This week, it has become abundantly clear that the Fed is "deeply divided." In speeches and public communications, FOMC committee members and Fed Chairman Ben Bernanke have revealed significant differences in their outlooks and intentions for the economy. Tim Duy writes that "The growing division makes it increasingly difficult to think of "the Fed" as a single entity with regards to policy intentions." This is an extremely important point, because most macroeconomic models do consider the Fed as a single entity, and would have different implications if they did not.
Monetary policy is often modeled as a dynamic game in which the two players are the central banker and the public. Typically, the central banker can choose what private information to reveal to the public. Monetary policymakers' preferences and reputational concerns determine their optimal communication strategy in the equilibrium of this dynamic credibility game (see for example Faust and Svensson 2001). Depending on the exact "rules of the game," the optimal strategy turns out to be something less than full information revelation. This game-theoretic political economy paradigm for thinking about monetary policy became hugely influential after seminal papers by Kydland and Prescott in 1977 and Barro in 1986, and has shaped the way economists think about the merits of central bank independence, rules versus discretion, transparency, and explicit inflation targets, with. Insights from this huge literature have been thoroughly integrated into the policymaking sphere.
In reality, of course, in almost every country, monetary policy is not made by a single representative agent, but rather by a committee of very non-representative agents, each with their own, sometimes conflicting, preferences and reputational concerns. With multiple central bankers, monetary policy is a dynamic game between more than two players-- which makes computing optimal strategies dauntingly complex. Strategic behavior between members of the committee will influence each member's communication strategy with the public and with each other. And the public, aware of these strategic interactions, will have quite a complex task computing their best response.
I'm not quite sure where we go from here. One of the most brilliant and famous game theorists, John Nash, proved that non-cooperative games with an arbitrary finite number of players have a Nash equilibrium. But actually finding such an equilibrium is a huge challenge (plus, the non-cooperative assumption is kind of restrictive.) A pair of computer scientists at Berkeley and Stanford note that "even less is known about computing equilibria in multi-player games than in the (still mysterious) special case of two-player games." Even more telling is the title of another paper by Berkeley computer scientists: "Three-Player Games are Hard."