Can the government talk cheap? Communication, announcements, and cheap talk.

I. Introduction

A great deal of attention has recently been focused on communication in economic contexts. While some models of credible announcements have been based on reputational forces or penalties imposed by third parties, as in Sobel |21~ and Cothren |4~, several recent papers have addressed the possibility of communication through "cheap talk," that is, talk in situations with no explicit penalties for deception. Perhaps the most prominent application of cheap talk in the literature is the recent model of Federal Reserve announcements by Stein |22~.(1)

This paper will show that cheap talk equilibria of the sort modeled by Stein and others often depend in a fundamental way on implausible discontinuities in public responses to government announcements. That is, cheap talk equilibria are frequently only possible if certain infinitesimally small changes in government announcements are capable of causing large, discontinuous changes in public expectations and behavior.

Intuitively, if public expectations are a continuous function of government announcements, then the government can "fine tune" these expectations. The government will therefore often be tempted to deviate from the cheap talk equilibrium announcements in order to manipulate expectations. This causes the equilibrium to unravel.

Discontinuous public reactions sometimes prevent these equilibria from unraveling because they convert continuous choice problems, which allow manipulative fine tuning, into discrete choice problems, where such fine tuning is impossible. These discontinuities are implausible, however, and this may often rule out cheap talk as a realistic model of governmental policy announcements.(2)

While attention in the next two sections is focused on Stein's paper for concreteness, a similar criticism also applies to some, though not all, of the other cheap talk models in the literature. As will be argued below, cheap talk models seem to fall into two major categories:

(a) models which, like Stein's, depend upon discontinuous reactions to convert continuous into discrete choice problems, and

(b) models in which cheap talk plays essentially a coordination role, usually in some sort of intrinsically discrete choice setting.

In models of type (b), the announcer cannot fine tune reactions since reactions are discrete by assumption. Cheap talk is therefore frequently more plausible in this case. However, if the discrete choices in models of type (b) are simply used as an approximation to a continuous choice reality, then cheap talk in such models may still depend implicitly on implausibly discontinuous reactions. In such cases, the arguments in this paper are still relevant.

Section II below describes Stein's model, and section III draws attention to the discontinuities required in public expectations. Section IV discusses the plausibility of these discontinuities. Section V then discusses other cheap talk models in the literature, and section VI concludes. An appendix generalizes the discussion in section III.

II. Stein's Model

The details of Stein's model are unimportant, since a variety of different models can lead to the same class of policy dilemmas. However, to keep the discussion concrete, the following gives a general idea of how Stein models the Federal Reserve's policy problem. Stein begins with a two period model in which the Fed has target interest rates with normalized values of zero in both periods, and target exchange rates of T in both periods. The public knows the target interest rate of zero, but does not know the target exchange rate T.

The Fed has one policy instrument in Stein's model, the second period money supply |M.sub.2~. A high money supply in period 2 pushes the exchange rate up but pushes the interest rate down. The Fed therefore chooses the second period money supply |M.sub.2~ = T/2 to balance off second period interest rate and second period exchange rate targets. However, the Fed wants the public in period 1 to expect |M.sub.2~ to equal T, since this will cause the first period exchange rate to equal the target level of T. Specifically, the Fed would like to manipulate |Mathematical Expression Omitted~ to minimize |Mathematical Expression Omitted~ where |Mathematical Expression Omitted~ is the public's first period expectation of |M.sub.2~. Equivalently, the Fed would like to manipulate |T.sup.e~ to minimize |(|T.sup.e~ - 2T).sup.2~ where |T.sup.e~ is the public's expectation regarding the exchange rate target.

The important thing in this model is that the Fed is planning on a money supply of |M.sub.2~ = T/2, but it wants the public to expect |M.sub.2~ to equal T. Or, expressed in terms of exchange rate targets, if the Fed's exchange rate target is T, it wants the public to believe that its target is 2T. The Fed's dilemma then becomes, how can any announcement it makes in period 1 be credible, given that it has an incentive to deceive the public, and no penalty for doing so?

Stein argues that vague, but only vague announcements will be credible. His argument actually shows less, however. Specifically, he shows that if the Fed is somehow restricted to a certain discrete set of permissible announcements, then it will have an incentive to choose the accurate announcement, so its announcements will be believed. Thus, Stein's argument must assume some mechanism which will restrict the Fed to this discrete set of announcements. In the next section it is shown that the Fed will only restrict itself to this discrete set of announcements if it is compelled to do so by public expectations which are a discontinuous function of Fed announcements.

First, however, we summarize Stein's solution. Suppose possible exchange rate targets for the Federal Reserve board are uniformly distributed along the interval |Mathematical Expression Omitted~. Also, suppose that |Mathematical Expression Omitted~ is partitioned using |Mathematical Expression Omitted~, with

|a.sub.i + 1~ = 6|a.sub.i~ - |a.sub.i - 1~. (1)

Then Stein shows that the Fed will honestly report the interval into which its target T will fall. That is, if the Fed is restricted to make announcements of the form:

"T is in the interval ||a.sub.i~, |a.sub.i + 1~~" for some i, or

"T is in the interval ||-a.sub.i~, |-a.sub.i - 1~~" for some i, (2)

then it will announce the correct interval.(3) Equilibria of this kind are called "partition equilibria" |5~.

The intuitive logic of this result is as follows. If the Fed is forced to choose from the discrete set of announcements in equation (2), then any deviation from the truth in a given direction will push expectations too far in that direction, so the Fed prefers telling the truth to lying. As Stein puts it, "if the Fed wants to lie, it has to tell big lies, rather than small ones. And . . . such big lies can be less attractive than telling the truth" |22, 38~. The question remains, however, what prevents the Fed from deviating from the announcements in (2), and so, telling small lies? This is the issue addressed in the next section.

Stein does not himself solve equation (1). However, using standard methods for solving difference equations |20~, it can be shown that

|Mathematical Expression Omitted~

(this formula can easily be checked by substitution into (1); also, it is easy to see that |a.sub.0~ = 0 and |Mathematical Expression Omitted~).(4)

III. A Closer Look at the "Cheap Talk" Equilibrium

It is now shown that the sort of mechanism modeled by Stein requires the public's expectations to be a discontinuous function of government announcements. That is, the cheap talk equilibrium breaks down entirely if small differences in government announcements can cause only small differences in public expectations.(5) For concreteness, we focus on the Stein model of Federal Reserve announcements. The general case is examined in the Appendix.

The key point is that, to determine whether an equilibrium is self enforcing we must, in the spirit of Kreps and Wilson's |13~ sequential equilibria, indicate what public expectations will be "off of the equilibrium path." That is, we must indicate, not only what the public does if the Fed chooses one of the intervals indicated in (2) above, but also what the public would do if the Fed made some other announcement. Stein's equilibrium is then shown to be self enforcing only if the public's reactions to any other announcement are so undesirable to the Fed, that the Fed would never choose any announcements other than those indicated in equation (2). Such undesirable reactions, in turn, are shown to depend upon beliefs which are implausibly discontinuous.(6)

Suppose that the Fed makes an announcement of the form "our exchange rate target T is between a and b" (or a |is less than or equal to~ T |is less than or equal to~ b, or T |is an element of~ |a, b~). To consider all possible announcements, not just "equilibrium" announcements, we must let a and b vary continuously. Finally, let the public's expectation of T given the announcement "T |is an element of~ |a, b~" be

|T.sup.e~ = f(a, b). (4)

This formulation forces us to model the public's reaction to all possible announcements, not just equilibrium announcements.

It will now be shown that the public expectations function f is either constant or discontinuous. That is, either Fed announcements have no effect on public expectations, or public expectations are discontinuously sensitive to certain infinitesimally small changes in Fed announcements.

For suppose that f is continuous and nonconstant. Then as a and b vary, the range of possible values of f(a, b) must form a closed bounded interval |A, B~. Using technical jargon, since the domain of f is a compact connected set (the set of all pairs (a, b) with |Mathematical Expression Omitted~), the range must be a compact connected subset of the real line, that is, an interval of the form |A, B~ (see, e.g., Theorems 3.4, 3.21, and 3.19 in Armstrong |1~).

Thus, the interval |A, B~ represents the set of all T's which the Fed can lead the public to expect. That is, f(a, b) |is an element of~ |A, B~ for all a and b, and for any number T# in |A, B~ there is an announcement |a, b~ which leads the public to expect T to be T#, so f(a, b) = T#.

Now, the Fed wants the public to expect T to be |T.sup.e~ = 2T, where T is its true exchange rate target. Thus, the Fed chooses the announcement "T in |a, b~" to minimize

|(|T.sup.e~ - 2T).sup.2~ = |(f(a, b) - 2T).sup.2~. (5)

This yields an optimal announcement function which expresses the announcement interval |a, b~ as a function of the target T, as in

|a, b~ = h(T). (6)

The function h is not necessarily unique. That is, there may be several different announced intervals |a, b~ which would all lead the public expectations of T to be the same optimal value.

However, while the optimal announcement function h(T) is not necessarily unique, it turns out that the composite function f(h(T)) is unique. This composite function is therefore very convenient to work with. The function f(h(T)) expresses public expectations as a function of an announcement chosen optimally as a function of T. That is, if the Fed's target is T, then its optimal announcement will be h(T), so the public will expect T to be f(h(T)). Briefly, f(h(T)) is the target which the Fed leads the public to expect, when the true target is T.

We now show that f(h(T)) takes the following form:

|Mathematical Expression Omitted~.

This may be seen as follows. First, the middle line of (7) simply says that whenever possible (i.e., when 2T |is an element of~ |A, B~), the Fed makes an announcement h(T) which leads the public to expect exactly what the Fed wants it to expect, i.e., 2T, so public expectations are f(h(T)) = 2T in this case. Similarly, the top line says that when 2T |is less than~ A, the Fed causes public expectations to be as close as possible to 2T, i.e., |T.sup.e~ = A, since A is the lowest value of T which the Fed can lead the public to expect. Thus, f(h(T)) = A in this case. Similarly, the bottom line indicates that f(h(T)) = B when 2T |is greater than~ B.

Thus, if 2T |is an element of~ |A, B~, then the Fed gets the public to expect exactly 2T, and if 2T |is not an element of~ |A, B~ the Fed leads public expectations to be as close as possible to 2T. The function f(h(T)) is shown in Figure 1.

Now suppose the public sees the Fed make the announcement "our target exchange rate T is in |a, b~" where f(a, b) = T# and A |is less than~ T# |is less than~ B. That is, the Fed makes an announcement which leads the public's expectation to be strictly between A and B. Assume that a rational public understands (7). Then the public knows that T# = f(h(T*)) = 2T*, where T* is the Fed's true exchange rate target. Therefore, the public knows the Fed's true target T* is T#/2, so the public expects T to be T#/2, rather than the value T# = f(a, b) given by the public's expectations function. The equilibrium therefore unravels. This contradicts the original assumption that f was continuous but nonconstant.

The expectations function f must therefore either be constant or discontinuous. If it is constant, then government announcements have no effect on public expectations. That is, the public ignores government announcements. If, on the other hand, the public expectations function f (a, b) is discontinuous, then public expectations must be capable of changing discontinuously in response to certain very small changes in government announcements, which seems implausible (see section IV).

Stein's solution does remain a possibility, though his solution can only be maintained if public expectations are a discontinuous function of announcements. His solution depends upon public expectations of the form

f(|a.sub.i~, |a.sub.i + 1~) = (|a.sub.i~ + |a.sub.i + 1~)/2 and f(|-a.sub.i~, |-a.sub.i - 1~) = -(|a.sub.i~ + |a.sub.i - 1~)/2 (8)

for the |a.sub.i~ given in equation (3).

However, it is not clear how to define f(a, b) for other values of a and b. Perhaps f(a, b) could be set equal to zero for other values (Fed announcement ignored). This would require the Fed to make vague statements, but choose its ranges very carefully. Alternatively, the public could simply match the Fed's announcement to the closest approximating interval from (2). This would require public expectations to be constant for wide variations in Fed announcements, but then change dramatically in response to other very small changes in announcements. Each possibility requires discontinuous reactions by the public to certain slight changes in Fed announcements. The likelihood of such discontinuities is briefly discussed in the next section.

IV. Are Discontinuous Reactions Plausible?

The previous section showed that cheap talk equilibria of the type developed by Stein |22~ depend upon public reactions which are discontinuous functions of government announcements. Moreover, the proof suggests that, if the set of possible public reactions is sufficiently dense, then cheap talk will unravel, even if the public reaction function is discontinuous. As above, the announcer would be tempted to fine tune public reactions, and so, reveal its true information, causing the equilibrium to fall apart.

Thus, if one believes that cheap talk of the sort modeled by Stein actually exists in the economy, then one must conclude that public reactions are highly discontinuous. It would then be an interesting test of the theory if one could identify these discontinuities empirically.

Alternatively, if one believes that such discontinuities are implausible, this would tend to rule out cheap talk equilibria of the sort modeled by Stein.(7) My own opinion is that such knife-edge reactions should be very unusual, and should only occur when agents are acutely aware of their own knife-edge behavior. I know of no compelling evidence for the existence of such behavior.

Furthermore, the equilibrium also requires perfect unanimity in public reactions. Heterogeneous public interpretations of Fed announcements would tend to yield continuity in the overall public expectations function f(a, b), because any small change in announcements would cause at most a few people to revise their expectations discontinuously, and so, cause only a small change in average expectations. The continuous public expectations function could, then, be manipulated by the Fed in the manner suggested above. For an argument along these lines, see Conlon |3~.

On the other hand, it could be argued that the limited number of words in the English language allows a cheap talk equilibrium to be maintained without the conscious effort of the Fed or the public. However, the public must still be extremely suspicious of attempts by the Fed to fine tune its language. Otherwise the Fed will be tempted to adopt terms like "somewhat large," "fairly large," "quite large," etc., to manipulate the public in the manner suggested above, and the equilibrium would unravel.

It could also be argued that the implausibility of discontinuous expectations should be weighed against the possibility that the uninformative (f constant) equilibrium may not be neologism-proof or announcement-proof (see, e.g., Farrell |7~, or the discussion in Farrell and Gibbons |9~ or Matthews et al. |15~). That is, the Fed could do something like explain a Steinlike equilibrium to the public, and then announce an interval. However, such neologisms may be more problematic in the above continuous state space case than in, e.g., the two-state model of Farrell and Gibbons.(8) This is clearly an issue that merits further study.

Finally, it should be noted that there are contexts in which cheap talk equilibria may be more plausible than they are in the case of government policy announcements. One example is diplomatic language, which is very rigid, and intended to communicate to a very specialized, highly trained audience. The argument above suggests that the rigidity of diplomatic language may serve a very specific purpose, since it limits the ability of the speaker to manipulate the listener through talk, and therefore preserves some of the communicative value of the language.

V. Other Cheap Talk Models in the Literature

The previous sections treated Stein's model of Federal Reserve announcements at some length. In this section, we indicate how our concerns apply to other cheap talk models in the literature. Our discussion is necessarily brief, since it is not possible to examine each of these papers at length. The interested reader is encouraged to consult the original papers.

Cheap talk models generally seem to fall into two major categories:

(a) models which, like Stein's, depend upon discontinuous reactions to convert continuous into discrete choice problems, and

(b) models in which cheap talk plays essentially a coordination role, usually in some sort of intrinsically discrete choice setting.

However, if the discrete choices in models of type (b) are actually used as an approximation to a continuous choice reality, then cheap talk in such models may still depend implicitly on implausibly discontinuous reactions. We now discuss a second model of type (a), and then briefly describe some models of type (b).

Another Model which Depends on Discontinuities

Perhaps the model which most resembles Stein's in its dependence on discontinuous reactions is Matthews |14~. The following gives a brief description of the model (for details see the original Matthews paper). Suppose that Congress and the President are considering the funding level of some new program. The Congress's most preferred level of funding is common knowledge, but the President's most preferred (target) level of funding is known only to the President, but not to Congress.

The timing of the process is as follows: First the President makes some sort of announcement to try to influence Congress. Then Congress chooses a level of spending and offers a bill to the President. The President, finally, decides whether to veto the bill or not.

This situation is a lot like the Stein model considered above. The President is the speaker, and Congress is the audience. The President would often like to deceive Congress about her target level of spending, in order to coax a more favorable compromise bill out of Congress. Thus, in the Matthews model, as in the Stein model, the speaker wants to mislead the audience, and faces no direct penalty for deception.

In the Matthews model, as in Stein's this leads to partition equilibria. Therefore, as in Stein's model, successful communication depends on rigid behavior. In the one nontrivial communication equilibrium, the President simply makes a threat to veto or not. The President does not vary the intensity of the threat in proportion with her feelings on the subject, even if she has access to a very rich language. Similarly, Congress responds to any veto threat with the same compromise bill. That is, Congress does not respond to "stronger" threats with more favorable compromises.

As in Stein's model, this rigidity is necessary in order to prevent the President from telling "small" lies. Thus, suppose that Congress's behavior was a continuous function of the intensity of the veto threat, so that a "mild" threat would cause Congress to yield less, and a "strong" threat would cause Congress to yield more. Then the President would be tempted to fine tune Congressional reactions, and, in the process, would reveal her true target spending level. This would cause Congress to deviate from its putative reaction function, and the equilibrium would unravel, just as it does in the Stein case. Thus, the Matthews model depends upon the same sorts of rigid behavior on the audience's part as does the Stein model.(9)

Models in which Cheap Talk Plays a Coordination Role

In the Matthews model, as in the Stein model, the state space (i.e., the space of things that the speaker knows, but the listeners do not know) is a continuous set. Thus, in Stein's model, the Fed's exchange rate target was drawn from a uniform distribution in the interval |Mathematical Expression Omitted~, while in the Matthews model, the President's most preferred spending level was also drawn from a continuous distribution on an interval. The listeners' action spaces in the two models were also continuous.

Thus, if the speaker has access to a sufficiently rich language in these models, and if audience reaction functions are continuous, then the speaker will be able to fine tune the listener's reactions. Therefore, if the speaker wants to deceive the audience, the equilibrium unravels.

However, if the state and action spaces are discrete, then cheap talk may be more robust. Thus, consider the two person game in figure 2. In this game, there are two possible states of the world, |t.sub.1~ and |t.sub.2~, corresponding to the two rows in the figure. One person, the "speaker," knows the state of the world. The other person, the "receiver," can take one of two possible actions, L and R, corresponding to the two columns in the figure. The first (second) number in the cell gives the speaker's (receiver's) payoff as a function of the state of the world and the receiver's action. Thus, if the state of the world is |t.sub.1~, and the receiver chooses action L, then the speaker gets payoff 3 and the receiver gets 2. Note that the speaker cannot control the state of the world, but he/she can communicate with the receiver.

In this game, one very plausible equilibrium is for the speaker to always truthfully reveal the state of the world, and for the receiver to choose L if the speaker says the state is |t.sub.1~, and choose R if the speaker says the state is |t.sub.2~.

In this game, the speaker has no incentive to manipulate the receiver, since truthful revelation already induces the receiver to act in exactly the way that the speaker wants. In a sense, there is no conflict of interest in this game, and cheap talk plays essentially a coordination role. Thus, even though talk is cheap, the equilibrium seems fairly robust.

Several papers in the literature model cheap talk in roughly this way. That is, they assume discrete state and/or action spaces, and model cheap talk as essentially a form of coordination. However, if the discrete state and action spaces in these models are seen as approximations to continuous state and action spaces, then these models may depend, implicitly, on the same sorts of discontinuous reactions as the Stein and Matthews models.(10) Thus, even in this sort of model, thought must be given to the appropriateness of cheap talk equilibria in actual applied situations.(11)

By contrast, cheap talk may sometimes play a coordinating role in situations where the discreteness of choices arises naturally in the model. For example, Farrell and Saloner |10~ show that cheap talk can facilitate coordination when agents are faced with the discrete choice of whether or not to switch from an old standard to a new one (e.g., from the English to the metric system of measurements). Similarly, Farrell |6~ shows that two firms may coordinate using cheap talk when they are each considering the discrete choice of whether or not to enter a market which is too small for both of them. Other models that may fall into this category are Ordeshook and Palfrey |18~, and Forges |11~.

Perhaps the most striking example of cheap talk as a coordination device is in Matthews and Postlewaite |16~. In this model, a seller with private information can use cheap talk to choose from a continuum of possible equilibria, depending on her type. In one equilibrium, cheap talk completely reveals the speaker's type. However, the subsequent play following the announcement is an equilibrium, and, in fact, the equilibrium most preferred by the speaker, even though the speaker's type is fully revealed. Thus, the revealed information does not cause the equilibrium to unravel, in contrast to the Stein case discussed above.(12)

VI. Conclusion

The reasoning above suggests that "cheap talk" equilibria of the sort considered by Crawford and Sobel |5~, Stein |22~, and Matthews |14~ may provide implausible models of communication. If we want to discipline our model building efforts by assuming that the public's expectations are a continuous function of government announcements, we must conclude that "cheap talk" will have no effect on audience reactions in these models.

The analogy with Samuelson's |19~ correspondence principal may be illuminating in this regard. Just as the correspondence principal rules out equilibria which are dynamically unstable, in the same way, the assumption of continuity of public expectations rules out "cheap talk" models of policy announcements, since if expectations are a continuous function of government announcements, then the government will attempt to manipulate the public, and the equilibrium will break down.

One must therefore conclude that models in which government policy announcements convey information to a rational public must either posit some sort of reputation building process, as in Sobel |21~, or assume that some third party imposes costs of some sort for dishonest policy announcements, as in Cothren |4~. Similarly, models that seek to explain vague announcements should look for foundations other than cheap talk. One obvious source of vagueness, for example, might simply be government uncertainty about future plans.

Finally, if one believes that cheap talk is important, even when the speaker would like to deceive the listener, it seems to me that one must either (i) argue that agents actually face a discrete choice framework of a sort which facilitates cheap talk, or (ii) show that the sorts of discontinuities discussed in this paper actually exist empirically. Results of type (ii) would be as interesting as they would be surprising.

Appendix

In this appendix we will extend the argument in the paper to the case in which government plans, |G.sup.p~, and the public expectations desired by the government, |G.sup.de~, are arbitrary (well behaved) functions of the state of the world, x, and show that nontrivial cheap talk is usually not possible with well behaved public expectations functions.

Suppose that the government's policy plan, G, depends on the state of the world, x, as in G = |G.sup.p~(x). Let x be drawn from some compact, connected probability space X. In addition, suppose that the value of G which the government wants the public to expect also depends on the state of the world as in |G.sup.de~ = |G.sup.de~(x), where "de" stands for "desired |public~ expectations." Assume that |G.sup.p~ and |G.sup.de~ are continuous.

A credibility problem arises unless the expected value of the government's policy plan |G.sup.p~(x), given that the government wants the public to expect |G.sup.de~(x) = G*, is exactly the value, G* which the government wants the public to expect, or,

E||G.sup.p~(x) such that |G.sup.de~(x) = G*~ = G*. (A1)

Equation (A1) says that, on average, the government will not bias the public's expectations systematically one way or the other. If (A1) holds, then the government can simply announce that it wants the public to believe G*, and a rational public will know that |G.sup.p~(x) will be G* plus an unpredictable noise term.

A conflict arises when (A1) is almost never true. Thus, for example, in Stein's case, with G = |M.sub.2~ and x = T, we have |G.sup.p~ = T/2 and |G.sup.de~ = T, so

E||G.sup.p~(x) such that |G.sup.de~(x) = G*~ = E|T/2 such that T = |M*.sub.2~~ = |M*.sub.2~/2 = G*/2. (A2)

Thus (A1) is only true for G* = 0 in Stein's case, and the government faces a credibility problem. To generate a similar conflict in the general case, assume

Equation (A1) holds for at most a finite set of G*, (A3)

so the government usually has an incentive to systematically deceive the public.

Suppose the government uses announcements to manipulate public expectations. Also, suppose as in Stein's case that the government chooses its announcements, |Alpha~, out of a continuous, compact and connected set S of possible announcements (in Stein's case, possible announcements take the form "T is in the interval |a, b~" with |Mathematical Expression Omitted~). Finally, assume that public expectations as a function of government announcements are given by the function |G.sup.e~ = f(|Alpha~).

The announcement function f is either constant or discontinuous. For suppose f is continuous but not constant. Since S is compact and connected, so is the set f|S~ of all values of G which it is possible for the government to lead the public to expect. Since this compact connected set is a subset of the real line, it is an interval of the form |A, B~. Thus, it is possible for the government to get the public to expect G to be any value between A and B.

Now, the government chooses an announcement function |Alpha~ = h(x) to minimize

|(|G.sup.e~ - |G.sup.de~(x)).sup.2~ = |(f(|Alpha~) - |G.sup.de~(x)).sup.2~. (A4)

Thus, h(x) gives the optimal announcement for the government to make, as a function of the state x (note that h is not necessarily unique). An argument similar to that in the text shows that f(h(x)) is given by

|Mathematical Expression Omitted~.

Thus, when A |is less than~ |G.sub.de~(x) |is less than~ B, the government's announcement perfectly reveals the value, |G.sup.de~(x), of G which the government wants the public to expect. Therefore, if the government's announcement initially causes the public to expect |G.sup.p~ to be G* = f(|Alpha~) with A |is less than~ G* |is less than~ B, then the public will rethink, remember (A5) conclude that |G.sup.de~(x) = G*, and so instead expect |G.sup.p~ to be

E||G.sup.p~(x) such that |G.sup.de~(x) = G*~, (A6)

which is different from G* for most G* in (A, B), by (A3). This contradicts f(|Alpha~) = G*, so our original assumption that the public expectations function f was nonconstant and continuous must be wrong.

1. Other cheap talk models will be discussed in section V below.

2. Alternatively, one could interpret the results in this paper as suggesting that public expectations actually do respond to announcements in a discontinuous manner. This seems unlikely. For example, if there is any uncertainty about the underlying parameters, and opinions differ, then average public expectations will tend to be a continuous function of announcements. For a further discussion, see section IV.

3. The argument goes as follows: If the Fed announces that its target exchange rate T is in the interval ||a.sub.i~, |a.sub.i + 1~~, and is believed, then the public expects the target exchange rate to be (|a.sub.i~ + |a.sub.i + 1~)/2. When the Fed's actual target is T = |a.sub.i~, the Fed would like the public to expect |T.sup.e~ = 2|a.sub.i~, but it is indifferent between public expectations of |T.sup.e~ = (|a.sub.i - 1~ + |a.sub.i~)/2 and |T.sup.e~ = (|a.sub.i~ + |a.sub.i + 1~)/2, assuming the |a.sub.i~ satisfy equation (1). Thus, the Fed is indifferent between the announcements ||a.sub.i - 1~, |a.sub.i~~ and ||a.sub.i~, |a.sub.i + 1~~. If the target exchange rate T is in the interior of the interval ||a.sub.i~, |a.sub.i + 1~~, however, then the Fed prefers the announcement ||a.sub.i~, |a.sub.i + 1~~ to the announcement ||a.sub.i - 1~, |a.sub.i~~. Proceeding in this way shows that the Fed will always announce the correct interval. See Stein |22~ for details.

4. Incidently, equation (3) can be used to show that the formula in Stein's Proposition 2 is incorrect. It should read |lim.sub.i|right arrow~|infinity~~|a.sub.i~/|a.sub.i + 1~ = |(3 + |square root of 8~).sup.-1~ which equals 0.1715728 . . . Stein's incorrect formula (involving a geometric sum) simplifies to 35/204 = 0.1715686 . . . These two numbers are identical if rounded off to five digits as Stein does, but the formulas do give slightly different numbers.

5. This actually can be shown to follow from Crawford and Sobel's |5~ solution. However, the following line of reasoning is less complicated. It also draws on familiar arguments about the difficulty of manipulating rational agents.

6. Kreps and Wilson |13, 864~ argue that "making explicit the construction of beliefs off the equilibrium path enables discussion of which beliefs are 'plausible' and which are not". Thus, for example, this paper shows that out-of-equilibrium beliefs in a solution such as Stein's must be discontinuous. It then argues that this discontinuity is implausible, so that Stein's cheap talk equilibrium provides an unrealistic model of public reactions to government announcements. See Kreps |12~, which reviews a growing literature on equilibrium refinements based on out-of-equilibrium beliefs.

7. Note that since messages do not enter the utility function directly, one cannot rule out discontinuous reaction functions a priori. Thus, ruling out discontinuous reactions reflects beliefs about "likely," as opposed to strictly rational behavior.

8. The simplest credible neologism seems to be, e.g., "|Mathematical Expression Omitted~." An argument similar to that in note 3 shows that the Fed would prefer this statement to the no-communication expectation |T.sup.e~ = 0 precisely when T is in |Mathematical Expression Omitted~. Thus, this neologism is credible, though, as above, the public should be suspicious unless it is confident that everyone would reject misleading neologisms such as "|Mathematical Expression Omitted~."

The statement "|Mathematical Expression Omitted~" would also be a "weakly credible announcement," in the terminology of Matthews, Okuno-Fujiwara, and Postlewaite |15~. However, it would not be a "credible announcement," using the Matthews et al. terminology, because there are other weakly credible announcements which the Fed would sometimes prefer to make, even if |Mathematical Expression Omitted~. Thus, the Fed's choice of one such announcement over another may reveal too much information about the Fed's true target. This suggests that the equilibrium with no communication may be acceptable under the "announcement-proof" criterion, which is the criterion preferred by Matthews, Okuno-Fujiwara and Postlewaite |15~.

It is also worth noting that in communication games of this sort, no equilibrium is generally neologism proof |7~.

9. Certain considerations, however, may make the equilibrium here somewhat more plausible than in Stein's case. First, since the decision whether or not to veto is a discrete choice, this may give the two-message equilibrium a certain saliency, with Congress simply ignoring the intensity of the threat, and so, the President making no statements about this intensity.

Second, congressional debate, committee structure, etc., may help congresspeople to focus on the discontinuous reactions necessary to support a nontrivial cheap talk equilibrium.

Third, Congress reacts to any Presidential statement with a vote. If we assume the median voter hypothesis, then the median Congressional reaction may be a discontinuous function of announcements, in accordance with the equilibrium, even if Congressional reactions to announcements are not unanimous; all that is necessary is that the median Congress-person act in accordance with the equilibrium (see Conlon |3~). On the other hand, with logrolling, the median voter hypothesis may be invalid |17, 82-6~, so the equilibrium may require essentially unanimous reactions.

10. For example, Austen-Smith |2~ introduces discreteness by allowing agents to observe a single realization of a discrete distribution. If the distribution had been continuous, by contrast, then cheap talk may have required partition equilibria.

Similarly, Farrell and Gibbons |9~ simply posit a discrete choice problem. The argument above thus suggests that the lessons of the Farrell and Gibbons paper may extend with some difficulty to the continuous choice case.

11. On the other hand, a continuous state and action space model may sometimes be seen as an approximation to a discrete state and action space model in which the state and action spaces form a fine grid. The study of cheap talk in such models therefore merits additional study. Matthews, Okuno-Fujiwara, and Postlewaite |15~ provide an extremely interesting treatment of cheap talk in a wide range of discrete state and action space models. Their work shows, among other things, that cheap talk is by no means unproblematic, even in the discrete choice framework.

12. For another example of cheap talk between a buyer and seller, see Farrell and Gibbons |8~.

References

1. Armstrong, Mark A. Basic Topology. New York: Springer-Verlag, 1983.

2. Austen-Smith, David, "Information Transmission in Debate." American Journal of Political Science, February 1990, 124-52.

3. Conlon, John R. "Robustness of Cheap Talk with a Large Audience." Paper presented at the North American Summer Meetings of the Econometric Society, Seattle, June, 1992.

4. Cothren, Richard, "Asymmetric Information, Optimal Money Growth Targets, and Fed Secrecy in a Monetary Policy Game," Journal of Macroeconomics, Fall 1990, 599-609.

5. Crawford, Vincent and Joel Sobel, "Strategic Information Transmission." Econometrica, November 1982, 1431-51.

6. Farrell, Joseph, "Cheap Talk, Coordination, and Entry." Rand Journal of Economics, Spring 1987, 34-39.

7. -----, "Meaning and Credibility in Cheap-Talk Games," forthcoming in Games and Economic Behavior.

8. ----- and Robert Gibbons, "Cheap Talk Can Matter in Bargaining." Journal of Economic Theory, June 1989, 221-37.

9. ----- and -----, "Cheap Talk with Two Audiences." American Economic Review, December 1989, 1214-23.

10. ----- and Garth Saloner, "Standardization, Compatibility, and Innovation." Rand Journal of Economics, Spring 1985, 70-83.

11. Forges, Francoise, "Equilibria with Communication in a Job Market Example." Quarterly Journal of Economics, May 1990, 375-98.

12. Kreps, David M. "Out-of-Equilibrium Beliefs and Out-of-Equilibrium Behavior," in The Economics of Missing Markets, Information, and Games, edited by Frank Hahn. Oxford: Clarendon Press, 1990.

13. ----- and Robert Wilson, "Sequential Equilibria." Econometrica, July 1982, 863-94.

14. Matthews, Steven A., "Veto Threats: Rhetoric in a Bargaining Game." Quarterly Journal of Economics, May 1989, 347-69.

15. -----, Masahiro Okuno-Fujiwara, and Andrew Postlewaite, "Refining Cheap Talk Equilibria." Journal of Economic Theory, December 1991, 247-73.

16. ----- and Andrew Postlewaite, "Pre-play Communication in Two-Person Sealed-Bid Double Auctions." Journal of Economic Theory, June 1989, 238-63.

17. Mueller, Dennis C. Public Choice II. Cambridge: Cambridge University Press, 1989.

18. Ordeshook, Peter C., and Thomas R. Palfrey, "Agendas, Strategic Voting, and Signaling with Incomplete Information." American Journal of Political Science, May 1988, 441-66.

19. Samuelson, Paul A. Foundations of Economic Analysis. New York: Atheneum, 1965.

20. Sargent, Thomas J. Macroeconomic Theory, Second Edition. Orlando: Academic Press, 1987.

21. Sobel, Joel, "A Theory of Credibility." Review of Economic Studies, July 1985, 557-73.

22. Stein, Jeremy C., "Cheap Talk and the Fed: A Theory of Imprecise Policy Announcements." American Economic Review, March 1989, 32-42.