Updating toward the signal
In economic theory, it is frequently assumed that agents infer the value of some random variable based on imperfect observation. For example, bidders in auctions infer the unknown value of an object from private signals (cf. [\textit{R. Wilson}, Rev. Econ. Stud. 44, 511--518 (1977; Zbl 0373.90012)]). Modelers often assume that the posterior expectation lies between the prior expectation and the observed signal for every possible signal value. We will call this property updating toward the signal (UTS). This property was verified in some cases such as under normal prior with a normally distributed and independent signal error. However, UTS is not verified in general, although many authors take this property as necessary for the results they present (cf. [\textit{D. A. Moore} and \textit{P. J. Healy}, Psychol. Rev. 115, 502--517 (2008)]). The authors also study a weaker updating requirement, where the posterior mean must lie in the same direction as the signal, relative to the prior mean. They refer to this property as UDS. Finally, mean reinforcement (MR) is also considered. It requires that the posterior mean equal the prior mean when the signal equals the prior mean. They show that if the prior and signal error densities are both symmetric and quasiconcave then UTS will occur. Moreover, if for a given prior, UTS occurs for all symmetric and quasiconcave error densities, then the prior must be symmetric and quasiconcave. Similar characterizations are derived for UDS and MR. Examples are used to verify that the results are tight. In the economics literature, this paper is similar in spirit to [\textit{P. R. Milgrom}, Bell J. Econ. 12, 380--391 (1981)], though the current article seeks to order the posterior mean relative to the prior mean while Milgrom's goal is to order various posterior distributions according to their signal realizations. Demonstrations of some results make use of changes of variables and Bayes's rule. They finally show their main results by applying them to the so-called portfolio allocation problem (cf. [\textit{K. J. Arrow}, Essays in the theory of risk-bearing. Amsterdam-London: North-Holland Publishing Company (1970; Zbl 0215.58602), Chapter 3], and [\textit{H. M. Markowitz}, J. Financ. 7, 77--91 (1952)], among others). To do this, they consider a risk-averse investor allocating a fixed endowment of wealth between a risky asset and a risk-free asset. Initially, he makes a portfolio allocation decision based only on his prior beliefs about the risky asset's return. Then, he receives an informative signal and is asked if he would like to revise his investment decision. They define three different notions of how the investor's signal might reinforce his prior investment decision that are equivalent to MR, UDS, and UTS, respectively. Applying two of the main results of the paper gives two characterizations in the context of the portfolio allocation problem. Its meaning is that a modeler who assumes that good signals always reinforce a decision to invest (for all uniform error distributions) and bad signals always reinforce a decision not to invest, implicitly assumes symmetry and perhaps quasiconcavity of the prior distribution.
scientific article; zbMATH DE number 6076612
scientific article