Operation Deadstick: During the invasion of Normandy in World War II, British forces were tasked with...

 

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Operation Deadstick: During the invasion of Normandy in World War II, British forces were tasked with capturing two bridges behind the German front line. The decision as to how many troops to send on this mission (and how many troops the Germans should leave behind to guard the bridges) surely depended on the perceived benefits of keeping the bridges intact and the costs of getting troops behind enemy lines for the British (and taking troops away from the front for the Germans). Let's illustrate this in the following simplified strategic setting: There is just one bridge, and the commanders of each side choose to send zero, one, or two platoons to this bridge. We will assume that the commander devoting more troops to the fight always wins and the benefit from winning is captured in w. In addition, the cost of losing the bridge battle is 4, the payoff for a draw (i.e., when both commanders devote the same number of platoons) is 0, and the cost of each platoon committed to the battle is 1. (a) Write the normal form of the resulting game in terms of w, I, and the troop cost. (b) What has to be true about the values of w and I in order for the commanders to devote all their available forces? (c) For what values of w and I will the commanders devote no resources to the fight? Operation Deadstick: During the invasion of Normandy in World War II, British forces were tasked with capturing two bridges behind the German front line. The decision as to how many troops to send on this mission (and how many troops the Germans should leave behind to guard the bridges) surely depended on the perceived benefits of keeping the bridges intact and the costs of getting troops behind enemy lines for the British (and taking troops away from the front for the Germans). Let's illustrate this in the following simplified strategic setting: There is just one bridge, and the commanders of each side choose to send zero, one, or two platoons to this bridge. We will assume that the commander devoting more troops to the fight always wins and the benefit from winning is captured in w. In addition, the cost of losing the bridge battle is 4, the payoff for a draw (i.e., when both commanders devote the same number of platoons) is 0, and the cost of each platoon committed to the battle is 1. (a) Write the normal form of the resulting game in terms of w, I, and the troop cost. (b) What has to be true about the values of w and I in order for the commanders to devote all their available forces? (c) For what values of w and I will the commanders devote no resources to the fight?