“Strategically Valuable Information”, Josh Cherry and Lones Smith (mimeo, 2014)

In a 1953 classic paper, Blackwell showed that experiment A is sufficient for B if
and only if a decision maker can attain a larger set of payoffs with A than with B in
any decision problem. In this paper, we ask the Blackwell equivalence question in a
strategic setting. In other words, does there exist a partial order on information held
by players in a game that reflects “more” or “better” information, which coincides precisely with the ability to induce more equilibrium payoff vectors in all Bayesian games? If so, we say that it is “strategically more valuable”. In this paper, we define a meaningful sense in which information structures can be compared by how “strategically informative” they are. Combining the two notions, we answer our original question in the affirmative: There exists an intuitive definition and characterization of the partial order more strategically informative, and it is equivalent to the partial order more strategically valuable. The conditions we provide are easily checked, are useful in an array of economic settings, and have straightforward geometric interpretations.

Our main theorem applies to a wide variety of economic environments of interest endowed with commonly used information structures. For example, sunspots are a frequently used tool in general equilibrium theory. Our results provide a natural partial ordering on sunspot equilibria, regardless of the environment in which they operate. The centerpiece application is to repeated games with private monitoring where the more strategically informative order ranks monitoring structures. Consequently, we can show when a change in monitoring structure will weakly expand the set of sequential equilibria. This mirrors a classic result of ? for repeated games with imperfect public monitoring.