How game theory beat the Nazis
Game theory is commonly used to model conflicts, and this paper, https://www.jstor.org/stable/166693?seq=1#metadata_info_tab_contents, uses game theory to model the decision-making behind a battle in France between Allied and German forces in August 1944. Set near Avranches, American General Omar Bradley was given the task of eliminating enemy presence in the area. The American forces had just broken through their beachhead by the English Channel, exposing the Western flank of the German Ninth Army. Bradley’s forces consisted of the First and Third Armies; the First was in the North and the Third was in the South, with a gap between the two. The German commander, von Kluge, could either attack West to secure his flank and split the southern American First Army from the Third Army by attacking the gap between them, or withdraw to the East and set up defensive positions in the Seine River. The Third Army was seeping inland and and the First Army as keeping the German Ninth Army busy in the North. Bradley had to decide what to do with the four divisions he had on reserve. He considered three courses of action: order his reserve to defend the gap in the American forces, send them to attack eastward to harass the Ninth Army and prevent its withdrawal, or leave it in an uncommitted position to provide flexibility in both defending the gap or attacking east if necessary. To model this in Game Theory, the Germans could either choose to attack the gap or not to attack the gap. With two different choices for von Kluge and three for Bradley, there are 6 different outcomes:
- von Kluge attacks gap, Bradley defends gap. Outcome: U.S. forces repulse attack to hold gap. Expected payoff ranking: 5th
- von Kluge attacks gap, Bradley attacks east. Outcome: German forces cut through American forces. Expected payoff ranking: 6th
- von Kluge attacks gap, Bradley does commit to any action. Outcome: U.S. forces repulse attack and combined attacks by reserve and First Army could encircle the Germans. Expected payoff ranking: 1st
- von Kluge retreats, Bradley attacks east. Outcome: strong pressure on German withdrawal. Expected payoff ranking: 2nd
- von Kluge retreats, Bradley does not commit to any action. Outcome: U.S. forces have moderate pressure on German withdrawal. Expected payoff ranking: 3rd.
- von Kluge retreats, Bradley defends gap. Outcome: weak pressures on German withdrawal. Expected payoff ranking: 4th
In a battle like this, both players are moving simultaneously, so players do not have time to react to the other player’s moves; they can only anticipate what the other player’s moves can possibly be and minimize their loss. There happens to be no Nash equilibrium in this situation; there is no pair of moves such that each move is a best response to the other player’s move. There are also no strictly dominated strategies, strategies that yield worse payoff than at least one other no matter what the other player chooses to do. So how did these generals figure out what to do?
Bradley reasoned that if he did not commit, the worst payoff he could receive was that U.S. forces have moderate pressure on German withdrawal. If he did commit to an attack eastward, his worst payoff is that he lose many, many men. He decided to hold his four divisions in reserve for another day since that yields the best worst case payoff he could receive
From von Kluge’s perspective, by withdrawing, the worst payoff he could receive was that there was strong pressure on German withdrawal. By attacking however, the worst payoff he could receive was that U.S. forces repulse the attack and may circle the Germans, which is the worst payoff possible. So, he decided the best course of action was to withdraw, also minimizing his worst potential payoff. However, Hitler ordered von Kluge to attack the gap, and since refusing an order from Hitler was not an option, he attacked. Bradley’s four divisions he held on reserve were able to defend the gap successfully and launched an attack east the next day. von Kluge did not have enough time to withdraw, and his army was nearly completed encircled by the reserve divisions from the East and the First Army from the North; the payoffs anticipated by both sides if von Kluge attacked and Bradley remained uncommitted turned out to be painfully accurate and von Kluge committed suicide as a result of this devastating loss.
Hitler chose course of action that had the worse potential payoff for the Germans, though if Bradley did not choose to leave his forces uncommitted, the Germans could have had either the first and second highest payoff . Both generals in this case were employing a maxmin strategy, which in Game Theory is essentially a strategy that maximizes the minimum payoff one could receive. It seems logical to play so conservatively in a game where the moves are happening simultaneously and if there is no external information obtained by spies or special intelligence. Generals would be even more inclined to play conservatively if the worst payoffs involve losing many human lives.