbitrage conditions. Important for understanding why LMSR's properties are not accidental but are fundamental requirements.

DeFi and Constant Product Market Makers

Uniswap Whitepaper

Adams, H., Zinsmeister, N., & Robinson, D. (2020). "Uniswap v2 Core." Uniswap Whitepaper.

The technical specification of Uniswap v2, the most influential CPMM implementation. While designed for token swapping rather than prediction markets, the $x \cdot y = k$ invariant and its properties are directly applicable. Readable and concise.

Available at: https://uniswap.org/whitepaper.pdf

Uniswap v3: Concentrated Liquidity

Adams, H., Zinsmeister, N., Salem, M., Keefer, R., & Robinson, D. (2021). "Uniswap v3 Core." Uniswap Whitepaper.

Introduces concentrated liquidity, where liquidity providers can focus their capital in specific price ranges. The concepts of capital efficiency and range-limited liquidity provision have direct applications to prediction market AMM design.

Available at: https://uniswap.org/whitepaper-v3.pdf

Angeris and Chitra on AMM Properties

Angeris, G. & Chitra, T. (2020). "Improved Price Oracles: Constant Function Market Makers." Proceedings of the 2nd ACM Conference on Advances in Financial Technologies.

A mathematical analysis of constant function market makers (the general class that includes CPMM). Proves important properties about price bounds, arbitrage, and information aggregation. Bridges the gap between DeFi AMMs and prediction market theory.

Prediction Market Platforms and Implementations

Gnosis and LMSR Implementation

Aitken, R. (2017). "Gnosis Prediction Market Platform Uses Ethereum To 'Crowdsource' Wisdom." Forbes.

An accessible overview of how Gnosis implemented LMSR on the Ethereum blockchain. Covers the practical challenges of implementing a mathematical AMM in a decentralized environment, including gas costs and numerical precision.

Manifold Markets Design Decisions

Manifold Markets Documentation. "How Manifold Markets Works."

Technical documentation from Manifold Markets explaining their AMM mechanism and its evolution over time. Useful for understanding how theoretical AMM designs are adapted for real-world platforms with play money and user-created markets.

Available at: https://docs.manifold.markets/

Polymarket's Hybrid Approach

Polymarket Documentation. "How Polymarket Works."

Documentation on Polymarket's design, which combines order book matching with AMM backstop liquidity. Illustrates the hybrid approach discussed in Section 8.1.

Available at: https://docs.polymarket.com/

Textbooks and Surveys

Abernethy et al. on Market Making

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), Article 12.

Connects market making to online learning theory, showing that the LMSR cost function corresponds to an entropy regularizer. This deep theoretical connection helps explain why LMSR works so well and suggests new AMM designs based on different regularizers.

Prediction Markets: Theory and Applications

Arrow, K. J., Forsythe, R., Gorham, M., Hahn, R., Hanson, R., Ledyard, J. O., ... & Zitzewitz, E. (2008). "The Promise of Prediction Markets." Science, 320(5878), 877-878.

A brief but influential overview by leading economists arguing for the broader adoption of prediction markets. While not focused on AMMs specifically, it provides context for why AMM-based prediction markets matter.

Manski on Prediction Market Interpretation

Manski, C. F. (2006). "Interpreting the Predictions of Prediction Markets." Economics Letters, 91(3), 425-429.

An important paper cautioning against naively interpreting prediction market prices as probabilities. Relevant because AMM design affects how closely market prices correspond to true beliefs. Understanding this distinction is important for interpreting AMM-produced prices.

Technical References

Log-Sum-Exp Trick

Blanchard, P., Higham, D. J., & Higham, N. J. (2021). "Accurately Computing the Log-Sum-Exp and Softmax Functions." IMA Journal of Numerical Analysis, 41(4), 2311-2330.

A thorough treatment of the numerical issues in computing log-sum-exp and softmax (which is the LMSR price function). Essential reading for anyone implementing LMSR in production, covering overflow, underflow, and precision loss.

Proper Scoring Rules

Gneiting, T. & Raftery, A. E. (2007). "Strictly Proper Scoring Rules, Prediction, and Estimation." Journal of the American Statistical Association, 102(477), 359-378.

The definitive reference on proper scoring rules, which are the mathematical foundation of LMSR. Understanding why the logarithmic scoring rule is proper helps explain why LMSR has such nice theoretical properties.

Advanced Topics

Combinatorial Prediction Markets

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.

Analyzes the computational complexity of combinatorial prediction markets, where markets are run over combinations of events. Shows that exact combinatorial LMSR is computationally intractable but proposes approximation techniques.

Dynamic Market Making

Das, S. & Magdon-Ismail, M. (2008). "Adapting to a Market Shock: Optimal Sequential Market-Making." Advances in Neural Information Processing Systems, 21.

Explores how an AMM should adapt when the market experiences a sudden shift (e.g., breaking news). Relevant to the discussion of dynamic AMMs in Section 8.11.

AMMs and Information Aggregation

Ostrovsky, M. (2012). "Information Aggregation in Dynamic Markets with Strategic Traders." Econometrica, 80(6), 2595-2647.

A theoretical analysis of how well market mechanisms aggregate information from strategic traders. Shows conditions under which prediction markets (including AMM-based ones) converge to the true probability.

Automated Market Makers with Fees

Othman, A. & Sandholm, T. (2011). "Liquidity-Sensitive Automated Market Makers via Homogeneous Risk Measures." Proceedings of the 7th International Workshop on Internet and Network Economics.

Extends AMM design to include fee structures that can make the market maker self-sustaining. Analyzes the trade-off between fee revenue and the theoretical properties (like path independence) that fees may break.

Online Resources

Robin Hanson's Blog (Overcoming Bias)

https://www.overcomingbias.com/

Hanson's long-running blog where he regularly discusses prediction markets, scoring rules, and information aggregation. Many important ideas about AMM design first appeared here as informal posts before becoming formal papers.

Vitalik Buterin on AMMs

Various blog posts at https://vitalik.eth.limo/

Buterin has written extensively about AMMs from the DeFi perspective, with several posts relevant to prediction markets. His analysis of different constant function market makers is particularly clear and insightful.

Metaculus Track Record

https://www.metaculus.com/questions/track-record/

While Metaculus does not use an AMM (it uses aggregated probability distributions), its track record page demonstrates the calibration standards that AMM-based platforms should aspire to. Useful as a benchmark for evaluating AMM accuracy.