Bayes–Nash equilibria of the generalized second-price auction

Bayes-Nash Equilibria in Generalized Second Price Auctions with Allocative Externalities

“…In section 5 we give examples where the monotonicity of the bidding function breaks down and efficient equilibria fail to exist. However, we believe that cases of non-existent results in our model have quite different features and are practically marginal compared to the fixed-CTR model in Gomes and Sweeney (2014). First, as illustrated in section 5, they happen only for distributions with extreme parameters, e.g., Beta distributions with parameters a = 2 while b > 70 approximately 9 or uniform distributions with partial support on [0, b] and b < 0.15 approximately, which all require strong assumptions on the primitives.…”

“…They prove the existence of symmetric Nash-equilibria and conduct an empirical analysis on bidders' willingness-to-pay. Still under the assumption of fixed CTRs but in an incomplete information setting, Gomes and Sweeney (2014) derive the symmetric efficient Bayes-Nash equilibrium bidding strategy and the optimal reserve price when equilibrium exists. Their most striking result is that in an incomplete information setting a symmetric model with a wide range of parameters may have no symmetric equilibria for any distributions.…”

“…9 See Appendix B for the definition of the Beta distribution. 10 See Proposition 2 in Gomes and Sweeney (2014). effects are indeed economically and statistically significant". In summary, cases of non-existence results are likely to be of marginal importance in practice once allocative externalities are appreciated by bidders.…”

“…Figure 1: Non-monotone candidate equilibrium bidding strategy in the fixed-CTR model of Gomes and Sweeney (2014) substitutes since the meaning of keyword is precision. 4 A keyword typically implies a specific interest and a user typically has a unique demand for a certain product.…”

An experimental study of sponsored-search auctions

“…The frequencies of adaptive optimality begin from 0.68 (periods 2-5) to 0.74 (periods 11-15) in the DI-11 treatment and from 0.80 (periods 2-5) to 0.87 (periods [11][12][13][14][15] in the DI-20 treatment. When we examine this more finely according to the value ranking, the highest-value bidders exhibit the lowest level of adaptive optimality and the lowest-value bidders show the highest level of adaptive optimality in each period.…”