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JenniferRM comments on Mixed strategy Nash equilibrium - Less Wrong
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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.
In some sense, I agree with you. The problem as posed had a clear answer that was calculable by a known method (if one had done the requisite reading in game theory). The thing I particularly liked about Rayhawk's post was the link to the a library of game theory software and tools for the construction and analysis of finite extensive and strategic games: gambit. That link was the kind of novel and useful pointer that is one of the many reasons I have for reading LW :-)
On the other hand, I find that the world frequently fails to present situations to me that are intelligible to the point that I can build a payoff matrix and run the numbers. So, as a simple exercise standing in for a more complex world there was potentially much more to say about the puzzle. In that vein I particularly liked Nominull's fast and frugal answer:
I expect that I would find it very difficult to mimic Rayhawk's application of gambit in the bulk of real life circumstances. Nominull's heuristic (which incidentally produced one of the options from the optimal mixed strategy) seems more generally applicable. I can imagine using Nominull's heuristic in much fuzzier contexts for much lower data gathering costs and getting pretty good results thereby. Not that I've tested it or anything... but it's the sort of thing I'll be looking for an opportunity to try out in the real world someday, and see if it helps :-)