Information Validates the Prior: A Theorem on Bayesian Updating and Applications

Abstract

We develop a result on expected posteriors for Bayesians with heterogenous priors, dubbed information validates the prior (IVP). Under familiar ordering requirements, Anne expects a (Blackwell) more informative experiment to bring Bob’s posterior mean closer to Anne’s prior mean. We apply the result in two contexts of games of asymmetric information: voluntary testing or certification, and costly signaling or falsification. IVP can be used to determine how an agent’s behavior responds to additional exogenous or endogenous information. We discuss economic implications. (JEL C11, D82, D84)

Conveying Value via Categories

“…Galperti (2019) and Kartik, Lee and Suen (2020) also study environments with heterogeneous priors across agents. Both these papers study problems different from ours, but some results in Kartik, Lee and Suen (2020) have parallels for the specific question of when full pooling or separation of qualities is optimal. Kolotilin (2018) and Dworczak and Martini (2019) study optimal signaling structures in environments where the receiver's action depends only on their posterior mean.…”

“…We can take a step further and think of even more primitive examples which map into the state-based-preference or divergent-prior reduced form. Contributions such as Rayo (2013), Kartik, Lee and Suen (2016) and Kartik, Lee and Suen (2020) have also studied problems that map in this way on to "distorted priors." 3.1.…”

A dynamic model of censorship

“…Also, Edmond (2013) and Redlicki (2017) study a signal-jamming model in global games. 5 Dye (2017), Daughety and Reinganum (2018), and Kartik, Lee, and Suen (2021) study disclosure of verifiable information in other applications where concealing information is costly. 6 Other papers use a similar framework to study reputation, but also consider the firm's exit decision (Board and Meyer-ter Vehn (2022)), the promotion of reputation (Hauser (2017)), and the design of optimal monitoring (Varas, Marinovic, and Skrzypacz (2020)).…”

Conveying value via categories

“…Some of Rayo's (2013) results parallel ours; for example, his characterization of the optimality of full separation is equivalent to ours in Proposition 3. Kartik, Lee, and Suen (2021) find related characterization of full separation in an environment with a different restriction on the set of experiments available to the sender.…”

“…88 This literal interpretation is also taken by Alonso and Camara (2016), who extend the “concavification” argument of Kamenica and Gentzkow (2011) to a context with heterogeneous priors. Further, see Van den Steen (2004, 2009, 2010, 2011), Che and Kartik (2009), Galperti (2019), de Clippel and Zhang (2020), and (Kartik, Lee, and Suen (2017, 2021), who also consider environments with heterogeneous priors. …”