On the Uniqueness of Bayesian Coarse Correlated Equilibria in Standard First-Price and All-Pay Auctions (2401.01185v4)
Abstract: We study the Bayesian coarse correlated equilibrium (BCCE) of continuous and discretised first-price and all-pay auctions under the standard symmetric independent private-values model. Our study is motivated by the question of how the canonical Bayes-Nash equilibrium (BNE) of the auction relates to the outcomes learned by buyers utilising no-regret algorithms. Numerical experiments show that in two buyer first-price auctions the Wasserstein-$2$ distance of buyers' marginal bid distributions decline as $O(1/n)$ in the discretisation size in instances where the prior distribution is concave, whereas all-pay auctions exhibit similar behaviour without prior dependence. To explain this convergence to a near-equilibrium, we study uniqueness of the BCCE of the continuous auction. Our uniqueness results translate to provable convergence of deterministic self-play to a near equilibrium outcome in these auctions. In the all-pay auction, we show that independent of the prior distribution there is a unique BCCE with symmetric, differentiable, and increasing bidding strategies, which is equivalent to the unique strict BNE. In the first-price auction, we need stronger conditions. Either the prior is strictly concave or the learning algorithm has to be restricted to strictly increasing strategies. Without such strong assumptions, no-regret algorithms can end up in low-price pooling strategies. This is important because it proves that in repeated first-price auctions such as in display ad actions, algorithmic collusion cannot be ruled out without further assumptions even if all bidders rely on no-regret algorithms.
- Approximating the cut-norm via Grothendieck’s inequality. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pages 72–80, 2004.
- E.J. Anderson and P. Nash. Linear Programming in Infinite-dimensional Spaces: Theory and Applications. A Wiley-Interscience publication. Wiley, 1987.
- Bayes correlated equilibrium and the comparison of information structures in games. Theoretical Economics, 11(2):487–522, 2016.
- Learning equilibria in symmetric auction games using artificial neural networks. Nature Machine Intelligence, 3:687–695, 2021.
- Computing Bayes-Nash equilibrium in auction games via gradient dynamics. Operations Research, to appear, 2023.
- Simultaneous Bayesian auctions and computational complexity. In Proceedings of the Fifteenth ACM Conference on Economics and Computation, EC ’14, page 895–910, New York, NY, USA, 2014. Association for Computing Machinery.
- Auctions with unique equilibria. In Proceedings of the Fourteenth ACM Conference on Electronic Commerce, EC ’13, page 181–196, New York, NY, USA, 2013. Association for Computing Machinery.
- Xi Chen and Xiaotie Deng. Settling the complexity of two-player Nash equilibrium. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’06, page 261–272, USA, 2006. IEEE Computer Society.
- Xi Chen and Binghui Peng. Complexity of equilibria in first-price auctions under general tie-breaking rules. arXiv preprint arXiv:2303.16388, 2023.
- Stephen A. Clark. An infinite-dimensional LP duality theorem. Mathematics of Operations Research, 28(2):233–245, 2003.
- New complexity results about Nash equilibria. Games and Economic Behavior, 63(2):621–641, 2008. Second World Congress of the Game Theory Society.
- The complexity of computing a Nash equilibrium. SIAM Journal on Computing, 39(1):195–259, 2009.
- Nash convergence of mean-based learning algorithms in first price auctions. In Proceedings of the ACM Web Conference 2022, pages 141–150, 2022.
- Correlated and coarse equilibria of single-item auctions. In Yang Cai and Adrian Vetta, editors, Web and Internet Economics, pages 131–144, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg.
- Convergence analysis of no-regret bidding algorithms in repeated auctions. volume 35, pages 5399–5406, 2021.
- On the complexity of equilibrium computation in first-price auctions. In Proceedings of the 22nd ACM Conference on Economics and Computation, EC ’21, page 454–476, New York, NY, USA, 2021. Association for Computing Machinery.
- Françoise Forges. Correlated equilibrium in games with incomplete information. Revue économique, 74(4):529–539, 2023.
- The theory of learning in games, volume 2. MIT press, 1998.
- Kaito Fujii. Bayes correlated equilibria and no-regret dynamics, 2023.
- Duality overlap in infinite linear programs. Journal of Mathematical Analysis and Applications, 41(2):333–335, 1973.
- John C. Harsanyi and Reinhard Selten, editors. A general theory of equilibrium selection in games. MIT Press, 1st edition, 1988.
- Uncoupled dynamics do not lead to Nash equilibrium. American Economic Review, 93(5):1830–1836, 2003.
- No-regret learning in Bayesian games. In Proceedings of the 28th International Conference on Neural Information Processing Systems - Volume 2, NIPS’15, page 3061–3069, Cambridge, MA, USA, 2015. MIT Press.
- Hans G. Kellerer. Measure theoretic versions of linear programming. Mathematische Zeitschrift, 198(3):367–400, 1988.
- Vijay Krishna, editor. Auction Theory. Academic Press, San Diego, 2nd edition, 2010.
- Bidder collusion at first-price auctions. Review of Economic Design, 15(3):177–211, 2011.
- The Slater conundrum: duality and pricing in infinite-dimensional optimization. SIAM Journal on Optimization, 26(1):111–138, 2016.
- Learning in games with continuous action sets and unknown payoff functions. Math. Program., 173(1–2):465–507, jan 2019.
- The importance of being discrete: on the inaccuracy of continuous approximations in auction theory, 2020.
- Alexander Shapiro. On duality theory of conic linear problems. Nonconvex Optimization and its Applications, 57:135–155, 2001.
- Lin Xiao. Dual averaging methods for regularized stochastic learning and online optimization. Journal of Machine Learning Research, 11(88):2543–2596, 2010.
- H. Peyton Young. Strategic learning and its limits. Oxford University Press, 2004.