  This paper compares mathematical models for automated market makers including
logarithmic market scoring rule (LMSR), liquidity sensitive LMSR (LS-LMSR),
constant product/mean/sum, and others. It is shown that though LMSR may not be
a good model for Decentralized Finance (DeFi) applications, LS-LMSR has several
advantages over constant product/mean based automated market makers. However,
LS-LMSR requires complicated computation (i.e., logarithm and exponentiation)
and the cost function curve is concave. In certain DeFi applications, it is
preferred to have computationally efficient cost functions with convex curves
to conform with the principle of supply and demand. This paper proposes and
analyzes constant circle/ellipse based cost functions for automated market
makers. The proposed cost functions are computationally efficient (only
requires multiplication and square root calculation) and have several
advantages over widely deployed constant product cost functions. For example,
the proposed market makers are more robust against front-runner (slippage)
attacks.
