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shokwave comments on Nash Equilibria and Schelling Points - Less Wrong
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This is a minor quibble, but while reading I got stuck at this point:
followed by a description of a game that didn't seem to have a Nash equilibrium and confirming text "Here there is no pure Nash equilibrium." and "So every option has someone regretting their choice, and there is no simple Nash equilibrium. What do you do?"
I kept re-reading this section, trying to work out how to reconcile these statements since it seemed like you have just offered an irrefutable counterexample to John Nash's theorem. It could use a bit of clarification (maybe something like "This game does have a Nash equilibrium, but one that is a little more subtle" or something similar.
Other than that I'm finding this sequence excellent so far.
There is no pure equilibrium, but there is a mixed equilibrium.
A pure strategy is a single move played ad infinitum.
A mixed strategy is a set of moves, with each turn's move randomly selected from this set.
A pure equilibrium is one where every player follows a pure strategy, and a mixd equilibrium is one where at least some players follow a mixed strategy.
Both pure equilibriums and mixed equilibriums are Nash equilibriums. Nash's proof that every game has an equilibrium rests on his previous work where he and von Neumann invented the concept of a mixed equilibrium and proving that it satisfies the criteria.
So this game has no pure equilibrium, but it does have a mixed one. Yvain goes on to describe how you calculate and determine that mixed equilibrium, and shows that it is the attacker playing Podunk 1/11th the time, and Metropolis 10/11th the time.
EDIT: The post explains this at the end:
Yvain: I would strongly recommend including a quick explanation of mixed and pure strategies, and defining equilibriums as either mixed or pure, as a clarification. At the least, move this line up to near the top. Excellent post and excellent sequence.