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

Abstract

While auction theory views bids and valuations as continuous variables, real-world auctions are necessarily discrete. In this paper, we use a combination of analytical and computational methods to investigate whether incorporating discreteness substantially changes the predictions of auction theory, focusing on the case of uniformly distributed valuations so that our results bear on the majority of auction experiments. In some cases, we find that introducing discreteness changes little. For example, the first-price auction with two bidders and an even number of values has a symmetric equilibrium that closely resembles its continuous counterpart and converges to its continuous counterpart as the discretisation goes to zero. In others, however, we uncover discontinuity results. For instance, introducing an arbitrarily small amount of discreteness into the all-pay auction makes its symmetric, pure-strategy equilibrium disappear; and appears (based on computational experiments) to rob the game of pure-strategy equilibria altogether. These results raise questions about the continuity approximations on which auction theory is based and prompt a re-evaluation of the experimental literature. * For comments and helpful suggestions, we would especially like to thank

Going... going... wrong: a test of the level-k (and cognitive hierarchy) models of bidding behaviour

“…Finally, we argue that level-k can never coincide with equilibrium; with the implication that level-k bidding functions must cycle as k → ∞. To prove this, note that, by Proposition 4 of Rasooly and Gavidia-Calderon (2020), the discrete all-pay auction with uniform values does not possess a symmetric equilibrium in pure strategies. Of course, it does have a symmetric equilibrium (see our Proposition 1), so this must be in mixed strategies.…”

“…Second, symmetric equilibria are analytically more tractable. Third, based on computational experiments reported in Rasooly and Gavidia-Calderon (2020), it does not seem as though discrete auction games like those described above possess asymmetric equilibria which are substantially different from their symmetric equilibrium. Therefore, one might hope that the restriction to symmetric equilibria is 'almost' without loss of generality.…”

“…Turning to uniqueness, we now generalise three lemmas from Rasooly and Gavidia-Calderon (2020) to the case of mixed strategies. Let σ denote an arbitrary symmetric equilibrium strategy; and for any v ∈ X, let s(v) ⊆ X denote the set of bids that are submitted with positive probability at value v (i.e.…”

“…In particular, the level-2 player would not possess a best response: for any number > 0 that they might bid, there exists a feasible bid < that would yield them a higher expected pay-off.7 This is invaluable from an experimental perspective since 'brute force' approaches to equilibrium computation take several hours even for auctions with a handful of possible values and essentially intractable for auctions of the size considered in our experiment(Rasooly and Gavidia-Calderon, 2020). In contrast, our algorithm is able to compute the symmetric equilibria of our experimental games in less than a second.…”

A heuristic for estimating Nash equilibria in first-price auctions with correlated values

“…Research on first-price auctions received renewed impetus in 2019 when Google switched its display advertising market place to first-price [16,17]. This led to a surge of interest in new topics such as computational complexity [18], numerical approximation [19] or more recently, the use of neural networks to compute the auction's equilibrium [20].…”