must obtain the desired action by choosing what information to reveal. Each player’s utility depends on the state of the world and on both players’ decisions that eventually lead to action a. Nash equilibrium We look for an equilibrium where each player decides optimally, assuming that the other player also decides optimally. Players are rational, although R has only limited information. Expectations get realized, and there is no incentive to deviate from this situation. Theorem Crawford and Sobel characterize possible Nash equilibria. There are typically multiple equilibria, but in a finite number. Separating, which means full information revelation, is not a Nash equilibrium. Babbling, which means no information transmitted, is always an equilibrium outcome. When interests are aligned, then information is fully disclosed. When conflict of interest is very large, all information is kept hidden. These are extreme cases. The model allowing for more subtle case when interests are close, but different and in these cases optimal behavior leads to some but not all information being disclosed, leading to various kinds of carefully worded sentences that we may observe. More generally: There exists N* > 0 such that for all N with 1 ≤ N ≤ N*, there exists at least an equilibrium in which the set of induced actions has cardinality N; and moreover there is no equilibrium that induces more than N* actions. Messages While messages could ex-ante assume an infinite number of possible values µ(t) for the infinite number of possible states of the world t, actually they may take only a finite number of values (m1, m2, . . . , mN). Thus an equilibrium may be characterized by a partition (t0(N), t1(N). . . tN(N)) of the set of types [0, 1], where 0 = t0(N) < t1(N) < . . . < tN(N) = 1. This partition is shown on the top right segment of Figure 1. The ti(N)’s are the bounds of intervals where the messages are constant: for ti-1(N) < t < ti(N), µ(t) = mi. Actions Since actions are functions of messages, actions are also constant over these intervals: for ti-1(N) < t < ti(N), α(t) = α(mi) = ai. The action function is now indirectly characterized by the fact that each value ai optimizes return for the R, knowing that t is between t1 and t2. Mathematically (assuming that t is uniformly distributed over [0, 1]), → Quadratic utilities: Given