Is there any way on any exchange to pick who you trade with? For example, if I want to buy a block from someone who is selling a block, and I'm willing to pay the higher price of skipping over the nearest asks, can I do it?
Preference trading in options is pretty standard. You can direct your order flow. It's subject to NBBO and all exchange rules. Your broker also have to make it available on their router. Also economics of fees/commission will change.
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That is the very reason institutions use dark pools. To set a dp up or join them as a retailer might be difficult 'thou. Some Brokers like IB let you use Iceberg orders which do exactely that and let you access some dark pools too. Alternatively: If you find a person who is willing to buy XYZ from you for price X you could make a swap agreement that you will transfer your stocks from your broker/bank to his and cash settlement is done seperately. However it is a slow process and fees might be enormous depending on the brokers/banks.
The size of the iceberg is a huge tell in an after-hours earnings report trade fwiw. On either side. They merit watching, and if large enough, they'll certainly let you know what will happen the next day with the stock. How one plays it is up to them, but that's why you have to watch. An off the radar stock is one thing, but a stock like Google is something else. Always always always.... watch the iceberg orders in the after-hours.