Goldman Sachs is spending $100M to shave milliseconds off stock trades

Orders are filled in the order that they arrive to the exchange (if there's more than one order at the same price). A global mis-pricing would still be subject to a race to exploit it - whoever submits first gets the fill, even if it happens in the future (in the next tick).

What you're proposing is turning continuous trading into a fast series of auctions, like what happens for every ticker on every exchange at the opening. This would have the disadvantage of no clear bid/ask - how can you be sure that the parties do not withdraw their offers before the next tick? And surely you must allow for offer withdrawals.

Worked in an HFT shop and we were down to optimizing nanoseconds. Lots of folks had war stories about spending months to shave off tens of nanoseconds off pieces used millions of times per day.

Yes, that's the model. So, tell me. Would any profit be possible in an ideally functioning market? If so, how is any item worth more than the total cost of all inputs including externalities? If not, how is such a dynamic system attracted to an ideal state, given that those who would make the market ideally functional are best placed to gain from market inefficiency and dysfunction? Given an answer, do you have a sound game theory explanation for this?

Additionally, what is the nyquist limit for such an ideally realised market, if it is indeed to be modelled as a recursive sampled approximator and how is this derived? Given an infinitely recursive network of arbitrarily connected market agents, is any such calculation convergent? If so, why? If not, how does the market ever converge to any appropriate price - a price which accurately reflects the market conditions excluding pricing operations and market costs which aren't directly related to the production of the instrument in question?

Keep in mind that, if the market is functioning ideally no market participant will exceed the nyquist rate as all participants knowledge of market conditions converges to zero. How is any sampling rate, excluding zero, convergent? If not, how is any such market realisable? If so, what is the loss function between ideal model and realisable, perfectly imperfect real world implementation? What is the minimum profit, if not zero, and why?

However, it does seem that arbitrage opportunities decrease when such high-speed trading is occurring and, does so even more quickly the faster trading speed and market sampling are increased. How can we account for this, if not by increased market efficiency?

I conjecture that, by ever increasing the sampling rate and the speed at which transactions complete, markets are not being made more efficient. Instead, I hypothesise that, as markets directly effect the price of the instrument reflexively, the feedback latency produced creates relative local pockets of perceived value - which are only profitable trades in relation to local information asymmetry. As the vast majority of high-speed trading holds market positions on extremely short time scales, shifting exposure constantly, this profit is immediately realised locally resulting in the gradual diffuision of this inefficiency as the increase in price of all instruments. This is a direct result of the cost of trading being factored directly into the agent's local acceptable sale price of held instruments. Every local agent trading action is ideal, but the global market is a divergently inefficient one. Indeed, it is a market in which its pricing inefficiency is maximally concealed from all market participants.

In a sense, I conjecture that the estimator is not functioning to increase market efficiency but is, instead amplifying local inefficiency globally, in effect, much like a charge pump would operate in a voltage multiplier circuit. In essence such a scheme acts to conceal increased market cost and overhead (including the profit of market participants) into market instrument pricing. However, it does so in an extremely small and diffuse way so as to make the rise in price of a single instrument, as a result of this activity, extremely difficult to detect as all instruments increase similarly on the same time scale.

This behaviour appears to be similar in nature to 'salami slicing', an often effective embezzlement technique - except that, instead of exploiting an information asymmetry created by lack of interest in small quantities in the part of auditing accountants, it exploits the information asymmetry created by the speed of light itself.

Of course, the faster the sampling rate, the more efficient the described amplification process would take place. Does this effect correlate between markets with differing but estimable information asymmetry? If there is no correlation, this hypothesis is invalid. It would seem to be an area ripe for research and analysis of market data.

Do you see any technical issue with this conjecture by which we may discount it immediately?

You have to consider some form of multiplier effect that comes into play. Because the prices of assets are derivatives of others (incl. many, many other variables, of course) these millisecond savings end up turning into seconds, even minutes en masse.