The first question that came to my mind when I first examined Quoridor, and probably that of many other players as well, is: what kind of game is this really?
It is obvious that Quoridor has something to do with connection games, but it is not a paradigmatic connection game where one forms a chain between one side of the board to the other before one’s opponent. In fact, it seems at first glance to be the exact opposite of that: a game where you try to prevent your opponent from forming a connection. But of course this is the wrong way of looking at things, since the rules explicitly forbid you from doing this (and that would be a much less interesting game!).
Finally I got the key idea from mulling about the Shiller Principle. I realized that the “paths to goal” count is a more generalized concept than actual concrete paths. At the beginning of the game, one may say that the number of possible concrete paths one can take to the goal is extremely large (I won’t say infinite, if only playing a game of infinite length is, well, impossible).
And if we start from that premise, then one can further theorize that laying down fences lowers the number of actual paths, and that the mechanism by which one wins is by whittling options down for your opponent, and whittling away the most attractive options, leaving them with fewer, and worse, options. So in essence it seems that the “reverse” view is correct, insofar as you start with near-infinite connections and try to lower that number, instead of starting with no connections and trying to form one.
Of course, no idea is entirely new, and someone already understood that before me:
[T]he number of ways for a player to reach his target edge decreases as more and more fences are placed on the board: Quoridor begins in a maximal state of connective potential that converges as the game progresses.
Cameron Browne, in Connection Games: Variations on a Theme
But I don’t find this perspective to be the most useful in terms of understanding how this game works. I want to suggest a different, and weirder, idea: Quoridor as a resource management game.
Going back to the Shiller Principle, it is true that at the beginning there is only one optimal path (or zero, depending on whether you started first). As the game develops, there may be a number of Shiller paths possible, but the main reason why it matters is because of fenceable gaps (i.e. because those paths can be closed down).
My idea is that, perhaps not explicitly but as part of strategy, fenceable gaps are a resource that is generated and spent during play. Depending where the gap is, it can be a resource or a liability to either player, and part of the unpredictability of this game is that one can easily be turned into the other.
From this viewpoint, any average game presents an abundance of riches. On the diagram to the right, fenceable gaps are numbered in red, and gaps fenceable in two are numbered in dark yellow.
Some of these are more of a resource for one side than the other. Pink’s direct line to the goal involves either 1 or 3-6, while Green’s direct line to the goal involves 2-1-7-10, 6-9 or 6-8-5. On the whole, we can see that Pink has a big advantage because its paths involve fewer fenceable gaps, which means the opponent will run out of resources much faster. As a result, Pink also has only six Schiller paths while Green has ten.
Other fenceable gaps are resources in that they force moves to the opponent: Pink here understandably closes down 4 on its next move, which drops his Shiller path count to a mere two, forcing Green to close down 1 as it is the least defensible path.