# 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.

• Not exactly rigorous, but this might be helpful: arxiv.org/abs/1002.2284 Aug 14, 2019 at 1:13

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? $$O \implies O$$

• Is it truly necessary for "all market participants to agree on a singular pricing model" for EMH to be true? I'm pretty sure that if all market participants agree on a singular pricing model, except one participant that has 1\$ to invest and a different pricing model, then the market will still converge to the price that everyone agreed on. In general, intuitively, if "most of the capital" agrees that the price is in a pretty narrow band, then the market will probably converge. I'm trying to find a formal theorem that formalizes this intuition. It's non-trivial, therefore not a tautology.
– greg
Aug 14, 2019 at 18:18

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.

• Ah, this answer might lead to an answer to my question! I got to looking at Fisher markets. A Fisher market with m products is kind of like a stock market with stock of m companies, where the utilities of the buyers represent the beliefs of the market participants about the price. Now, if I take the calculation method for Fisher markets and simplify it to these particular types of utility functions, I might get a simple theorem proving Price Discovery. I'd love to find it in the literature rather than proving it myself, but I think this is quite doable to do myself. Will update.
– greg
Aug 14, 2019 at 18:33