My answer was too simple. And I had to read
the real answer three times before I finally understood it. It's a devilishly clever problem.
Two countries, call them A and B, with armies of randomly determined strengths somewhere between 0 (very weak) and 1 (very strong), are deciding whether or not to declare war on each other to win a stash of gold. Each country knows its own strength, but not that of the other. They declare “war” or “peace” simultaneously. If either declares war, they go to war, and the country with the stronger army wins. The optimal strategy? Always declare war!
Why? Suppose the optimal strategy is to declare war when your strength is greater than some number X and that, because each player is facing an identical situation at the beginning of the game, both players play according to this strategy. Here’s where the game theory comes in. If Country B declares war if and only if its strength is greater than X, Country A would do well to declare war whenever its strength is greater than X/2. Why? There are two cases: In those situations where B’s strength (unknown to A) is greater than X, it doesn’t make any difference what A does, since B will declare war and A will be forced to fight. But in those situations where B’s strength is less than X, A profits by lowering its threshold to X/2. In half of those scenarios, it will be stronger than B, win the war, and double its gold cache.
But if A is declaring war when its strength is greater than X/2, then B will do well to declare war when its strength is greater than X/4, but then A would do well to declare war when its strength is greater than X/8, and so on. It’s bloodshed all the way down. They don’t stop until they’re both declaring war in every situation. Put another way, the only equilibrium “threshold” strength is 0. The strengths are always greater than 0, so both countries will always declare war.
I like it, but also hate myself for not getting it.