From Nash to dependency equilibria

• Nash equilibrium depends on the assumption of causual independence of players' decisions.
• Nash equilibrium is the compbination of strategies s of A and t of B where none of the players can improve the outcome by unilaterally changing the strategy.
  • If a player uses a mixed strategy, it can't be better than any simple strategy.
  • It can also be seen as an equilibrium of opinions: all substrategies of a mixed strategy are equally good.
• We assumed that the choice of strategies is independent, but what if it's not, what if A's choice is correlated with B's choice. Then probability distribution over strategy choices doesn't factor into two individual probability distributions.
• If we talk about joint probability distribution, we get a broader concept of dependency equilibrium of which Nash equilibrium is a degenerate case.

Example: single-shot prisoner's dilemma -- the only Nash eq is mutual defection, but there's a dependecy equilibrium of mutual cooperation if the players can assume large enough chance of cooperation in response to cooperation.

• The choices of strategies can't cause each other. But they might have a common cause.
• It's important to note that decision is a process that takes some time and the action might not immediately follow it.
  • We can model decision with a more detailed reflexive model that includes desires and beliefs.
• Given this, the common cause of strategy choices would be the joint formation of the players' decision situation.
• If the players maintain the dependency between their choices, they can reach a dependence equilibrium, if they so choose. Or they can break the dependency and be left with the choices of a Nash equilibria.

We can also modify the original game to include coordination -- this will make dependency equilibria of the original game into Nash equilibria of the coordinated game. In general it's rational to cooperate if we are sufficiently sure of the rationality of the other player and their knowledge of our rationality (then we can expect cooperation).

See also: Aumann's correlated equilibria.

Q/A: - This doesn't quite work under an assumption of libertarian free will? - The player can always break the dependency.