A strategy profile is a set of strategies, one for each player. Informally, a strategy profile is a Nash equilibrium if no player can do better by unilaterally changing their strategy. To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: “Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?”
If any player could answer “Yes”, then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to the other players’ strategies in that equilibrium.
What is required for something to be a Nash equilibrium is that no one could do better by min-maxing assuming others pessimize their utility. That minmax value is if you set temperature to min, others max it.
This seems incorrect. The Wiki definition of Nash equilibrium posits a scenario where the other players’ strategies are fixed, and player N chooses the strategy that yields his best payoff given that; not a scenario where, if player N alters his strategy, everyone else responds by changing their strategy to “hurt player N as hard as possible”. The Wiki definition of Nash equilibrium doesn’t seem to mention minimax at all, in fact (except in “see also”).
In this case, it seems that everyone’s starting strategy is in fact something like “play 99, and if anyone plays differently, hurt them as hard as possible”. So something resembling minimax is part of the setup, but isn’t part of what defines a Nash equilibrium. (Right?)
Looking more at the definitions...
The “individual rationality” criterion seems properly understood as “one very weak criterion that obviously any sane equilibrium must satisfy” (the logic being “If it is the case that I can do better with another strategy even if everyone else then retaliates by going berserk and hurting me as hard as possible, then super-obviously this is not a sane equilibrium”).
It is not a definition of what is rational for an individual to do. It’s a necessary but nowhere near sufficient condition; if your decisionmaking process passes this particular test, then congratulations, you’re maybe 0.1% (metaphorically speaking) on the way towards proving yourself “rational” by any reasonable sense of the word.
Ok, let’s see. Wikipedia:
This is sensible.
Then… from the Twitter thread:
This seems incorrect. The Wiki definition of Nash equilibrium posits a scenario where the other players’ strategies are fixed, and player N chooses the strategy that yields his best payoff given that; not a scenario where, if player N alters his strategy, everyone else responds by changing their strategy to “hurt player N as hard as possible”. The Wiki definition of Nash equilibrium doesn’t seem to mention minimax at all, in fact (except in “see also”).
In this case, it seems that everyone’s starting strategy is in fact something like “play 99, and if anyone plays differently, hurt them as hard as possible”. So something resembling minimax is part of the setup, but isn’t part of what defines a Nash equilibrium. (Right?)
Looking more at the definitions...
The “individual rationality” criterion seems properly understood as “one very weak criterion that obviously any sane equilibrium must satisfy” (the logic being “If it is the case that I can do better with another strategy even if everyone else then retaliates by going berserk and hurting me as hard as possible, then super-obviously this is not a sane equilibrium”).
It is not a definition of what is rational for an individual to do. It’s a necessary but nowhere near sufficient condition; if your decisionmaking process passes this particular test, then congratulations, you’re maybe 0.1% (metaphorically speaking) on the way towards proving yourself “rational” by any reasonable sense of the word.
Does that seem correct?
This is specifically for Nash equilibria of iterated games. See the folk theorems Wikipedia article.