Given this situation there is no “price”. If there are only 2 players, and A doesn’t want to sell to B, you can’t force them to (except in very specific situations of malfeasance). If there is no seller (or no buyer) you cannot [make a market](https://www.investopedia.com/terms/m/makeamarket.asp). It’s that simple.
If it was listed on a stock exchange most exchanges would require a [Market Maker](https://www.investopedia.com/terms/m/marketmaker.asp) (someone who always has an offer to buy and sell).
This is to avoid situations like your scenario.
However if for some reason your scenario did occur that level of inactivity would result in [No Quote](https://www.investopedia.com/terms/n/noquote.asp).
Closely held companies – which is what you’re describing – aren’t listed on a stock exchange and don’t have a simple valuation of their shares. Person B’s interest is valued at whatever they can get someone else to pay them for it. Until they have a buyer lined up for their shares, the “value” is whatever Person B thinks it is.
Person B generally can’t force Person A to sell their shares. If Person A is running the business in a way that is detrimental to Person B, then it is *possible* that Person B may be able to convince a court to force Person A to buy Person B’s shares. How the share are valued in that situation depends on the state where the company is incorporated and how big it is.
There are plenty of accounting firms that will value companies based on their assets and income. The more common option is valuation would be that both Person A and Person B will hire one of those firms to value the company and try to convince the judge that their valuation is correct. But there are a lot of other possibilities as to how a court can determine the proper valuation of a company in that situation.
Fwiw, there are other ways to value a company. Market cap is just 1 of them.
You can do a technical analysis of its Financials and compare it to other similar companies.
Or you can do a discounted cash flow valuation based an expected future performance.
In an efficient market all 3 things should be similar, varying only by the assumptions you must make in each. In reality… to actually lay out the assumptions to support their market cap can be a sobering realization.
On many cases such as you describe, there is a way in the company bylaws to resolve this. An example is the “double ended shotgun”. Person A offers $100/share. Person B can accept that offer (A pays B and A then owns all the shares) OR they can choose to instead reverse the buyout and choose to buy A’s shares at the price A offered (so B end up with all the shares). This approach forces A to offer a fair price, even overpaying so that B doesn’t switch around the shotgun.
Most companies that start off with this sort of ownership structure will have these sorts of bylaws. If not, there’s no way to resolve it.
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