Short answer: I already addressed this. Is your point that I didn’t emphasize it enough?
One thing should be kept in mind. A Nash equilibrium strategy, much like a minimax strategy, is “safe”. It makes sure your expected payoff won’t be too low no matter how clever your opponent is. But what if you don’t want to be safe—what if you want to win? If you have good reason to believe you are smarter than your opponent, that he will play a non-equilibrium strategy you’ll be able to predict, then go ahead and counter that strategy. Nash equilibria are for smart people facing smarter people.
Long answer:
The distribution of sword/armor choices in an MMO will not be the Nash equilibrium with overwhelming probability. In fact, it probably won’t be anywhere close if the choice is at all complicated.
Correct. Do you know what the distribution is? Can you gather statistics? Do you understand the mentality of your opponents so well that you can predict their actions? Can you put the game in some reference class and generalize from that? If any of the above, knock yourself out. Otherwise there’s no justification to use anything but the equilibrium.
If you are playing an MMO with random pvp, populated by people the people who play MMOs, and you choose the Nash equilibrium, you will probably do worse than me.
If you assume I will continue to use the NE, even after collecting sufficient statistics to show that the players follow some specific distribution (as opposed to “not Nash equilibrium”), then this is a strawman. If you’re just saying you’re very good at understanding the mind of the MMOer then that’s likely, but doesn’t have much to do with the post.
But saying that someone who talks about what their opponents are likely to do is “wrong” is itself quite wrong.
I did not criticize the rare swords&armor posts that actually tried to profile their opponents and predict their actions. I criticized the posts that tried to do some fancy math to arrive at the “optimal” solution inherent to the game, and then failed to either acknowledge or reconcile the fact that their solution is unstable.
Reasoning about the distribution of strategies your opponents play is the correct approach to this problem, not reasoning about the Nash equilibrium.
One should first learn how to do Nash equilibrium, then learn how to not do Nash equilibrium. NE is the baseline default choice. If someone doesn’t understand the problem well enough to realize that any suggested deterministic strategy will be unstable and that the NE is the answer to that, what hope does he have with the harder problem of reasoning about the distribution of opponents?
If 90% of American generals in the past have chosen to attack West, it is probably wrong to defend East with the “optimal” probability.
Even under the strong assumption that there is some clear reference class about which you can collect these statistics, this is true only if one of the following conditions holds:
The attacker hasn’t heard about those statistics.
The attacker is stupid.
The attacker is plagued with the same inside-view biases that made all his predecessors attack west.
The plausibility of each depends on the exact setting. If one holds, and we know it, and we really do expect to be attacked west with 90% probability, then this is no longer a two-player game. It’s a one-player decision theory problem where we should take the action that maximizes our expected utility, based on the probability of each contingency.
But if all fail, we have the same problem all over again. The attacker will think, “paulfchristiano will look at the statistics and defend west! I’ll go east”. How many levels of recursion are “correct”?
I didn’t really mean to insult your post (although I apparently did :). I was probably just as surprised as you at many of the comments in that thread. I agree that you should understand NE if you want to say anything useful about games (and that they are basically the complete story for two-player zero sum games from a theoretical perspective). The one thing I object to is the sentiment that “if you don’t know exactly what is going on, you should just play NE.” After re-reading your latest post this a bit of a straw man. I agree totally that if you know nothing else then you have nothing to do but play the NE, which is all that you actually said. However, you can put any game that you are faced with into the reference class of “games humans play,” and so I don’t think this fact is very relevant most of the time. In particular, if the question was “what should you do in an MMO with these properties” then there are many acceptable answers other than play the NE. It may or may not be the case that anyone in the thread in question actually gave one.
In particular, because I can put all games I have ever played into the reference class “games that I have ever played,” I can apply, over the course of my entire life, an online learning algorithm which will allow me to significantly outperform the Nash equilibrium. In practice, I can do better in many games than is possible against perfectly rational opponents.
The distribution of sword/armor choices in an MMO will not be the Nash equilibrium with overwhelming probability. In fact, it probably won’t be anywhere close if the choice is at all complicated.
Correct. Do you know what the distribution is? Can you gather statistics? Do you understand the mentality of your opponents so well that you can predict their actions? Can you put the game in some reference class and generalize from that? If any of the above, knock yourself out. Otherwise there’s no justification to use anything but the equilibrium.
To expand on palfchristiano’s quibble with “if you don’t know exactly what is going on, you should just play NE”: The technically correct phraseology for that part may be too complicated for this post. But it would be closer to “if your belief that you can predict your opponent’s non-ideal behavior, multiplied by your expected gain from exploitation of that prediction, exceeds your expected loss from failing to correctly predict your opponent’s behavior, go ahead.”
Short answer: I already addressed this. Is your point that I didn’t emphasize it enough?
Long answer:
Correct. Do you know what the distribution is? Can you gather statistics? Do you understand the mentality of your opponents so well that you can predict their actions? Can you put the game in some reference class and generalize from that? If any of the above, knock yourself out. Otherwise there’s no justification to use anything but the equilibrium.
If you assume I will continue to use the NE, even after collecting sufficient statistics to show that the players follow some specific distribution (as opposed to “not Nash equilibrium”), then this is a strawman. If you’re just saying you’re very good at understanding the mind of the MMOer then that’s likely, but doesn’t have much to do with the post.
I did not criticize the rare swords&armor posts that actually tried to profile their opponents and predict their actions. I criticized the posts that tried to do some fancy math to arrive at the “optimal” solution inherent to the game, and then failed to either acknowledge or reconcile the fact that their solution is unstable.
One should first learn how to do Nash equilibrium, then learn how to not do Nash equilibrium. NE is the baseline default choice. If someone doesn’t understand the problem well enough to realize that any suggested deterministic strategy will be unstable and that the NE is the answer to that, what hope does he have with the harder problem of reasoning about the distribution of opponents?
Even under the strong assumption that there is some clear reference class about which you can collect these statistics, this is true only if one of the following conditions holds:
The attacker hasn’t heard about those statistics.
The attacker is stupid.
The attacker is plagued with the same inside-view biases that made all his predecessors attack west.
The plausibility of each depends on the exact setting. If one holds, and we know it, and we really do expect to be attacked west with 90% probability, then this is no longer a two-player game. It’s a one-player decision theory problem where we should take the action that maximizes our expected utility, based on the probability of each contingency.
But if all fail, we have the same problem all over again. The attacker will think, “paulfchristiano will look at the statistics and defend west! I’ll go east”. How many levels of recursion are “correct”?
I didn’t really mean to insult your post (although I apparently did :). I was probably just as surprised as you at many of the comments in that thread. I agree that you should understand NE if you want to say anything useful about games (and that they are basically the complete story for two-player zero sum games from a theoretical perspective). The one thing I object to is the sentiment that “if you don’t know exactly what is going on, you should just play NE.” After re-reading your latest post this a bit of a straw man. I agree totally that if you know nothing else then you have nothing to do but play the NE, which is all that you actually said. However, you can put any game that you are faced with into the reference class of “games humans play,” and so I don’t think this fact is very relevant most of the time. In particular, if the question was “what should you do in an MMO with these properties” then there are many acceptable answers other than play the NE. It may or may not be the case that anyone in the thread in question actually gave one.
In particular, because I can put all games I have ever played into the reference class “games that I have ever played,” I can apply, over the course of my entire life, an online learning algorithm which will allow me to significantly outperform the Nash equilibrium. In practice, I can do better in many games than is possible against perfectly rational opponents.
Okay, then it looks like we are in agreement.
To expand on palfchristiano’s quibble with “if you don’t know exactly what is going on, you should just play NE”: The technically correct phraseology for that part may be too complicated for this post. But it would be closer to “if your belief that you can predict your opponent’s non-ideal behavior, multiplied by your expected gain from exploitation of that prediction, exceeds your expected loss from failing to correctly predict your opponent’s behavior, go ahead.”