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Link between Logarithmic market scoring rule and Automated market makers

In most articles I read about AMMs, there is a link or a mention to Hanson’s work about the Logarithmic scoring rule. But I don’t clearly see the link between AMM’s and the LMSR. Can someone help me ? Like, on Uniswap are used some constant product function, and in some articles (see below) are links to LMSR (Hanson, 2002). Thank you
Here are some articles talking about constant functions market makers : https://arxiv.org/pdf/2003.10001.pdf
You can see than there is a reference to Hanson’s work in the intro.
here is the article about Hanson’s Work:
http://mason.gmu.edu/~rhanson/mktscore.pdf

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