The importance of being discrete: on the inaccuracy of continuous approximations in auction theory

While auction theory views bids and valuations as continuous variables, real-world auctions are necessarily discrete. In this talk, we use a combination of analytical and computational methods to investigate the accuracy of this assumption, focusing on the case of uniformly distributed valuations so that our results bear on the majority of auction experiments. In the first price auction with two bidders, we find that there is a unique symmetric equilibrium that closely resembles the continuous equilibrium and converges to the continuous equilibrium as the discretisation goes to zero. However, with more than two bidders, there is no symmetric equilibrium at all – in sharp contrast to the continuous case. Furthermore, there is no symmetric equilibrium in the all-pay auction (regardless of the number of bidders); and computational experiments suggest that there is no pure strategy equilibrium whatsoever. These results cast doubt on the continuity approximations on which auction theory is based and prompt a re-evaluation of the experimental auction literature.