CHAPTER 15 OF 15

Appendix: the pricing formula

Chapter 6 describes FIRM's pricing in plain language. This appendix gives the exact math for readers who want it.

FIRM’s demo engine prices YES and NO shares using a logarithmic market scoring rule (LMSR), a standard bounded-loss automated market maker used across the prediction-market industry.

Cost function

For outstanding YES shares qYES, outstanding NO shares qNO, and a liquidity parameter b, the market maker's cost function is:

C(q) = b · ln( e^(qYES / b) + e^(qNO / b) )

The cost of any trade is the change in C(q) between the market's state before and after the trade.

Price function

The instantaneous price of YES is the cost function's gradient with respect to YES shares, which works out to a logistic (softmax) function of the net position:

P(YES) = e^(qYES / b) / ( e^(qYES / b) + e^(qNO / b) )

P(YES) and P(NO) always sum to 1. As the difference between qYES and qNO grows, price moves smoothly toward the edges, fastest near 50 percent and flattest near the extremes, and never actually reaches exactly 0 or 100 percent.

Worked example

With a liquidity parameter b = 220, a real value used in FIRM's live demo pricing, the YES price moves like this as net shares outstanding (qYES − qNO) changes:

Bounded worst case

LMSR's defining property is a hard ceiling on the market maker's worst-case loss, reached only if every share outstanding ends up on one side:

Maximum loss = b · ln(2)

With b = 220 and a $1 payout per winning share, that ceiling is approximately $152.49 per market, before fees. This is the reason FIRM can always quote a price on both sides of every market without taking on open-ended risk: the most the market maker can ever lose on one market is a known, fixed number, calculable before the market even opens.

Fees

A flat fee, currently 1 percent, applies to buys and sells. Buy fees are added on top of the trade's cost; sell fees are subtracted from the trade's proceeds. This fee is a current default and can vary by market.