threearmies - The Three Armies Against Two Problem Problem...

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The Three Armies Against Two Problem Problem 118.3: General A is defending territory accessible by two mountain passes against an attack by General B. General A has three divisions at his disposal and General B has two divisions. Each general allocates his divisions between the two passes. General A, the defender, wins the battle at the pass if he assigns at least as many divisions to the pass as B, the attacker. General A wins the battle if and only if he wins the battle at both passes. Find all of the mixed strategy Nash equilibria. This problem turns out to be more difficult than I first thought when I assigned it. Establishing that there is no mixed strategy equilibrium in which General B has a positive probability of sending one division to each pass seems to require a fairly elaborate argument. Also, it is interesting to note that although General A would never use 3 divisions to defend 1 pass if there were any chance that General B would send 1 division to each pass, there are Nash equilibria in which he sends 3 divisions to one or both passes with positive probability. Proposed Answer: Let us denote the two passes as the north pass, N, and the south pass, S. Armies that are not sent to N are sent to S. General B has three possible pure strategies. Send 2 armies to N. Send 1 army N Send 0 armies N.
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