LMSR for the layman
Gensyn recently released Delphi, a model prediction market where evaluations are transparent and priced on-chain using a Logarithmic Market Scoring Rule (LMSR). AI models compete in evaluations, and participants are able to trade positions as performance data emerges.
While there are already good resources explaining the mathematics behind LMSR, including a blog post by Gabriel, if you've never interacted with a prediction market before, much of this can sound opaque. This post is written for readers starting from zero, explaining how markets work and why LMSR exists.
What are prediction markets?
Prediction markets turn beliefs into prices. Each market revolves around a question with a clear outcome. For each outcome, the market offers a contract, costing between $0 and $1. At market close, a contract pays $1 if the outcome happens, and $0 if it doesn't. When the market is open, the cost of a contract reflects the probability an event will happen.
Prediction markets exploit the same forces the Efficient Market Hypothesis (EMH) describes. EMH states that prices reflect available information. Consider a market Will Cthulhu become Prime Minister of Greenland?.
The answer is obviously no. If this market were priced at 50/50, it would not be efficient. Anyone could buy NO shares, profiting from public, obvious information and push the price toward certainty. It's not that mispricings can't exist, it's that they don't persist. Markets correct themselves because traders are incentivized to exploit errors.
Studies show that prediction markets are effective at predicting future events because of their ability to adapt to new information more quickly than other forecasting and polling methods. The difference between polls and prediction markets mirrors the difference between centralized and decentralized systems. Polls collect opinions and rely on a central rule to aggregate them. Prediction markets decentralize that process; participants decide for themselves how much weight their belief deserves by risking capital. No authority assigns credibility, or decides who participates.
How traditional markets work
Most prediction markets today work like traditional financial exchanges, utilizing an order book. In an order book market, buyers and sellers must match up. Buyers post bids ("I want to buy YES at $0.42"), and sellers post asks ("I want to sell YES at $0.44). Trades only happen when the two meet, and the current price is simply the most recent trade that's occurred. This works well when there are many active traders and liquidity is deep. When activity is low, or there's shallow liquidity (tiny, scattered orders), prices can stall, and may stop reflecting what people actually believe.
LMSR was designed to solve this problem. Instead of waiting for buyers and sellers to match up, LMSR acts a participant that is always willing to trade.
How an LMSR works
LMSR replaces order books with the idea that the market is always available to trade. Instead of waiting for another participant to take the other side of your bet, you trade against the LMSR itself.
Because participants trade against the LMSR, prices don't come from buyers and sellers matching up. Instead, prices come from a cost function, a formula that determines how much it costs to move a market's belief.
When a trader buys shares of an outcome, they don't pay another person, they pay the market. The amount they pay is the difference between the market's total cost before and after the trade. The total cost is the amount of money the market has collected so far from all traders combined.
The total cost exists so the market can charge traders when they move prices and guarantee it can always pay out winners. The details about total cost are handled internally by the LMSR. Traders see the current price, and how much it costs to move the market.
Consider a market Will it rain tomorrow?, and the current price of YES is $0.60, which you read as a 60% chance it will rain. You believe rain is more likely than that, so you buy YES shares. Behind the scenes, the LMSR calculates the market's total cost before your trade and what it will be after. You pay the difference, and in exchange receive YES shares that will pay out to $1 if it does rain tomorrow, and $0 if it doesn't.
On the flip side, if you believe it's not likely to rain tomorrow, you can buy NO shares. Similarly, if you already bought YES shares and have since changed your mind about the probability of rain, you can sell your shares. The total cost of the market would be lower if you sold your YES shares, and the market will pay out the difference to you.
Because the LMSR doesn't depend on participants buying and selling with each other, the LMSR needs initial liquidity that is provided by the market creator. An LMSR is created with a parameter that controls how easily prices move. This parameter also determines the maximum amount of money the market maker can lose. To start the market, the creator provides an initial subsidy equal to this maximum loss. Over the life of the market, some traders pay in by moving prices the wrong way, and others get paid by moving prices the right way.
Wrapping up
After understanding LMSRs at a high level, use cases become more clear. Users can create markets to test their own hypotheses and see how others respond, all while knowing the maximum cost to run the market up front. Rather than producing a one-time result, these markets generate a continuously updating signal that reflects new information as it arrives.
In the future, evaluations of model performance could be decentralized and verified using Verde.
Imagine a market creator asks Which model identifies the most bugs in a JavaScript codebase?. The market creator pays an initial subsidy to bootstrap the market and incentivize information discovery. Another party of Evaluators could run evaluations on models, providing updates to the state of the market. In parallel, a separate group of evaluators could independently check the evaluations are correct. Both evaluators and verifiers could be incentivized through payments, while market participants are incentivized by the opportunity to profit from accurately anticipating the outcome. Together, this creates a system where forecasting, evaluation, and verification are decoupled but economically aligned.