Convoy protection

Why does gathering merchant ships into one large cluster make fewer of them sink?

Putting forty merchant ships into one formation looks like an obvious mistake. You have made a target forty times larger. If it is found, the attacker has forty chances instead of one. Spreading the ships across the ocean seems safer — a loss can only ever be one ship, and no single misfortune can be a catastrophe.

That was the intuition, and it was held by serious naval officers who argued against convoying on exactly those grounds. It is wrong, and understanding why requires noticing that the intuition answers a question about consequences while the problem is about encounters.

Start with how a ship is attacked at all. Before anything can be sunk, it must be found — and the ocean is enormous relative to the range at which a submarine can see. A submarine patrols a strip of water and detects whatever crosses it. The overwhelming majority of its time is spent finding nothing.

So ask what concentration does to the number of detection opportunities. Forty independent ships crossing the ocean present forty separate chances to be stumbled upon. Gather them into one formation and they present one. The formation is physically larger, but not forty times larger in the sense that matters: a convoy occupying a few miles of sea is still a point target against an ocean of thousands of miles, and a patrol line is only marginally more likely to intercept it than to intercept a single ship.

That is the central asymmetry, and it is worth stating plainly. Concentration divides the number of encounters by the number of ships, while multiplying the width of the target only slightly. The chance of being found falls almost in proportion to how many ships you put in each convoy.

Now the second effect, which is about defence. Suppose escorts must screen the perimeter of the formation. The area occupied by N ships at fixed spacing grows as N, so the perimeter grows as roughly the square root of N. A convoy of forty ships therefore needs only about twice the escort screen of a convoy of ten — for four times the cargo. Escort strength per ship improves as convoys get larger. This is the result that surprised people most at the time, because it says the answer to a shortage of escorts is bigger convoys rather than smaller ones.

Third, concentration converts the escort from a defender into a hunter. An attacker who approaches a convoy must come to where the escorts already are. Scattered shipping obliges escorts to search; convoying obliges the submarine to present itself.

Now put the intuition back in. Yes — when a convoy is found, more ships are at risk in that encounter, and losses come in clusters rather than singly. But the number of encounters has fallen by a much larger factor than the losses per encounter have risen, and an attacker has limited torpedoes and limited time before the escorts drive it off. The expected loss is encounters multiplied by losses per encounter, and concentration reduces the first term far more than it raises the second.

This is also why the historical argument was so hard to settle by experience: both sides could point to real events. Convoy losses were visibly dramatic when they happened. The ships that were never found generated no evidence at all.

"Bigger convoys are safer per ship" is the counter-intuitive part, and it has limits. The escort-perimeter argument favours large convoys, and the historical analysis supported it. But the benefit is not unbounded: very large convoys are slower to assemble, harder to manoeuvre, and hold ships in port waiting to sail, which costs delivered tonnage even if it saves ships. The optimum is a trade between sinkings avoided and cargo delayed, and it is not the same number in every campaign.

Convoying slows the whole to the speed of the slowest, and that is a real cost paid in cargo. A convoy sails at the pace its slowest ship can manage, and ships wait for assembly. Fast ships were sometimes routed independently for exactly this reason — a genuine exception, since a ship fast enough to outrun a submerged attacker does not need the protection as much.

Concentration and escort are separable, and it matters which is doing the work. Some of the benefit comes from the escorts and some from the mere fact of concentration. Unescorted convoys still reduce encounters, and that is the part of the effect that depends purely on the search geometry. Attributing the whole benefit to escorts overstates their contribution and understates the arithmetic.

On the historical record, I would be careful. Tonnage figures, sinking rates and the relative contribution of convoying, air cover, radio intelligence and improved sensors are all argued over, and the analyses were produced by parties with an interest in which measure had worked. The structural argument above stands on its own; the quantitative apportionment does not, and I am not going to give figures for it. No casualty figures appear here for the same reason.

The falsification test. If the mechanism is reduced encounters rather than better defence, then unescorted convoys should still lose a smaller fraction of ships than the same ships sailing independently, and losses per convoy encountered should be a small fraction of the convoy's size. If independently routed ships had shown the same loss rate as convoyed ones, the search-geometry account would be wrong and the benefit would have to be entirely about escorts.

• Why evasive routing and radio intelligence compounded with concentration rather than substituting for it.
• How search theory generalises this to any problem of finding sparse targets in a large space.
• Why fast ships were sometimes routed independently, and what that says about the limits of the argument.