How to update fair value estimates based on observed market quotes in prediction markets/trading games?

External question — Cross Validated Stack Exchange Author: bigstreet Original post: https://stats.stackexchange.com/questions/674163 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. Context: I'm playing a trading game (specifically Figgie ( https://www.figgie.com/how-to-play.html ), but this applies broadly) where: Each player has private information (their own hand of cards) Players submit bids and asks for different "assets" (card suits) The true value of each asset is unknown initially but can be inferred from trading activity and your distribution(initial) I need to continuously update my price estimates based on what others are bidding The Problem Suppose I initially estimate an asset's fair value at $3 based on my private information. Then I observe: Player A bids 6 (willing to buy at 6) Player B offers at 8 (willing to sell at 8) Player C bids $5 How should I update my fair value estimate to V_new? I'm looking for a principled framework (ideally with mathematical justification) for updating price estimates given observed market quotes, especially in contexts where: Information is distributed among players Quotes reveal players' private information I need to update my probability vector repeatedly based on updated beliefs