Competitive Information Design with Asymmetric Senders

Zhicheng Du, Wei Tang, Zihe Wang, Shuo Zhang

Abstract: We consider a competitive information design game in which there are multiple senders vie for the selection of a risk-neutral receiver by disclosing information about their individual state realizations. The receiver aims to select the sender who has the highest expected value of the state. We consider a general setting where the senders may be ex-ante heterogeneous, namely, while the senders' state realizations are independently distributed, they do not necessarily follow the same prior distributions. Each sender can only control the disclosure of information regarding his own state realizations, but there is no structural restriction on the set of the feasible information disclosing strategies.It is well-known that the sender's full flexibility in choosing any information to be revealed to the receiver can be modeled as each sender being able to choose any mean-preserving contraction (henceforth, MPC) of his own prior distribution. The receiver's expected value of each sender's state realization is then independently drawn according to that sender's designed MPC. We focus on the Nash equilibrium played by the senders.The first main result of this work is that we provide sufficient conditions under which an equilibrium always exists among senders' competition game. We prove that as long as there is no mass point in senders' prior distributions, an equilibrium always exists.We next characterize the necessary and sufficient conditions for the equilibrium structure. Using the properties of equilibrium, we introduce the virtual competitive function, which can be served as a powerful tool for verifying whether any given strategy profile is indeed an equilibrium. Our characterizations strictly generalize the symmetric equilibrium conditions provided in the symmetric environment studied in previous works. En route, we also show the non-uniqueness of the equilibrium through an construction of an example.The last contribution of this paper is the applications of our structural characterizations of a Nash equilibrium. In the first application, we show that by utilizing the established verification conditions, we are able to fully characterize the equilibrium of a general two-sender game when the senders' priors are strict uni-modal. In the second application, we revisit the symmetric setting when the senders share the same prior distribution, we show that symmetric equilibrium is unique among all asymmetric strategy profiles.The full version is available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4861024.