The sleeping general

This is a reformulation of the sleeping beauty problem discussed in Motl’s blog, and hereby reshaped in the way of russian game theory books.

What we have is a country that can be attacked from two different border pass, the Headgate Pass and the Tailgate Pass. Due to the geography, the cost of moving the attacking army down to the Tailgate Pass and the poor booty usually looted from the Tailgate zone of the country causes that always that the enemy chooses this pass, it always try two attacks. Attacks are really guerrilla razzias, or «algaradas», they never take the country, just run looting around.

The sleeping condition follows the original idea: an attack is always launched when everybody in the sieged country has forgotten where the previous attack come from. It is more an issue of a «sleeping prince» than a «sleeping beauty» here. Still, this lack of memory of the attack is at the same time the signal that a next one can come, and then the defending general must decide where to put the defensive army, if either in the Headgate or in the Tailgate pass.

The game theory problem usually would finish here and ask for the best attacking and defensive strategies. But we have spies at play, at least the defending general has. They know the peculiarity of the Tailgate attack, and moreover the attacked country has got the knowledge of the method used to decide which pass to raise an army at: basically, the attacker council of elders launches a fair coin; if heads they raise the army at the Headgate Pass, if tail they raise it at the Tailgate Pass.

So, which is the probability of the elders’ coin to land a tail? which is the probability of being attacked across the Headgate Pass? Where should the defenders put their army?