The Informativeness Of Signals

Abstract

This paper examines a class of signaling games with multi-dimensional private information to study how the prior, joint distribution of the private information variables affect a signal's effectiveness in revealing information about these variables. To illustrate the general problem investigated in this paper, consider the following signaling situation: A corporate insider decides on an action a corporation will take. After this action has been observed, the stock market revises the equilibrium value of the corporation's shares. To the stock market, private information about the stock value is most important. Unfortunately, the insider may also have private information about other factors that influence the firm's action, and she may have private information about how the action affects her own private utility. Therefore, to infer what the action implies about the firm value, the stock market must disentangle all the factors that caused the action. For instance, a dividend decision may be conservative because the insider expects future cash flows to be moderate, or because the corporation has a temporary liquidity shortfall, or the firm has a need for high liquidity, or the insider likes to keep cash available for perks, or because she is particularly risk averse, or some combination. Some of these factors are relevant to the firm value, some to private utility, some to both, and some to neither. But all factors may still affect the decision. The insider's private information about all these factors must therefore be considered when drawing inferences from her decision. We use numerical analysis to show that the signal is more informative about a particular variable (the value or quality of the sender) the lower the prior standard deviation of the other variable(s). Also, the signal is less informative the more the variables are correlated. Average signals are less informative than very high or very low signals. There exist combinations of standard deviations of the variables, and correlations between them, so that the signal is completely uninformative although there is substantial private information present.