A few comments. In terms of A&A as a board game, a complaint I’ve sometimes read is that the problem is actually the opposite one: that games go on forever because the losing side has no reason to quit, especially if it’s only losing by a small margin relative to the winning side.
In terms of actual wars, I can’t think of any examples of a major war in which one side threw up its hands and gave up the fight as soon as it lost a bit of momentum. It’s actually the opposite that tends to happen in major conflicts: the greater the level of death and destruction, the more the participants tend to dig in their heels and press on with the fight – WWI being a classic example of a war that got out of control on a scale never anticipated by the participants, who by 1915 found themselves trapped in a conflict they could neither end nor win. Apart from Russia, which quit in late 1917 because of regime change, all the participants kept slugging it out until late 1918. And in WWII, Germany and Japan both kept fighting long past the point where it was clear to everyone that they were going to lose. It’s not rational, but there are typically a combination of reasons for why it happens anyway:
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The principle of “don’t throw good money after bad”, whereby you cut your losses in a losing situation before things can get worse, sounds like a rational thing to do in war, but it can easily get overruled by another principle which sometimes gets invoked by generals and politicians: “If we quit now, all our previous losses will have been for nothing.” In WWI, this was often combined with the wishful thinking that “One more big push will bring victory,” which explains the horrendously costly Verdun-style big pushes of 1915…and 1916…and 1917…and 1918, only the last of which (on the Allied side) finally did bring victory.
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A related point is that total war demands total objectives. To give the example of WWI: when millions of people have died and when the entire economies of nations (and their civilian workers) have been mobilized, you can’t just sit down with your opponents, sort out the obscure Balkan rivalry that started it all, trade a couple of colonies and call it a day. The conflict becomes one of national survival. None of the regimes on the losing side survived WWI, and four empires were destroyed in the process.
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WWII is an interesting case. Germany kept going until its armies were almost literally fighting back to back down the centre of the country, with the Anglo-Americans on one side and the Russians on the other. Part of the reason, of course, was that Hitler refused to quit and that he still had the power to compel his armies to keep fighting. A less obvious reason was that the Wehrmacht, who knew perfectly well that the game was up, greatly preferred the prospect of surrendering to the Americans and the British rather than the Soviets, and wanted to buy time for that purpose. Japan is a different story. In its case, part of the reason for holding out more and more stubbornly as the Americans got closer and closer to Japan (just look at Iwo Jima and Okinawa) was to convince the Americans that an invasion of the Japanese home islands would be both necessary and horrifically costly in lives, and thereby to somehow convince the Americans to seek the alternatives of a negotiated settlement. That turned out to be a miscalculation: the Americans, who still remembered Pearl Harbor, were determined to defeat Japan at whatever the cost might be…and they had the atomic bomb up their sleeve. The other reason Japan held out was the death-before-dishonour tradition which the Japanese Army had carried over from the days of the Bushido code. In fairness, nobody likes to lose face and nobody likes to lose. One can sympathize on that basis with the careful wording of Hirohito’s rescript (essentially Japan’s declaration of surrender), which said that the war had “not necessarily developed to Japan’s advantage” – probably the biggest understatement in recorded history.