This is just to show off my newly-acquired skills from ICPSR summer school. If you aren't into game theory it will mean nothing to you - you have been warned!
The standard cheap-talk game, found e.g. in Morrow, has Sender's type t drawn from a uniform distribution on a unit interval: t ~ U[0,1]. Sender sends a message m from some set of possible messages M; then Receiver chooses an action a from that same unit interval. Receiver's utility declines monotonically, continuously and symmetrically with the distance of a from t, e.g. UR(a) = - (t-a)^2. In effect Receiver is trying to guess Sender's type. Sender's utility declines monotonically, continuously and symmetrically with distance of a from some point t+c, where c>0 and measures some conflict of interest. For example, Sender's utiliy might be US(a) = - (t+c-a)^2. So Sender wants Receiver to guess Sender's type, but be a little bit off.
It then turns out that if c is small enough, there are informative equilibria where different ranges of types send different messages (so that Receiver can infer something about Sender's type); and as c shrinks the maximum number of messages in a potential equilibrium increases. It's a nice prediction - communication gets easier when there are common interests to rely on.
But now consider this jolly little variation. (I should warn you at once that I can think of no substantive applications, but it's food for thought in any case.)
Suppose that Sender's type is still drawn from U[0,1], but now we think of it as being defined on a unit circle, with distance defined as the shortest possible way round the circle. So now, for example, choosing a = 1 when t=0 would be a very good action for Receiver, as this would give a-t=0.
Now there is no equilibrium in which Senders with t in different sub-intervals send different messages, no matter how small c is. (I am not sure I can prove that there is no informative equilibrium at all. Perhaps there's one in which a continuum of possible messages are sent, informing R about the value of t modulo c, or something similar.)
Suppose that there are n different sub-intervals around the circle; Senders with t in any of these sub-intervals send different messages. Let's take an arbitrary pair of adjacent sub-intervals (x,y) and (y,z), sending messages m1 and m2. As usual, since t is uniformly distributed, Receiver's best responses are a1 = (x+y)/2 and a2=(y+z)/2. And, as usual, because Sender's utility function is continuous w.r.t. a, Senders at the boundary where t = y must be indifferent between a1 and a2.
Therefore, it must be the case that the distance from a1 to y+c (Sender's ideal point when t=y) is equal to the distance from a2 to y+c. And so the boundary point y must be closer to the midpoint of (x,y) than to the midpoint of (y,z); and so the interval (y,z) must be larger than (x,y).
But this must be true for all intervals round the circle, which is clearly impossible. (If there is a finite number of intervals, n, then the nth interval must be bigger than the n-1th interval, which itself is bigger than the n-2th interval and so on down to the first; but it must also be smaller than the first interval which is adjacent to it. If there are an infinite number of intervals, then choose an arbitrary selection of 2 or more points on the circle and consider that the size of the intervals containing these points must continue to increase as you move clockwise around the circle. I think this works. Hey, it's a blog, and I'm only writing this to avoid real work.) QED.
Note that nothing here depends on the exact geometry of the "circle". The proof works for any interval with distance defined in an appropriately circular fashion. (But, as I said, what the hell are the substantive applications? A hunter is chasing an elephant, which itself is chasing a bear around a big lake; he wants his friend to slay both elephant and bear, but his friend only wants to tackle the elephant. Oh Lord. Answers on a postcard?)