, In the following, we prove the regret bound of the PDTFP dynamics (without the vanishing regret condition) in smooth auctions

, Observe that for any reserve price vector r, |REV(r, b) ? REV(r, b)| ? m · 2 ?M where r is a reserve price vector such that r ij is the largest multiple of 2 ?M smaller than r ij for every i, j. Therefore, one can approximate the revenue up to any arbitrary precision by restricting the reserve price values in the discretization

, Note that K ? L = L. We slightly abuse notation by denoting REV j (1 S , b) as REV j (r, b), Consider the lattice L = { · 2 ?M : 0 ? ? 2 M } n×m

A. Agrawal, P. Klein, and R. Ravi, When trees collide: An approximation algorithm for the generalized steiner problem on networks, SIAM Journal on Computing, vol.24, issue.3, pp.440-456, 1995.

S. Albers, Energy-efficient algorithms, Commun. ACM, vol.53, issue.5, pp.86-96, 2010.

S. Anand, N. Garg, and A. Kumar, Resource augmentation for weighted flow-time explained by dual fitting, Proc. 23rd ACM-SIAM Symposium on Discrete Algorithms, pp.1228-1241, 2012.

M. Andrews, S. Antonakopoulos, and L. Zhang, Minimumcost network design with (dis)economies of scale, Proc. 51th Symposium on Foundations of Computer Science (FOCS), pp.585-592, 2010.

M. Andrews, A. F. Anta, L. Zhang, and W. Zhao, Routing for power minimization in the speed scaling model, IEEE/ACM Trans. Netw, vol.20, issue.1, pp.285-294, 2012.

A. Antoniadis, S. Im, R. Krishnaswamy, B. Moseley, V. Nagarajan et al., Hallucination helps: Energy efficient virtual circuit routing, Proc. 25th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp.1141-1153, 2014.

S. Arora, E. Hazan, and S. Kale, The multiplicative weights update method: a meta-algorithm and applications, Theory of Computing, vol.8, pp.121-164, 2012.

. Robert-j-aumann, Subjectivity and correlation in randomized strategies, Journal of mathematical Economics, vol.1, issue.1, pp.67-96, 1974.

Y. Azar, U. Bhaskar, L. Fleischer, and D. Panigrahi, Online mixed packing and covering, Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pp.85-100, 2013.

Y. Azar, R. Nikhil, Z. Devanur, D. Huang, and . Panigrahi, Speed scaling in the non-clairvoyant model, Proc. 27th ACM on Symposium on Parallelism in Algorithms and Architectures, pp.133-142, 2015.

Y. Azar, N. Buchbinder, S. Th-hubert-chan, . Chen, A. Ilan-reuven-cohen et al., Online algorithms for covering and packing problems with convex objectives, Proc. Symposium on Foundations of Computer Science (FOCS), 2016.

M. Balcan and N. J. Harvey, Learning submodular functions, Proc. Machine Learning and Knowledge Discovery in Databases, pp.846-849, 2012.

E. Bampis, A. V. Kononov, D. Letsios, G. Lucarelli, and M. Sviridenko, Energy efficient scheduling and routing via randomized rounding, Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp.449-460, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00946354

N. Bansal and A. Gupta, , 2017.

N. Bansal, T. Kimbrel, and K. Pruhs, Speed scaling to manage energy and temperature, J. ACM, vol.54, issue.1, 2007.

N. Bansal, N. Buchbinder, and J. Naor, Randomized competitive algorithms for generalized caching, SIAM Journal on Computing, vol.41, issue.2, pp.391-414, 2012.

N. Bansal, N. Buchbinder, and J. Naor, A primal-dual randomized algorithm for weighted paging, Journal of the ACM, vol.59, issue.4, p.19, 2012.

N. Bansal, D. Ho-leung-chan, K. Katz, and . Pruhs, Improved bounds for speed scaling in devices obeying the cube-root rule. Theory of Computing, vol.8, pp.209-229, 2012.

R. Bar-yehuda and S. Even, A linear-time approximation algorithm for the weighted vertex cover problem, Journal of Algorithms, vol.2, issue.2, pp.198-203, 1981.

M. Beckmann, C. B. Mcguire, and . Winsten, Studies in the economics of transportation, 1956.

M. Benaïm and M. Faure, Consistency of vanishingly smooth fictitious play, Mathematics of Operations Research, vol.38, issue.3, pp.437-450, 2013.

K. Bhawalkar and T. Roughgarden, Welfare guarantees for combinatorial auctions with item bidding, Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pp.700-709, 2011.

K. Bhawalkar, S. Gollapudi, and K. Munagala, Coevolutionary opinion formation games, Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pp.41-50, 2013.

K. Bhawalkar, M. Gairing, and T. Roughgarden, Weighted congestion games: the price of anarchy, universal worst-case examples, and tightness, ACM Transactions on Economics and Computation, vol.2, issue.4, p.14, 2014.

V. Bilo, A unifying tool for bounding the quality of non-cooperative solutions in weighted congestion games, International Workshop on Approximation and Online Algorithms, pp.215-228, 2012.

A. Blum and J. D. Hartline, Near-optimal online auctions, Proc. 16th Symposium on Discrete Algorithms, pp.1156-1163, 2005.

A. Blum, K. Even-dar, and . Ligett, Routing without regret: On convergence to nash equilibria of regret-minimizing algorithms in routing games, Theory of Computing, vol.6, issue.1, pp.179-199, 2010.

A. Blum, A. Gupta, Y. Mansour, and A. Sharma, Welfare and profit maximization with production costs, Foundations of Computer Science (FOCS), 2011 IEEE 52nd Annual Symposium on, pp.77-86, 2011.

S. Boyd and L. Vandenberghe, Convex Optimization, 2004.

W. George and . Brown, Iterative solution of games by fictitious play. Activity analysis of production and allocation, vol.13, pp.374-376, 1951.

S. Bubeck, B. Michael, J. R. Cohen, and . Lee, Yin Tat Lee, and Aleksander Madry. k-server via multiscale entropic regularization, Proc. 50th Symposium on Theory of Computing, 2018.

N. Buchbinder and J. Naor, Online primal-dual algorithms for covering and packing, Mathematics of Operations Research, vol.34, issue.2, pp.270-286, 2009.

N. Buchbinder and J. Naor, The design of competitive online algorithms via a primal-dual approach, Foundations and Trends in Theoretical Computer Science, vol.3, issue.2-3, pp.93-263, 2009.

N. Buchbinder and J. Naor, Online primal-dual algorithms for covering and packing, Mathematics of Operations Research, vol.34, issue.2, pp.270-286, 2009.

N. Buchbinder, K. Jain, and J. Naor, Online primal-dual algorithms for maximizing ad-auctions revenue, European Symposium on Algorithms, pp.253-264, 2007.

N. Buchbinder, J. S. Naor, R. Ravi, and M. Singh, Approximation algorithms for online weighted rank function maximization under matroid constraints, International Colloquium on Automata, Languages, and Programming, pp.145-156, 2012.

N. Buchbinder, S. Chen, and J. Naor, Competitive analysis via regularization, Proc. 25th Symposium on Discrete Algorithms, pp.436-444, 2014.

N. Buchbinder, M. Feldman, and R. Schwartz, Online submodular maximization with preemption, Proc. 26th Symposium on Discrete Algorithms, pp.1202-1216, 2015.

N. Buchbinder, S. Chen, J. Naor, and O. Shamir, Unified algorithms for online learning and competitive analysis, Math. Oper. Res, vol.41, issue.2, pp.612-625, 2016.

I. Caragiannis, Better bounds for online load balancing on unrelated machines, Proc. 19th ACM-SIAM Symposium on Discrete Algorithms, pp.972-981, 2008.

I. Caragiannis, C. Kaklamanis, P. Kanellopoulos, M. Kyropoulou, B. Lucier et al., Bounding the inefficiency of outcomes in generalized second price auctions, Journal of Economic Theory, vol.156, pp.343-388, 2015.

N. Cesa, -. Bianchi, and G. Lugosi, Prediction, Learning, and Games, 2006.

N. Cesa-bianchi, C. Gentile, and Y. Mansour, Regret minimization for reserve prices in second-price auctions, IEEE Transactions on Information Theory, vol.61, issue.1, pp.549-564, 2015.

C. Chekuri and S. Khanna, On multidimensional packing problems, SIAM journal on computing, vol.33, issue.4, pp.837-851, 2004.

J. Cohen, C. Dürr, and N. Thang, Smooth inequalities and equilibrium inefficiency in scheduling games, Internet and Network Economics, pp.350-363, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00820139

R. Cominetti, R. José, N. E. Correa, and . Stier-moses, The impact of oligopolistic competition in networks, Operations Research, vol.57, issue.6, pp.1421-1437, 2009.

M. Conforti and G. Cornuéjols, Submodular set functions, matroids and the greedy algorithm: tight worst-case bounds and some generalizations of the rado-edmonds theorem, Discrete applied mathematics, vol.7, issue.3, pp.251-274, 1984.

A. S. José-r-correa, N. E. Schulz, and . Stier-moses, A geometric approach to the price of anarchy in nonatomic congestion games, Games and Economic Behavior, vol.64, issue.2, pp.457-469, 2008.

C. Daskalakis and V. Syrgkanis, Learning in auctions: Regret is hard, envy is easy, 57th Annual Symposium on Foundations of Computer Science (FOCS), pp.219-228, 2016.

R. Nikhil, Z. Devanur, and . Huang, Primal dual gives almost optimal energy efficient online algorithms, Proc. 25th ACM-SIAM Symposium on Discrete Algorithms, 2014.

R. Nikhil, K. Devanur, and . Jain, Online matching with concave returns, Proc. 44th ACM Symposium on Theory of Computing, pp.137-144, 2012.

M. Dudik, N. Haghtalab, H. Luo, R. E. Schapire, V. Syrgkanis et al., Oracle-efficient online learning and auction design, Proc. 58th Symposium on Foundations of Computer Science (FOCS), pp.528-539, 2017.

B. Edelman, M. Ostrovsky, and M. Schwarz, Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords. The American economic review, vol.97, pp.242-259, 2007.

M. Feldman, H. Fu, N. Gravin, and B. Lucier, Simultaneous auctions are (almost) efficient, Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pp.201-210, 2013.

M. Feldman, N. Immorlica, B. Lucier, T. Roughgarden, and V. Syrgkanis, The price of anarchy in large games, Proc. 48th Symposium on Theory of Computing (STOC), pp.963-976, 2016.

M. Feldman, J. Naor, and R. Schwartz, A unified continuous greedy algorithm for submodular maximization, Symposium on Foundations of Computer Science (FOCS), pp.570-579, 2011.

M. Feldman, O. Svensson, and R. Zenklusen, Online contention resolution schemes, Proc. 27th Symposium on Discrete Algorithms, pp.1014-1033, 2016.

D. Fotakis, A primal-dual algorithm for online non-uniform facility location, Journal of Discrete Algorithms, vol.5, issue.1, pp.141-148, 2007.

D. Fotakis, On the competitive ratio for online facility location, Algorithmica, vol.50, issue.1, pp.1-57, 2008.

Y. Freund and R. E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Journal of computer and system sciences, vol.55, issue.1, pp.119-139, 1997.

D. Fudenberg and . David-k-levine, Consistency and cautious fictitious play, Journal of Economic Dynamics and Control, vol.19, issue.5-7, pp.1065-1089, 1995.

D. Fudenberg and . David-k-levine, The theory of learning in games, 1998.

D. Fudenberg and J. Tirole, Game theory, 1991.

X. Michel, . Goemans, and . David-p-williamson, A general approximation technique for constrained forest problems, SIAM Journal on Computing, vol.24, issue.2, pp.296-317, 1995.

X. Michel, . Goemans, J. A. Nicholas, S. Harvey, V. Iwata et al., Approximating submodular functions everywhere, Proc. 20th Symposium on Discrete algorithms, pp.535-544, 2009.

G. Greiner, T. Nonner, and A. Souza, The bell is ringing in speedscaled multiprocessor scheduling. Theory of Computing Systems, vol.54, pp.24-44, 2014.

A. Gupta, R. Krishnaswamy, and K. Pruhs, Online primaldual for non-linear optimization with applications to speed scaling, Proc. 10th Workshop on Approximation and Online Algorithms, pp.173-186, 2012.

J. Hannan, Approximation to bayes risk in repeated play, Contributions to the Theory of Games, vol.3, pp.97-139, 1957.

T. Harks, Stackelberg strategies and collusion in network games with splittable flow. Theory of Computing Systems, vol.48, pp.781-802, 2011.

H. Hassani, M. Soltanolkotabi, and A. Karbasi, Gradient methods for submodular maximization, Advances in Neural Information Processing Systems, pp.5841-5851, 2017.

A. Hassidim, H. Kaplan, Y. Mansour, and N. Nisan, Nonprice equilibria in markets of discrete goods, Proc. 12th ACM Conference on Electronic Commerce, pp.295-296, 2011.

E. Hazan and S. Kale, Extracting certainty from uncertainty: Regret bounded by variation in costs, Machine learning, vol.80, issue.2-3, pp.165-188, 2010.

E. Hazan and S. Kale, Online submodular minimization, Journal of Machine Learning Research, vol.13, pp.2903-2922, 2012.

E. Hazan and T. Koren, The computational power of optimization in online learning, Proc. 48th Symposium on Theory of Computing, pp.128-141, 2016.

E. Hazan, Introduction to online convex optimization, Foundations and Trends R in Optimization, vol.2, issue.3-4, pp.157-325, 2016.

. Aj-hoffman, On simple combinatorial optimization problems, Discrete Mathematics, vol.106, pp.285-289, 1992.

Z. Huang and A. Kim, Welfare maximization with production costs: a primal dual approach, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.59-72, 2015.

S. Im, J. Kulkarni, K. Munagala, and K. Pruhs, Selfishmigrate: A scalable algorithm for non-clairvoyantly scheduling heterogeneous processors, Proc. 55th IEEE Symposium on Foundations of Computer Science, 2014.

S. Im, N. Kell, J. Kulkarni, and D. Panigrahi, Tight bounds for online vector scheduling, Foundations of Computer Science (FOCS), 2015 IEEE 56th Annual Symposium on, pp.525-544, 2015.

K. Rishabh, S. Iyer, J. A. Jegelka, and . Bilmes, Curvature and optimal algorithms for learning and minimizing submodular functions, Advances in Neural Information Processing Systems, pp.2742-2750, 2013.

A. Kalai and S. Vempala, Efficient algorithms for online decision problems, Journal of Computer and System Sciences, vol.71, issue.3, pp.291-307, 2005.

R. Kleinberg and T. Leighton, The value of knowing a demand curve: Bounds on regret for online posted-price auctions, Proc. 44th Symposium on Foundations of Computer Science, pp.594-605, 2003.

P. Kling and P. Pietrzyk, Profitable scheduling on multiple speedscalable processors, ACM Transactions on Parallel Computing, vol.2, issue.3, p.19, 2015.

E. Koutsoupias and C. Papadimitriou, Worst-case equilibria, Computer science review, vol.3, issue.2, pp.65-69, 2009.

A. Krause and D. Golovin, Submodular function maximization, Tractability: Practical Approaches to Hard Problems, pp.71-104, 2014.

V. Krishna, Auction theory, 2009.

R. Krishnaswamy, V. Nagarajan, K. Pruhs, and C. Stein, Cluster before you hallucinate: approximating node-capacitated network design and energy efficient routing, Proc. 46th Symposium on Theory of computing, pp.734-743, 2014.

J. Kulkarni and V. Mirrokni, Robust price of anarchy bounds via lp and fenchel duality, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.1030-1049, 2015.

S. Lahaie, M. David, A. Pennock, . Saberi, V. Rakesh et al., Sponsored search auctions. Algorithmic game theory, pp.699-716, 2007.

R. Paes-leme, V. Syrgkanis, and É. Tardos, Sequential auctions and externalities, Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pp.869-886, 2012.

N. Littlestone and M. K. Warmuth, The weighted majority algorithm. Information and computation, vol.108, pp.212-261, 1994.

G. Lucarelli, K. Nguyen, and . Thang, Online non-preemptive scheduling in a resource augmentation model based on duality, European Symposium on Algorithms, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01334219

P. Maillé, E. Markakis, M. Naldi, D. George, B. Stamoulis et al., Sponsored search auctions: an overview of research with emphasis on game theoretic aspects, Electronic Commerce Research, vol.12, issue.3, pp.265-300, 2012.

K. Makarychev and M. Sviridenko, Solving optimization problems with diseconomies of scale via decoupling, Foundations of Computer Science (FOCS), pp.571-580, 2014.

A. Mehta, A. Saberi, U. Vazirani, and V. Vazirani, Adwords and generalized online matching, Journal of the ACM (JACM), vol.54, issue.5, p.22, 2007.

I. Menache and M. Singh, Online caching with convex costs, Proceedings of the 27th ACM symposium on Parallelism in Algorithms and Architectures, pp.46-54, 2015.

A. Meyerson, Online facility location, Proceedings. 42nd IEEE Symposium on Foundations of Computer Science, pp.426-431, 2001.

H. Moulin and J. Vial, Strategically zero-sum games: the class of games whose completely mixed equilibria cannot be improved upon, International Journal of Game Theory, vol.7, issue.3-4, pp.201-221, 1978.

. Roger-b-myerson, Optimal auction design. Mathematics of Operations Research, vol.6, issue.1, pp.58-73, 1981.

U. Nadav and T. Roughgarden, The limits of smoothness: A primal-dual framework for price of anarchy bounds, International Workshop on Internet and Network Economics, pp.319-326, 2010.

V. Nagarajan and X. Shen, Online covering with sum of qnorm objectives, Proc. 44th Colloquium on Automata, Languages and Programming (ICALP), 2017.

F. John and . Nash, Equilibrium points in n-person games, Proc. Nat. Acad. Sci. USA, vol.36, pp.48-49, 1950.

. Kim-thang-nguyen, Lagrangian duality in online scheduling with resource augmentation and speed scaling, Proc. 21st European Symposium on Algorithms, pp.755-766, 2013.

H. Christos, K. Papadimitriou, and . Steiglitz, Combinatorial Optimization: Algorithms and Complexity. Courier Corporation, 1998.

H. Peters, Game theory: A Multi-leveled approach, 2015.

W. Robert and . Rosenthal, A class of games possessing pure-strategy nash equilibria, International Journal of Game Theory, vol.2, issue.1, pp.65-67, 1973.

T. Roughgarden, Frontiers in mechanism design, Lecture, vol.17, 2014.

T. Roughgarden, Intrinsic robustness of the price of anarchy, Journal of the ACM, vol.62, issue.5, p.32, 2015.

T. Roughgarden, The price of anarchy in games of incomplete information, ACM Transactions on Economics and Computation, vol.3, issue.1, p.6, 2015.

T. Roughgarden and F. Schoppmann, Local smoothness and the price of anarchy in splittable congestion games, Journal of Economic Theory, vol.156, pp.317-342, 2015.

T. Roughgarden and É. Tardos, How bad is selfish routing, Journal of the ACM (JACM), vol.49, issue.2, pp.236-259, 2002.

T. Roughgarden and É. Tardos, Bounding the inefficiency of equilibria in nonatomic congestion games, Games and Economic Behavior, vol.47, issue.2, pp.389-403, 2004.

T. Roughgarden and J. R. Wang, Minimizing regret with multiple reserves, Proc. 2016 ACM Conference on Economics and Computation, pp.601-616, 2016.

T. Roughgarden, V. Syrgkanis, and E. Tardos, The price of anarchy in auctions, Journal of Artificial Intelligence Research, vol.59, pp.59-101, 2017.

S. Shalev, -. Shwartz, and Y. Singer, A primal-dual perspective of online learning algorithms, Machine Learning, vol.69, pp.115-142, 2007.

S. Shalev-shwartz, Online learning and online convex optimization. Foundations and Trends R in Machine Learning, vol.4, pp.107-194, 2012.

M. Sviridenko, J. Vondrák, and J. Ward, Optimal approximation for submodular and supermodular optimization with bounded curvature, Mathematics of Operations Research, 2017.

V. Syrgkanis and E. Tardos, Bayesian sequential auctions, Proceedings of the 13th ACM Conference on Electronic Commerce, pp.929-944, 2012.

V. Syrgkanis and E. Tardos, Composable and efficient mechanisms, Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pp.211-220, 2013.

V. Syrgkanis, A. Krishnamurthy, and R. Schapire, Efficient algorithms for adversarial contextual learning, International Conference on Machine Learning, pp.2159-2168, 2016.

. Hal-r-varian, Position auctions, international Journal of industrial Organization, vol.25, issue.6, pp.1163-1178, 2007.

V. Rakesh and . Vohra, Mechanism design: a linear programming approach, vol.47, 2011.

J. Vondrák, Polyhedral techniques in combinatorial optimization, Lecture notes, 2010.

J. Vondrák, Submodularity and curvature: The optimal algorithm, 2010.

J. Wardrop, Some theoretical aspects of road traffic research, vol.1, pp.325-362, 1952.

. David-p-williamson, The primal-dual method for approximation algorithms, Mathematical Programming, vol.91, issue.3, pp.447-478, 2002.

P. David, D. B. Williamson, and . Shmoys, The design of approximation algorithms, 2011.