During the Allied invasion of France in 1944, the Germans had to decide where to

Question

During the Allied invasion of France in 1944, the Germans had to decide where to place their defenses. They had three choices: they could concentrate their defenses at Calais, concentrate at Normandy, or split their defenses between both locations. The Allies had two choices: they could attack at Calais or at Normandy. Assume that this is a zero-sum game and that the possible outcomes are ranked as in the following matrix (where larger positive numbers represent outcomes more favorable for the Allies). German Calais Normandy Split Allies Calais 6, -6 2, -2 5, -5 Normandy 1, -1 4, -4 3, -3 (a) Are there any pure-strategy Nash equilibria for this game if the Germans and Allies move simultaneously? (b) Suppose the Germans move first. Draw the game tree for this game and find the rollback equilibrium outcome. (c) Suppose the Allies move first. Draw the game tree for this game and find the rollback equilibrium outcome.

Explanation / Answer

This question is a typical question of Game Theory.

To understand this we must first consider two assumptions:

1) both the allies and the germans are rational and both of them have knowledge of the results of the invasion.

2) The outcomes of the invasion are reasonable and correct and thus there shall not be any ex-post regret.

If these two conditions are satisfied then we believe that a Nash-Equlibrium shall exist and there shall be an optimal strategy to solve this problem.

let us construct the payoffs of the invasion as (Allies,Germans), N: Normandy, C: Calais

(N,N) = (0,0) probability of allied success = 75%

(N,C) = (80,-80)  probability of allied success = 100%

(C,N) = (100,-100)  probability of allied success = 100%

(C,C) = (0,0)  probability of allied success = 20%

If the commander chooses C, he knows that the marshall will act rationally and will choose C instead of N, and the allied with 20% probability shall loose.

If the commander chooses N, he know that the marshall will act rationally and will choose N instead of C, and the allied with 75% probability shall win.

In totality the Marshall knows that the commander knows that he has higher probability of winning at if the invasion takes place at Normandy. So the marshall shall act rationally and defend at Normandy.

Thus the nash-equlibrium suggests that (N,N) is the best possible outcome

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