Information Acquisition in Global Games of Regime Change (Online Appendix)

This appendix contains the proofs of all the intermediate results that have been omitted from the paper, as well as extensions of several results reported in the paper. The appendix is divided into 6 sections. In Section 2 we show that for any distribution of precision choices , there are no non-monotonic equilibria in the second stage of the game. We also establish properties of the ex-ante utility function that we invoked in the proof of Theorem 1. In particular, we show that the ex-ante utility function is increasing and that its derivative is uniformly bounded from above and converges to zero. Finally, we show that there exists a precision level 2 ( ;1) such that players never …nd it optimal to choose a precision higher than . Section 3 is concerned with the results regarding the over-acquisition and under-acquisition of information. We …rst prove the technical result that we used to argue that the unique equilibrium of the game is generically ine¢ cient. We then analyze the conditions under which agents globally over-acquire and under-acquire information and explain why a complete characterization of these global e¤ects is not attainable. In Section 4 we prove intermediate results that have been invoked in the paper when analyzing strategic complementarities in information choices and the welfare implications of an increase in transparency. We also discuss what happens when we relax the assumption that T = 12 . In particular, we provide an extension to Proposition 5 in the main text to the case when T 6= 12 . We also discuss why it is di¢ cult to establish a full characterization of strategic complementarities and substitutabilities between private and public information when T 6= 12 . In Section 5 we consider the e¤ects of an increase in the precision of the prior on the probability of a successful investment and on welfare. In particular, we show that for any T the probability of a successful investment increases in when is su¢ ciently high and decreases when is su¢ ciently low. We also include …gures generated using numerical simulations that decompose the fT; g space into regions where the probability of successful investment is increasing or decreasing in . In Section 5:2; we evaluate numerically the e¤ect of an increase in on the ex-ante utility of an investor for di¤erent parameter values. These numerical results provide support to the claim that welfare is increasing in