Competitive Investment with Bayesian Learning: Choice of Business Size and Timing

Abstract

Motivated by challenges faced by firms entering an unknown market, we study a strategic investment problem in a duopoly setting. The favorableness of the market is unknown to both firms, but firms have prior information about it. A leader invests first by choosing its investment size. Then, in a continuous-time Bayesian setting, a competitive follower dynamically learns about whether the market is favorable or not by observing the leader’s earnings, and chooses its investment size and timing. In this setting, we characterize equilibrium strategies of firms. A distinctive feature of our model is that firms choose their investment sizes, and thus the follower’s observations about the favorableness of the market can be censored due to the leader’s investment size choice. It is generally accepted that if there is an increase in the likelihood of a favorable market, the firm’s expected discounted profit and its investment size increase. Our paper shows that, contrary to this common understanding, the leader’s equilibrium expected discounted profit and equilibrium investment size can strictly decrease when there is an increase in the likelihood of a favorable market. This is due to a non-trivial interplay between the leader’s investment size decision and the follower’s investment strategy.

Keywords: optimal stopping, Bayesian learning, competition, investment size, investment timing