Limits of Proving Price Discovery and Market Equilibrium
Summary
The discussion asks whether price discovery can be mathematically guaranteed for a market mechanism, including mechanisms that do not use a conventional order book. One response argues that a proof of the Efficient Market Hypothesis cannot follow without strong assumptions, such as participants sharing a common pricing model; otherwise, the claim risks reducing to a statement about opinions reflecting themselves.
Another response points to general equilibrium results associated with Arrow and Debreu and fixed-point methods. These results establish equilibrium under abstract assumptions, while the answer suggests that details of a particular trading mechanism may matter less when prices influence excess demand and excess demand feeds back into prices. The exchange does not specify the assumptions, prove convergence for a concrete mechanism, or resolve whether equilibrium existence guarantees price discovery in practice. Its value is mainly in separating philosophical claims about market beliefs from formal equilibrium analysis.
Key ideas
- A general proof of market price discovery requires explicit assumptions about participants and their pricing models.
- The Efficient Market Hypothesis is not established as a universal mathematical theorem in the discussion.
- General equilibrium results use abstract conditions and fixed-point arguments.
- Equilibrium existence does not by itself provide a demonstrated convergence result for a specific market mechanism.
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Full text
# How to Mathematically Prove Markets are Price-Discovering? # How to Mathematically Prove Markets are Price-Discovering? We all know that the Efficient Market Hypothesis is true if you're willing to make enough simplifying assumptions about the market participants. But where can I find a mathematical proof of this in the literature? Details: The Efficient Market Hypothesis, in its softest version, basically says that if enough people agree on the price of an asset, then when the asset is traded on an open market, the market will converge to that price. But where can I find a mathematical proof of this fact, of the form "For a market trading stock A, under assumptions X, Y and Z about all market participants and about the stock, the market is guaranteed to converge to a price that can be calculated as follows: ...". I ask this because I'm trying to prove that an unorthodox market mechanism (which is not two-sided markets using an order book) also has the Price Discovery property, and I'm not even sure under what conditions a traditional two-sided market has the Price Discovery Property. ## Answer by amdopt (score 1) https://quant.stackexchange.com/a/47115 There is no mathematical proof of EMH. You would need all market participants to agree on a singular pricing model for that to be possible. Without a singular, agreed-upon model, what you are asking for is proof that people's collective opinions represent people's collective opinions. This is called a tautology. Tautology There is an endless number of tautologies. Verifying a formula is a tautology is possible with propositional logic. Do people's opinions imply people's opinions? $O \implies O$ ## Answer by nbbo2 (score 1) https://quant.stackexchange.com/a/47116 There are famous proofs by Arrow & Debreu and others, based on the Kakutani Fixed Point Theorem, but they are at a very abstract and general level. I am not sure the details of market mechanism matter, as long as excess demand affects prices, and prices affect excess demand, there will be a convergence to a price equilibrium.
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