Inside Baseball: Uncommon Knowledge

Note: This is the first of what might be an irregular “column” of sorts, “Inside Baseball,” focusing on the minutiae of my research, as opposed to current events. 


The heart of game theory is “what would everyone else think if I do what I am about to do differently?”

This is slightly different than the standard “introduction to game theory” approach, where the focus is often on the related question, “what would everyone else do if I do what I am about to do differently?”  But while the difference is slight, it is fundamental.  Game theory is about beliefs, or more appropriately, about consistency of beliefs.

This point bedevils empirical applications (or, more crudely, “tests”) of game theory for at least two reasons.  First, we rarely, if ever, can measure beliefs in anything approximating a direct fashion.  There is a core concept in game theory that is amenable to this test, known as rationalizability, and—unsurprisingly to me as a game theorist—people frequently refute the claim that all actions are rationalizable.  But let’s leave that point to the side.

The second point is more important, to me at least.  At the heart of game theory is the idea that not only are beliefs consistent with one’s own actions (that’s rationalizability, in a nutshell) and consistent with others’ actions (that’s Nash equilibrium, very loosely), they are are consistent with each other.  That is, in any reasonable game theoretic notion of equilibrium, every person not only acts in accordance with his beliefs about what others will do, he or she also understands (“believes”) correctly what everyone else in the game believes, understands that the other players believe correctly about what the player in question believes, including that the player believes correctly about what the other players’ believe about what the player in question believes about their beliefs, and so forth….

This uncommon notion is an instance of what is referred to as common knowledge in game theory.

Well, this uncommon notion is simultaneously elegant and unambiguously empirically false.  For example, it flies in the face of the reality that Florida Gulf Coast University made it to the Sweet 16.

But more seriously, this point is exactly the point of game theory. Game theory is a theoretical enterprise and accordingly requires a priori constraints for the purpose of being meaningful.  And, since this constraint is theoretically elegant and epistemologically appealing, one must always keep in mind that game theory is an inherently philosophical endeavor.  While one can (and should) certainly employ the trappings of game theory for empirically-minded endeavors, the goal of equilibrium analysis is inherently normative or prescriptive.  In other words, game theory models ask “what can (in theory) be achieved in a world in which individuals are intimately involved with the interaction at hand?”

A key (and illuminating) point in this regard is the beginning point of this post: what can happen in equilibrium is, in most interesting settings (i.e., “games”), dependent on what each individual believes about what will happen—or, more fundamentally, what other involved individuals will believe—if he or she acts differently.

When you take this point seriously, you must realize that “testing” game theory models is an inherently ambiguous enterprise.  Suppose the model “works.”  Did it work for the “game theoretically correct” reasons?  Suppose the model doesn’t work.  Why did it fail?

These are important questions, and any answer to either one has no bearing on the “validity” of game theory. Rather, the fact that one could ask either of these questions, the context within which these questions is accordingly posed, is to the credit of game theory.  In a nutshell, every time equilibrium predictions fail, an empirical angel gets his or her wings thanks to game theoretic reasoning.

With that, I leave you with this.