Discussion of “strategies and interactive beliefs in dynamic games” and “calibration: Respice, adspice, prospice”

This discussion consists of three loosely related comments. The first two comments are intended to help readers with understanding and interpreting the results presented by Rakesh Vohra; the second comment also will express my perhaps very subjective view on this literature. The third comment raises a challenge to the chapter presented by Pierpaolo Battigalli. To shorten my exposition, I closely follow the terminology and notation fromthe chapters I discuss. Comment 1: The negative results presented by Vohra seem very surprising. For example, when I first listened to a lecture on this topic, I left the seminar room feeling certain that I simply had missed some crucial details. Only later, after reading the paper did, I realize that I had gotten all of the details right, but the result was still as surprising as it had seemed during the seminar. Inspecting the proofs of these negative results is helpful only with a substantial time investment because they refer to quite advanced mathematical tools. Thus, it is instructive to look at some examples. I present here the simplest relevant test, resembling calibration, that can be ignorantly passed by applying a simple forecasting strategy. We let S = {0, 1} be the state space. The test provides a verdict at infinity. The verdict is PASS if the observed frequency of outcome sr = 1 coincides with the average forecast fr of this outcome. More precisely, we let: and This test can be ignorantly passed by the algorithm that forecasts in period t + 1 outcome 1 with probability one if Rt≤ 0 and that otherwise forecasts outcome 0 with probability one.