Price Manipulability in First-Price Auctions

First-price auctions have many desirable properties, including uniquely possessing some, like credibility. However, first-price auctions are also inherently non-truthful, and non-truthfulness may result in instability and inefficiencies. Given these pros and cons, we seek to quantify the extent to which first-price auctions are susceptible to manipulation. In this work we adopt a metric that was introduced in the context of bitcoin fee design markets: the percentage change in payment that can be achieved by being strategic. We study the behavior of this metric for single-unit and $k$-unit auction environments with $n$ i.i.d. buyers, and seek conditions under which the percentage change tends to zero as $n$ grows large. To the best of our knowledge, ours is the first rigorous study of the extent to which large multi-unit first price auctions are susceptible to manipulation. We provide an almost complete picture of the conditions under which they are truthful in the large, and exhibit some surprising boundaries.