# Yvain comments on Nash Equilibria and Schelling Points - Less Wrong

35 29 June 2012 02:06AM

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Comment author: 29 June 2012 08:13:56AM 8 points [-]

This is a minor quibble, but while reading I got stuck at this point:

And since John Nash (remember that movie A Beautiful Mind?) proved that every game has at least one,

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.

Comment author: 30 June 2012 12:44:42AM 1 point [-]

Good point. I've clarified pure vs. mixed equilibria above.

Comment author: 01 July 2012 02:29:33PM *  0 points [-]

Here the answer should be obvious: it doesn't matter. Flip a coin. If you flip a coin, and your opponent flips a coin, neither of you will regret your choice. Here we see a "mixed Nash equilibrium", an equilibrium reached with the help of randomness.

Hmm, I'm still not finding this clear. If I flip a coin and it comes up heads so I attack East City, and my opponent flips a coin and it comes up to defend East City, so I get zero utility and my opponent gets 1, wouldn't I regret not choosing to just attack West City instead? Or not choosing to allocate 'heads' to West City instead of East?

Is there a subtlety by what we mean by 'regret' here that I'm missing?

Comment author: 01 July 2012 03:32:01PM *  3 points [-]

I usually understand "regret" in the context of game theory to mean that I would choose to do something different in the same situation (which also means having the same information).

That's different from "regret" in the normal English sense, which roughly speaking means I have unpleasant feelings about a decision or state of affairs.

For example, in the normal sense I can regret things that weren't choices in the first place (e.g., I can regret having been born), regret decisions I would make the same way with the same information (I regret having bet on A rather than B), and regret decisions I would make the same way even knowing the outcome (I regret that I had to shoot that mugger, but I would do it again if I had to). In the game-theory sense none of those things are regret.

There are better English words for what's being discussed here -- "reject" comes to mind -- but "regret" is conventional. I generally think of it as jargon.

Comment author: 01 July 2012 02:35:49PM 3 points [-]

Here we're not thinking of your strategy as "Attack East City because the coin told me." We're thinking of your strategy as "flip a coin". The same is true of your opponent: his strategy is not "Defend East City" but "flip a coin to decide where to defend"

Suppose this scenario happened, and we offered you a do-over. You know what your opponent's strategy is going to be (flip a coin). You know your opponent is a mind-reader and will know what your strategy will be. Here your best strategy is still to flip a coin again and hope for better luck than last time.

Comment author: 02 July 2012 02:13:09AM 1 point [-]

Okay, I think I get it. You're both mind-readers, and you can't go ahead until both you and the opponent have committed to your respective plans; if one of you changes your mind about the plan the other gets the opportunity to change their mind in response. But the actual coin toss occurs as part-of-the-move, not part-of-the-plan, so while you might be sad about how the coin toss plan actually pans out, there won't be any other strategy (e.g. 'Attack West') that you'd prefer to have adopted, given that the opponent would have been able to change their strategy (to e.g. 'Defend West') in response, if you had.

...I think. Wait, why wouldn't you regret staying at work then, if you know that by changing your mind your girlfriend would have a chance to change her mind, thus getting you the better outcome..?

Comment author: 03 July 2012 02:51:40AM 1 point [-]

I explained it poorly in my comment above. The mind-reading analogy is useful, but it's just an analogy. Otherwise the solution would be "Use your amazing psionic powers to level both enemy cities without leaving your room".