In what order and at what price would these orders be executed? Consider the case that both the buyer and the seller submitted orders to their brokers while the market was closed.
It seems logical to me that the mean price between the bid and ask spread would be used. Is that the case?
Does it depend on the brokerage?
This is governed not by the brokerage but by the exchange, specifically the "opening rotation" procedure. If there is a sell (limit) order at $10 and a buy order at $11 going into the open, the first trade would probably be designated at the midpoint ($10.50) for the lesser of the two quantities. But it is unlikely that these would be the only orders.
More typically, in an active stock, a spectrum of orders will pin down the opening price based on clearing the market. A hypothetical opening price of $X would involve executing all sell orders below $X (total shares an increasing function of X) and all buy orders above $X (total shares a decreasing function of X), along with any or all of the orders at exactly $X. Up to the price granularity at which orders are typically spaced, there will be a unique price (solution for X) at which an equal number of shares is bought and sold, and this is how the exchange determines the opening price.
If all buy orders are below all sell orders at the open, the security may not have an "opening price" at all (unless a market maker is required to produce one). The orders would simply initialize the day's order book, and any trades would occur if and when someone hits the evolving bid or ask. In the extreme, daily volume might be zero (as with many options).
EDIT: See here for the NYSE opening procedure. On this exchange, the opening price $X is the one where the maximum number of shares can be traded -- effectively equivalent to the supply-demand balancing described above. Ties are resolved by keeping as close as possible to a reference price (rather than a midpoint rule). Specific examples are given with "crossed" orders (buy above sell) as well as other scenarios.
A crossed market is when the bid price exceeds the ask price. This is an infrequent occurrence that is more likely in a fast market and lasts for mere seconds, if that. It arises from different prices on different exchanges being collated into NBBO. Here's my simple understanding of what happens:
Suppose the bid is $25.00 and the ask is is $25.01. If you place an order to buy for more than $25.01 you will be filled at $25.01, assuming there were enough shares at that price to fill your buy order.You are filled at the best ask price.
Now suppose the market briefly crosses and the B/A is $25.02 x $25.01. The seller is still at $25.01 and any routed orders will be sent to the exchange with the $25.01 ask price.
