*2.1. Logarithmic Market Scoring Rule*

The first automated market maker for prediction markets was introduced by Hanson [21,22]. It has been quite popular due to its simple analytical form and satisfying the main desirable properties for cost functions (convexity, bounded loss and translation invariance).

The Logarithmic Market Scoring Rule (LMSR) conservation function for *n* assets is defined as:

$$\mathcal{C}(\mathbf{x}) = b \log(\sum\_{i=1}^{n} \exp(\mathbf{x}\_i / b)) \tag{1}$$

where *b* > 0 is the liquidity parameter, it is strictly positive, constant and it is defined before the pricing of assets. *b* parameter controls the liquidity in the market—the higher the *b*, the less the price is shifted when assets are added. Moreover, it translates into the bigger maximum loss because the market maker's worst-case loss is the function of *b* which is *b* log *n*.

The derivative of the cost function *C*(*x*) is the price function in the LMSR:

$$p\_i(\mathbf{x}) = \frac{\exp\left(\frac{\mathbf{x}\_i}{\mathbf{b}}\right)}{\sum\_j \exp\left(\frac{\mathbf{x}\_j}{\mathbf{b}}\right)}\tag{2}$$

The LMSR is used in many settings, such as auctions, prediction markets, rating markets, etc. In decentralized finance, the LMSR has not been widely used for a few reasons: first, the LMSR does not satisfy the liquidity sensitivity property; second, it is quite easy and cheap to compromise the price of an asset when the LMSR is used as a cost function.

By allowing the parameter *b* to be the function of the outstanding quantities instead of being constant, the LMSR becomes liquidity sensitive—the Liquidity-Sensitive Logarithmic Market Scoring Rule (LS-LMSR), introduced by Othman [23]:

$$\mathcal{C}(\mathbf{x}) = b(\mathbf{x}) \log(\sum\_{i=1}^{n} \exp(\mathbf{x}\_i/b(\mathbf{x})) \tag{3}$$

where function *b* is as follows:

$$b(\mathbf{x}) = a \sum\_{i} \mathbf{x}\_i \tag{4}$$

where *α* is the parameter that is strictly positive and set before the pricing of assets. The possible maximum commission (also called vigorish *v*) depends on the *α* parameter and does not exceed *v* when *α* is set as follows [24]:

$$\alpha = \frac{v}{n \log n} \tag{5}$$

where *n* is the number of outcomes (assets in the pool for AMM-based DEX). Depending on what is the desired maximum commission *v*, the optimal parameter *α* can be easily calculated.

In decentralized finance, the LS-LMSR is used in applications such as Augur [25] and Gnosis [26].
