From Nash equilibria to collective behavior

https://twitter.com/ulaulaman/status/517303481565458432 by @ulaulaman about #Nash equilibria and their role in collective behavior

The Nash equilibrium is an important tool in game theory:

[It] is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy. If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.
Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Will's decision, and Will is making the best decision he can, taking into account Amy's decision. Likewise, a group of players are in Nash equilibrium if each one is making the best decision that he or she can, taking into account the decisions of the others in the game.

Nash equilibria may, for example, be found in the game of coordination, in the prisoner's dilemma, in the paradox of Braess⁽⁶⁾, or more generally in any strategy game. In particular, given a game, we can ask whether it has or not a Nash equilibrium: apparently deciding the existence of Nash equilibria is an intractable problem, if there is no restriction on the relationships among players. In addition for a strong Nash equilibrium, the problem is on the second level of the polynomial hierarchy, which is a scale for the classification problem based on the complexity of resolution⁽¹⁾.
In addition to this study about Nash equilibria, Gianluigi Greco (one of my high school classmates), together with Francesco Scarcello, also studied the Nash equilibria (in this case the forced equilibria) in graphical games, where graphical game is a game represented in a graphical manner, through a graph⁽²⁾.