“…9 Aradillas-López, Gandhi, and Quint (2013) make this assumption to identify bounds on certain counterfactuals in an English auction setting, and Aradillas-López, Gandhi, and Quint (2016) offer a partial test of this assumption using bid data from English auctions. This same assumption has also been made in several empirical articles on first-price auctions: Vuong (2000, 2009), Haile, Hong, and Shum (2003), Gillen (2010), and Aryal, Grundl, Kim, and Zhu (2018), for example, make this assumption to improve the efficiency of their estimator, to identify a risk aversion parameter, to distinguish private from common values, to identify a "level-k" model of strategic sophistication, and to estimate a model with ambiguity aversion, respectively, and Liu and Luo (2017) present a test of the assumption on first-price auction data. 10 Although we vary (below) what we assume about the observability of N, we will maintain the assumption it is exogenous.…”