# How to Set Specific Probabilities Using the LMSR?

How many shares do you need to buy/sell in the in a prediction market using the Logarithmic Market Scoring Rule (LMSR) to change the price/probability of an asset to a specific value?

This post assumes knowledge about prediction markets and market scoring rules. I think the original paper by Robin Hanson provides a good introduction.

There are basically two approaches to buying/selling the right amount of shares to reach a certain probability. Method #1 is the sledgehammer approach, Method #2 is elegant, but tedious for more complicated price functions.

## Method #1

Doing some kind of binary search until the price is “close enough”. I’ve written some example code in Clojure:

(defn set-to-prob [q i prob]
(loop [lower (magically-find-lower-bound q i prob)
upper (magically-find-upper-bound q i prob)]
(let [q_i'  (/ (+ upper lower) 2)
q'    (assoc q i q_i')
p'    (p i q')]
(if (close? 0.1 p' prob)
q'
(if (> p' prob)
(recur lower q_i')
(recur q_i'  upper))))))


Other than inelegance, this has the obvious flaw that it’s difficult to find reasonable upper and lower bounds for the number of outstanding shares.

I used a simple implementation for the upper (lower) bound that added (subtracted) a multiple m$m$ of the liquidity parameter b$b$, but it was always possible to force an infinite loop by using an input \vec{q}$\vec{q}$ so that \max_{i, j} \vert q_i - q_j \vert > m \, b$\max_{i, j} \vert q_i - q_j \vert > m \, b$.

## Method #2

It’s possible to solve analytically for the number of shares q_i$q_i$ to move the price to a certain probability p_i$p_i$.

I’ve actually found the solution in this GitHub repository, but there was no hint as to how the solution was arrived at (maybe obvious when you have better math skills 🙁). However, eventually I was able to figure it out 😏. So if you are like me and need to have every baby step laid out, here they are:

We know the price function for the LMSR is

p_i(\vec{q}) = \frac{e^{q_i/b}}{\sum_j{e^{q_j/b}}}

For a specific price/probability p_i$p_i$ for outcome i$i$, it must be the case that

p_i = \frac{e^{q_i/b}}{e^{q_i/b} + \sum_{j \neq i}{e^{q_j/b}}}

where \sum_{j \neq i}$\sum_{j \neq i}$ iterates over all indices in \vec{q}$\vec{q}$ except for i$i$.

Using some math magic:

\begin{aligned}
p_i \left(e^{q_i/b} + \sum_{j \neq i}{e^{q_j/b}}\right)           &= e^{q_i/b} \\[2em]
p_i\, e^{q_i/b} + p_i \sum_{j \neq i}{e^{q_j/b}}                  &= e^{q_i/b} \\[2em]
p_i \sum_{j \neq i}{e^{q_j/b}}                                    &= e^{q_i/b} - p_i\, e^{q_i/b} \\[2em]
p_i \sum_{j \neq i}{e^{q_j/b}}                                    &= e^{q_i/b}(1 - p_i) \\[2em]
\frac{p_i}{1 - p_i} \sum_{j \neq i}{e^{q_j/b}}                    &= e^{q_i/b} \\[2em]
\log{\left(\frac{p_i}{1 - p_i} \sum_{j \neq i}{e^{q_j/b}}\right)} &= \frac{q_i}b
\end{aligned}

q_i = b\ \log{\left(\frac{p_i}{1 - p_i} \sum_{j \neq i}{e^{q_j/b}}\right)}

which is the number of shares that element i$i$ in the quantity vector \vec{q}$\vec{q}$ needs to be changed to, in order for the price of contract i$i$ to reach price p_i$p_i$.

For the number of shares \Delta q_i$\Delta q_i$ to buy/sell, the current amount of shares q_{i,t}$q_{i,t}$ needs to be subtracted \Delta q_i = q_i - q_{i,t}$\Delta q_i = q_i - q_{i,t}$.

Now somebody needs to do the same for the the liquidity-sensitive LMSR

p_i(\vec{q}) = \alpha \log\left({\sum_j{e^{q_j/b(\vec{q})}}}\right) +
\frac{\sum_j{q_j \, e^{q_i/b(\vec{q})}} - \sum_j{q_j \, e^{q_j/b(\vec{q})}}}
{\sum_j{q_j} \sum_j{e^{q_j/b(\vec{q})}}}

where

b(\vec{q}) = \alpha \sum_j{q_j}

How hard can it be?