Monday, January 12, 2015

Iterated games

Now that I've completed my series on the evolution of Iterated Prisoner's Dilemma strategies, I want to say something about the importance of iterated games.

At the very end, I brought up another two player game, called the Battle of the Sexes.  Battle of the Sexes is often told through a particular narrative: a husband and wife negotiate over whether to go to the opera or the football game.  But this narrative is not an entirely accurate representation of the game-theoretic concept of Battle of the Sexes.  In theory, the players do not communicate anything, and go into the game without any knowledge of what the other player will choose.

The narrative that is usually used for the Prisoner's Dilemma is more precise.  The Prisoner's Dilemma narrative tells of two prisoners who are locked separately with no way to communicate.  That's the way that game theory operates.

There's actually some justification for this, even in real situations where players can communicate with each other.  Players can basically say whatever they want about what they will choose, but it's all just "cheap talk".  There's no way to enforce a correspondence between what players say and what they do.  Thus the correct strategy is to ignore everything the other player says.  (Of course in practice, that's now how people behave, but put it up to human irrationality.)

Consider a situation where two players have the ability to communicate only one piece of information: their intention to go to a football game or the opera.  They are allowed to switch their intention at any time.  After exactly a minute has passed, each player's intention becomes reality.  But everything you communicate in the first 59 seconds has no impact whatsoever, because you can just change your mind in the last instant.

But suppose that the game doesn't end at exactly one minute.  Maybe you don't have any idea when the game will end.  Then, it's like you and your spouse play the game over and over again, an unknown number of iterations, and it's only the last iteration which counts.  But since you don't know which is the last iteration, you should choose a strategy which optimizes the average outcome over a large number of iterations.  In fact it can be a very large number of iterations, if our reaction time is much shorter than our uncertainty about the end time.

I suggest that this latter situation is closer to what we experience in most realistic situations.  Thus, even if we're only playing one game of Prisoner's Dilemma or Battle of the Sexes, it's often like an iterated version of the game.  I've never heard a game theorist express this equivalence between "real time" games and iterated games, but that's my analysis.

I think this is a practical solution to the Prisoner's Dilemma in most realistic situations.  In a simple Prisoner's Dilemma, rational players (for some definition of rational) always defect.  But in an iterated Prisoner's Dilemma where the number of iterations is unknown, rational players may cooperate.  The keyword is "may", since as my simulations found, the iterated Prisoner's Dilemma is far from simple, even when considering an idealized situation.

A large portion of morality and politics are essentially dedicated to solving game-theoretic problems, particularly the Prisoner's Dilemma.  It is my vague hope that in the future we will develop new game theory "technology" which will have as much positive impact as anything new in physics.