[Photo by Rafael Garcin on Unsplash]

In C. S. Forester’s Hornblower and the Hotspur, Horatio Hornblower is a captain of a three-masted sloop (one of the smaller ships at the time), the Hotspur, in the (British) Royal Navy. It is 1803 and there is a temporary peace, the peace of Amiens, during which the Hotspur is patrolling some parts of the coast of France. The episode I want to study begins with a French ship, the Loire, a frigate that is a good deal bigger than the Hotspur, leaving her anchorage in the direction of the Hotspur. Captain Hornblower correctly suspects (through an interesting series of what one could describe as Bayesian probabilistic inferences) that war has been declared and, given the size disadvantage, sets a course to avoid a confrontation.

This situation can now be described as a game between two players, the two ships (or their two captains), with opposing preferences: The Loire wants to catch up with the Hotspur, and the Hotspur wants to evade the Loire. To finalize our model, we need to specify the available strategies. These are all the different directions that the ships could go in, taking into account some geographical (and weather) constraints. From the goals that the two captains have, we can derive payoffs for any strategy combination. These are presumably so that the Loire always prefers to go in the same direction as the Hotspur, and the Hotspur prefers to go in any direction that is different from the direction the Loire takes. A more careful reading of the book suggests a secondary payoff-relevant concern. Given the possibility of something (exogenously, as we like to say) happening (such as bad weather or the sudden appearance of another (most likely British) ship), the Loire would probably like to catch up with the Hotspur as quickly as possible, while the Hotspur would like to delay such an event as much as possible. This is highly relevant in the present case, because it quickly emerges that the Loire is the (slightly) faster ship.

Having thus verbally fully described the game between the two ships, we can turn to the equilibrium analysis. Equilibrium play seems very plausible in this case for two reasons. One, the game is relatively simple, and two, this is not the first time in the history of naval warfare that one ship tries to catch another ship. Together, these two observations make it quite likely that each of the two captains of the two respective navies (from their training and their experience) has learned to behave optimally given the other captain’s behavior. In short, it seems likely that they have learned Nash equilibrium behavior.

It seems that, at least in the present case, the best course of action for the Hotspur and, thus, the Hotspur’s equilibrium behavior, assuming (correctly) that the Loire would follow wherever the Hotspur goes, is to sail into the wind. I don’t know that much about sailing, but I have been given to understand that you cannot sail directly into the wind. You can only sail at some (maximally acute) angle against the direction that the wind is coming from. The geographical realities in the present case are such that the two ships cannot go in the same direction against the wind for too long before they would run aground just off the coast. The escaping ship, therefore, has to occasionally “tack”, that is, to turn to move along the opposite (maximally) acute angle against the wind, thereby going into the wind in a zigzag fashion. In equilibrium and, indeed, also in the book, the Loire tacks whenever the Hotspur does to lose as little time as possible. This highly predictable behavior now goes on for quite some time. During this time the officers of the Hotspur make fairly precise and worrying predictions as to how long they have before the Loire catches them.

But then there is an interesting twist. Some small isolated low clouds appear just above the water. Noticing these, Hornblower decides to delay tacking his ship beyond what would otherwise be seen as optimal until the Hotspur is hidden by one of these clouds. When the Hotspur comes out of the cloud on the other tack, Hornblower realizes to his surprise that the Loire is also already on the other tack. Her captain has wisely predicted the Hotspur’s movements and has tacked at the same time as the Hotspur, thereby not losing any time at all. Hornblower tells himself that he will not use this trick again, and when a second cloud covers the Hotspur, he does indeed not tack; he does not even think about tacking, in fact. After coming out of this cloud, Hornblower finds that, again to his surprise, the Loire has tacked. Presumably, the captain of the Loire thought that the Hotspur would be tacking and did the same. But in this case, the Loire made a mistake and lost some valuable time.

The presence of the clouds has changed the nature of the game. Without the clouds, that is, with full visibility, the game is essentially a sequential move game. The Loire can simply observe what the Hotspur does and make her choice afterwards. This makes it easy for the Loire to match her action to that of the Hotspur, while making it impossible for the Hotspur to prevent that. With the clouds, the game is better described by a simultaneous move game. While one of the ships is hidden from view within a cloud, neither captain can see the other captain’s move. The game they are now playing is an instance of the so-called matching-pennies game. Both captains have two choices: they can tack or not tack. The captain of the Loire would like the two ships’ actions to match; Hornblower would like them to mismatch. This game also has a Nash equilibrium, but it is in what game theorists call “mixed” strategies: this means both captains choose randomly whether to tack or not. Given the description in the book about Hornblower’s thought process, it does not sound like he is actually randomizing. All that is really needed, however, is that the two players’ actions are unpredictable for their opponent. And it seems that Hornblower’s thought process was not exactly what the captain of the Loire thought it was (at least not in the second cloud instance). In any case, with just the two data points that we have (there being only two instances with clouds), we cannot reject that the two captains are playing the equilibrium of this game.

The game without clouds had an equilibrium in pure strategies, in which both players were able to predict each other’s moves precisely. The game with clouds also had an equilibrium, but in mixed strategies. In this case, the two captains should not expect to be able to fully predict their opponent’s choice, but they should be able to predict the extent of unpredictability. They should know that there is essentially a 50-50 chance of either move and, therefore, should never really be super-surprised by anything the other captain does.

Donald Rumsfeld, as US Secretary of Defense, once puzzled the world a bit with a statement about the differences between “known knowns”, “known unknowns”, and “unknown unknowns”. One can give a purely decision-theoretic discussion of this statement, but one can also see this in game-theoretic terms. In the equilibrium of the game without clouds, every future move is known to the two captains. In the game with clouds, the captains (should) know that they don’t quite know their opponent’s next move (in the cloud). In another blog post, I will use another Hornblower story to discuss situations in which players don’t even know what they don’t know.