It's indeed a mystery to me why anyone bothered to post and discuss "solutions" different from Rayhawk's in the swords and armor thread. This stuff is like arithmetic: one right answer, nothing to argue about.

As a bonus, I'll give an introduction to the notion of "correlated equilibrium" invented by Aumann, using a model game invented by Shapley. Imagine you're playing a variant of Rock Paper Scissors where a win gives you 1 point, but a lose or a draw give 0 points. (So the game is no longer zero-sum - this is essential.) Obviously, if you use some strategy more than 1/3 of the time, the other guy may adjust to that; therefore the only Nash equilibrium is mixed where you both play each strategy with probability 1/3, which gets you both an expected payoff of 1/3. But the problem with this result is that sometimes the game ends in a draw and no one wins any money. So it would be mutually beneficial to somehow arrange that you never play the same strategy. But doesn't the uniqueness of the Nash equilibrium mean that any such arrangement would be unstable?

Well, here's how you do it. Suppose you both ask a trusted third party to randomly pick one of the six non-draw outcomes of the game, and then privately tell each of you which strategy to play (without telling you what they said to the other guy). For example, they might randomly pick "Rock Scissors", tell you to play Rock, and tell your opponent to play Scissors. In this freaky situation, even though no one's forcing you to follow the advice, doing so is an equilibrium! This means that neither of you can gain anything by deviating from the advice - provided that the opponent doesn't deviate. And your expected payoff is now 1/2, because draws cannot happen, which is better then the Nash equilibrium payoff of 1/3. This is called a "correlated equilibrium". It's one of the examples that show how even non-binding agreements, "cheap talk", can still make people better off.