Fortunately, I'm a snooty American intellectual. As such, I realise that three-sided conflicts have been a staple plot device in American films for more than thirty years, that games like Monopoly, Risk and poker provide America with lots of people who are familiar with the dynamics of multi-sided competition, and that nobody gets very far in American politics without being able to handle more shades of grey than a high-end laser printer. Moreover, as a snooty American intellectual who has spent a lot of time studying the military history of Europe, I also realize that the chief cause of our strategic myopia is an idea that we borrowed from European intellectuals, and that is the notion that war is necessarily a two-sided affair.
The great irony here is that the writers who gave us the idea that war had to be two-sided - Carl von Clausewitz, Antoine Jomini and a host of lesser lights who popularized their ideas - were products of an age when international conflict was anything but two-sided. Both Clausewitz and Jomini, for example, were veterans of the Napoleonic Wars, a series of struggles in which each participant switched partners with a frequency that would put a present-day movie star to shame. Both, moreover, based their theories on the wars and campaigns of a century (1680-1780) in which each of the five or six "great powers" that dominated the Continent saw itself as competing with every other "great power". Indeed, the struggle among the "great powers" in eighteenth century Europe is probably the best documented case study of sustained multi-sided conflict that we can find.
Why, then, did Clausewitz, Jomini and their second-rate immitators base their theories of war on the axiom that war was necessarily a two-sided affair? The reason, quite simply, was that they were soldiers rather than statesmen and, as such, looked at war from a bottom-up perspective. As befits a man who first went to war at the tender age of twelve, Clausewitz devoted most of his energies to matters that we would now categorize as "operational". This was even more true of the other theorists of the day. When mentioned at all, politics was treated as something my old economics teacher called an "exogenous variable", a phenomenon that, while might intrude into military calculations in a powerful way, was nonetheless alien to the subject at hand.
The relationship between multi-sided politics and two-sided war thought is nicely illustrated by an episode from the First Balkan War (1912-1913). Crown Prince Constantine, who commanded the Greek army in the field, wanted to use his forces to help the much-larger Bulgarian army defeat the main Turkish army. Bulgaria and Greece, after all, were allies, and shared the common goal of expelling the Turks from the Balkan peninsula. King George of Greece, however, had a different idea. While just as eager as anyone else to gain territory at the expense of the Turks, he was also worried about the territorial ambitions of his allies. The king therefore ordered his son to let the Bulgarians worry about the main Turkish army while the Greek field army took the shortest possible route to the choicest bit of real estate at issue, the city of Salonika.
Constantine made all sorts of noise about the necessity of keeping faith with allies and how the main Turkish army was the center of gravity of the campaign. In the end, however, he obeyed his orders. As a result, the relatively small detachments that the field armies of Bulgaria and Serbia (which was also taking part in the war against Turkey) sent to Salonika found themselves marching into in a city that was already occupied by a much larger Greek force.
As might be imagined, the Bulgarians, who were fighting alone against the main Turkish army, were not happy about this turn of events. They were even less happy when they discovered that the Serbian army had done something similar in central Macedonia, using forces that would have been very welcome in the bloody battles against the main Turkish army to occupy territory that Bulgaria had claimed as its own. Indeed, the Bulgarians, who had done the lion's share of the work of driving the Turks out of the Balkans, and who had consequently taken the lion's share of the casualties, were greatly annoyed. Thus, when it became clear that neither the Greeks nor the Serbs were —to give up the territory they had snatched while Bulgaria had been distracted, Bulgaria declared war on its former allies. (This was the start of the Second Balkan War, but that is a different story ...)
While the relationships among the countries that took part in the First Balkan War were extremely complex, the moral of the tale of Bulgaria's great blunder is painfully simple. A power that gets involved in a multi-sided conflict needs to follow the logic of multi-sided "political" conflict. If it doesn't, and persists in acting as if it is in a two-sided "military" conflict, then it will end up being the patsy of actors who, while far less powerful, understand the dynamics of the situation at hand.