Computer-driven American stock markets have become so complex that at any moment in time more than 800 different pricing possibilities are being offered to trading firms across 12 official exchanges, according to new research attempting to explain the tangled system.

The report was prepared by the Royal Bank of Canada, one of the most outspoken critics of the computer-driven American stock market. It will be released to clients this week in advance of a Senate hearing on Thursday that will examine how the structure became so convoluted and what might be done to improve it.

The complexity is a result, in part, of the constant jockeying among exchanges to win business from the biggest traders, many of which are so-called high-frequency trading firms that make money by capitalizing on small changes in prices.

RBC Capital Markets, the division of the bank which led the research, found that the New York Stock Exchange, Nasdaq and other exchanges make frequent small tweaks to their prices, influencing trading behavior in ways that are hard for even sophisticated investors to understand.