PPAD-Membership for Problems with Exact Rational Solutions: A General Approach via Convex Optimization
Authors: 
Aris Filos-Ratsikas, 
Kristoffer Arnsfelt Hansen, 
Kasper Høgh, 
Alexandros HollenderAuthors Info & Claims
Aris Filos-Ratsikas, Kristoffer Arnsfelt Hansen, Kasper Høgh, Alexandros HollenderAuthors Info & ClaimsPages 1204 - 1215
Abstract
We introduce a general technique for proving membership of search problems with exact rational solutions in PPAD, one of the most well-known classes containing total search problems with polynomial-time verifiable solutions. In particular, we construct a "pseudogate", coined the linear-OPT-gate, which can be used as a "plug-and-play" component in a piecewise-linear (PL) arithmetic circuit, as an integral component of the "Linear-FIXP" equivalent definition of the class. The linear-OPT-gate can solve several convex optimization programs, including quadratic programs, which often appear organically in the simplest existence proofs for these problems. This effectively transforms existence proofs to PPAD-membership proofs, and consequently establishes the existence of solutions described by rational numbers. Using the linear-OPT-gate, we are able to significantly simplify and generalize almost all known PPAD-membership proofs for finding exact solutions in the application domains of game theory, competitive markets, auto-bidding auctions, and fair division, as well as to obtain new PPAD-membership results for problems in these domains.
References
[1]
Kenneth J. Arrow and Gerard Debreu. 1954. Existence of an equilibrium for a competitive economy. Econometrica: Journal of the Econometric Society, 265–290. https://doi.org/10.2307/1907353
[2]
Robert J Aumann. 1974. Subjectivity and correlation in randomized strategies. Journal of mathematical Economics, 1, 1 (1974), 67–96. https://doi.org/10.1016/0304-4068(74)90037-8
[3]
Moshe Babaioff, Robert Kleinberg, and Christos H Papadimitriou. 2009. Congestion games with malicious players. Games and Economic Behavior, 1, 67 (2009), 22–35. https://doi.org/10.1016/j.geb.2009.04.017
[4]
Santiago Balseiro, Yuan Deng, Jieming Mao, Vahab Mirrokni, and Song Zuo. 2021. Robust auction design in the auto-bidding world. Advances in Neural Information Processing Systems, 34 (2021), 17777–17788.
[5]
Santiago Balseiro, Anthony Kim, Mohammad Mahdian, and Vahab Mirrokni. 2021. Budget-Management Strategies in Repeated Auctions. Operations research, 69, 3 (2021), 859–876. https://doi.org/10.1287/opre.2020.2073
[6]
Santiago R. Balseiro, Yuan Deng, Jieming Mao, Vahab S. Mirrokni, and Song Zuo. 2021. The landscape of auto-bidding auctions: Value versus utility maximization. In Proceedings of the 22nd ACM Conference on Economics and Computation (EC). ACM, 132–133. https://doi.org/10.1145/3465456.3467607
[7]
Santiago R. Balseiro and Yonatan Gur. 2019. Learning in repeated auctions with budgets: Regret minimization and equilibrium. Management Science, 65, 9 (2019), 3952–3968. https://doi.org/10.1287/mnsc.2018.3174
[8]
Xiaohui Bei, Ning Chen, Xia Hua, Biaoshuai Tao, and Endong Yang. 2012. Optimal proportional cake cutting with connected pieces. In Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI). AAAI Press, 1263–1269. https://doi.org/10.1609/aaai.v26i1.8243
[9]
Christian Borgs, Jennifer Chayes, Nicole Immorlica, Kamal Jain, Omid Etesami, and Mohammad Mahdian. 2007. Dynamics of bid optimization in online advertisement auctions. In Proceedings of the 16th international conference on World Wide Web (WWW). ACM, 531–540. https://doi.org/10.1145/1242572.1242644
[10]
Steven J. Brams and Alan D. Taylor. 1996. Fair Division: From cake-cutting to dispute resolution. Cambridge University Press.
[11]
L. E. J. Brouwer. 1911. Über Abbildung von Mannigfaltigkeiten. Math. Ann., 71 (1911), 97–115. http://eudml.org/doc/158520
[12]
Ioannis Caragiannis, Kristoffer Arnsfelt Hansen, and Nidhi Rathi. 2023. On the complexity of Pareto-optimal and envy-free lotteries. CoRR, abs/2307.12605 (2023), https://doi.org/10.48550/arXiv.2307.12605
[13]
Bhaskar Ray Chaudhury, Jugal Garg, Peter McGlaughlin, and Ruta Mehta. 2022. A Complementary Pivot Algorithm for Competitive Allocation of a Mixed Manna. Mathematics of Operations Research, 48, 3 (2022), 1630–1656. https://doi.org/10.1287/moor.2022.1315
[14]
Xi Chen, Xiaotie Deng, and Shang-Hua Teng. 2009. Settling the complexity of computing two-player Nash equilibria. Journal of the ACM (JACM), 56, 3 (2009), 1–57. https://doi.org/10.1145/1516512.1516516
[15]
Xi Chen, Christian Kroer, and Rachitesh Kumar. 2021. The Complexity of Pacing for Second-Price Auctions. In Proceedings of the 22nd ACM Conference on Economics and Computation (EC). ACM, 318. https://doi.org/10.1145/3465456.3467578
[16]
Xi Chen, Dimitris Paparas, and Mihalis Yannakakis. 2017. The Complexity of Non-Monotone Markets. J. ACM, 64, 3 (2017), 20:1–20:56. https://doi.org/10.1145/3064810
[17]
Vincent Conitzer, Christian Kroer, Debmalya Panigrahi, Okke Schrijvers, Nicolas E. Stier-Moses, Eric Sodomka, and Christopher A. Wilkens. 2022. Pacing equilibrium in first price auction markets. Management Science, 68, 12 (2022), 8515–8535. https://doi.org/10.1287/mnsc.2022.4310
[18]
Vincent Conitzer, Christian Kroer, Eric Sodomka, and Nicolas E. Stier-Moses. 2022. Multiplicative pacing equilibria in auction markets. Operations Research, 70, 2 (2022), 963–989. https://doi.org/10.1287/opre.2021.2167
[19]
Richard W. Cottle and George B. Dantzig. 1968. Complementary pivot theory of mathematical programming. Linear Algebra Appl., 1, 1 (1968), 103–125. https://doi.org/10.1016/0024-3795(68)90052-9
[20]
Richard W. Cottle, Jong-Shi Pang, and Richard E. Stone. 2009. The linear complementarity problem. SIAM. https://doi.org/10.1137/1.9780898719000
[21]
Partha Sarathi Dasgupta and Eric S. Maskin. 2015. Debreu’s social equilibrium existence theorem. Proceedings of the National Academy of Sciences, 112, 52 (2015), 15769–15770. https://doi.org/10.1073/pnas.1522640113
[22]
Constantinos Daskalakis, Alex Fabrikant, and Christos H. Papadimitriou. 2006. The Game World Is Flat: The Complexity of Nash Equilibria in Succinct Games. In Proceedings of the 33rd International Colloquium on Automata, Languages, and Programming (ICALP). Springer, 513–524. https://doi.org/10.1007/11786986_45
[23]
Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. 2009. The complexity of computing a Nash equilibrium. SIAM J. Comput., 39, 1 (2009), 195–259. https://doi.org/10.1137/070699652
[24]
Gerard Debreu. 1952. A social equilibrium existence theorem. Proceedings of the National Academy of Sciences, 38, 10 (1952), 886–893. https://doi.org/10.1073/pnas.38.10.886
[25]
Xiaotie Deng, Qi Qi, and Amin Saberi. 2012. Algorithmic solutions for envy-free cake cutting. Operations Research, 60, 6 (2012), 1461–1476. https://doi.org/10.1287/opre.1120.1116
[26]
B Curtis Eaves. 1976. A finite algorithm for the linear exchange model. Journal of Mathematical Economics, 3, 2 (1976), 197–203. https://doi.org/10.1016/0304-4068(76)90028-8
[27]
Kousha Etessami and Mihalis Yannakakis. 2010. On the complexity of Nash equilibria and other fixed points. SIAM J. Comput., 39, 6 (2010), 2531–2597. https://doi.org/10.1137/080720826
[28]
Ky Fan. 1952. Fixed-point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences, 38, 2 (1952), 121–126. https://doi.org/10.1073/pnas.38.2.121
[29]
Aris Filos-Ratsikas, Kristoffer A. Hansen, Kasper Høgh, and Alexandros Hollender. 2023. FIXP-Membership via Convex Optimization: Games, Cakes, and Markets. SIAM J. Comput., FOCS21–30–FOCS21–84. https://doi.org/10.1137/22M1472656
[30]
George Gamow and Marvin Stern. 1958. Puzzle-math. Viking Press.
[31]
Jugal Garg. 2017. Market equilibrium under piecewise Leontief concave utilities. Theoretical Computer Science, 703 (2017), 55–65. https://doi.org/10.1016/j.tcs.2017.09.001
[32]
Jugal Garg, Albert Xin Jiang, and Ruta Mehta. 2011. Bilinear games: Polynomial time algorithms for rank based subclasses. In Proceedings of the 7th International Workshop on Internet and Network Economics (WINE). Springer, 399–407. https://doi.org/10.1007/978-3-642-25510-6_35
[33]
Jugal Garg, Ruta Mehta, Milind Sohoni, and Vijay V. Vazirani. 2015. A Complementary Pivot Algorithm for Market Equilibrium under Separable, Piecewise-Linear Concave Utilities. SIAM J. Comput., 44, 6 (2015), 1820–1847. https://doi.org/10.1137/140971002
[34]
Jugal Garg, Ruta Mehta, and Vijay V. Vazirani. 2018. Substitution with satiation: A new class of utility functions and a complementary pivot algorithm. Mathematics of Operations Research, 43, 3 (2018), 996–1024. https://doi.org/10.1287/moor.2017.0892
[35]
Jugal Garg and Vijay V. Vazirani. 2014. On Computability of Equilibria in Markets with Production. In Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM, 1329–1340. https://doi.org/10.1137/1.9781611973402.98
[36]
Irving L. Glicksberg. 1952. A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points. Proc. Amer. Math. Soc., 3, 1 (1952), 170–174. https://doi.org/10.2307/2032478
[37]
Michel X. Goemans. 2015. Smallest compact formulation for the permutahedron. Mathematical Programming, 153, 1 (2015), 5–11. https://doi.org/10.1007/s10107-014-0757-1
[38]
Paul Goldberg, Alexandros Hollender, and Warut Suksompong. 2020. Contiguous cake cutting: Hardness results and approximation algorithms. Journal of Artificial Intelligence Research, 69 (2020), 109–141. https://doi.org/10.1613/jair.1.12222
[39]
Kristoffer Arnsfelt Hansen and Troels Bjerre Lund. 2018. Computational complexity of proper equilibrium. In Proceedings of the 19th ACM Conference on Economics and Computation (EC). ACM, 113–130. https://doi.org/10.1145/3219166.3219199
[40]
Joseph T. Howson. 1972. Equilibria of Polymatrix Games. Management Science, 18, 5-part-1 (1972), 312–318. https://doi.org/10.1287/mnsc.18.5.312
[41]
Kamal Jain. 2007. A polynomial time algorithm for computing an Arrow-Debreu market equilibrium for linear utilities. SIAM J. Comput., 37, 1 (2007), 303–318. https://doi.org/10.1137/S0097539705447384
[42]
Elena Janovskaja. 1968. Equilibrium points in polymatrix games. Lithuanian Mathematical Journal, 8, 2 (1968), 381–384. https://doi.org/10.15388/LMJ.1968.20224
[43]
David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. 1988. How easy is local search? Journal of computer and system sciences, 37, 1 (1988), 79–100. https://doi.org/10.1016/0022-0000(88)90046-3
[44]
Shizuo Kakutani. 1941. A generalization of Brouwer’s fixed point theorem. Duke Mathematical Journal, 8, 3 (1941), 457–459. https://doi.org/10.1215/S0012-7094-41-00838-4
[45]
Shiva Kintali, Laura J. Poplawski, Rajmohan Rajaraman, Ravi Sundaram, and Shang-Hua Teng. 2013. Reducibility among Fractional Stability Problems. SIAM J. Comput., 42, 6 (2013), 2063–2113. https://doi.org/10.1137/120874655
[46]
Max Klimm and Philipp Warode. 2020. Complexity and parametric computation of equilibria in atomic splittable congestion games via weighted block Laplacians. In Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms (SODA). ACM, 2728–2747. https://doi.org/10.5555/3381089.3381255
[47]
Daphne Koller, Nimrod Megiddo, and Bernhard Von Stengel. 1996. Efficient computation of equilibria for extensive two-person games. Games and Economic Behavior, 14, 2 (1996), 247–259. https://doi.org/10.1006/game.1996.0051
[48]
Carlton E. Lemke. 1965. Bimatrix equilibrium points and mathematical programming. Management science, 11, 7 (1965), 681–689. https://doi.org/10.1287/mnsc.11.7.681
[49]
Carlton E. Lemke and Joseph T. Howson, Jr. 1964. Equilibrium points of bimatrix games. Journal of the Society for industrial and Applied Mathematics, 12, 2 (1964), 413–423. https://doi.org/10.1137/0112033
[50]
Juncheng Li and Pingzhong Tang. 2023. Auto-bidding Equilibrium in ROI-Constrained Online Advertising Markets. arXiv preprint arXiv:2210.06107.
[51]
Nimrod Megiddo and Christos H. Papadimitriou. 1991. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81, 2 (1991), 317–324. https://doi.org/10.1016/0304-3975(91)90200-L
[52]
Frédéric Meunier and Thomas Pradeau. 2019. Computing solutions of the multiclass network equilibrium problem with affine cost functions. Annals of Operations Research, 274 (2019), 447–469. https://doi.org/10.1007/s10479-018-2817-z
[53]
Igal Milchtaich. 2000. Generic uniqueness of equilibrium in large crowding games. Mathematics of Operations Research, 25, 3 (2000), 349–364. https://doi.org/10.1287/moor.25.3.349.12220
[54]
Roger B. Myerson. 1978. Refinements of the Nash Equilibrium Concept. International Journal of Game Theory, 7, 2 (1978), 73–80. https://doi.org/10.1007/BF01753236
[55]
John F. Nash. 1950. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences, 36, 1 (1950), 48–49. https://doi.org/10.1073/pnas.36.1.48
[56]
Christos Papadimitriou and Binghui Peng. 2021. Public goods games in directed networks. In Proceedings of the 22nd ACM Conference on Economics and Computation (EC). ACM, 745–762. https://doi.org/10.1016/j.geb.2023.02.002
[57]
Christos Papadimitriou, Emmanouil-Vasileios Vlatakis-Gkaragkounis, and Manolis Zampetakis. 2023. The Computational Complexity of Multi-player Concave Games and Kakutani Fixed Points. In Proceedings of the 24th ACM Conference on Economics and Computation (EC). ACM, 1045. https://doi.org/10.1145/3580507.3597812
[58]
Christos H. Papadimitriou. 1994. On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence. J. Comput. System Sci., 48, 3 (1994), 498–532. https://doi.org/10.1016/S0022-0000(05)80063-7
[59]
Christos H. Papadimitriou and Tim Roughgarden. 2008. Computing correlated equilibria in multi-player games. J. ACM, 55, 3 (2008), 1–29. https://doi.org/10.1145/1379759.1379762
[60]
Ariel D. Procaccia. 2013. Cake cutting: Not just child’s play. Commun. ACM, 56, 7 (2013), 78–87. https://doi.org/10.1145/2483852.2483870
[61]
Jack Robertson and William Webb. 1998. Cake-cutting algorithms: Be fair if you can. CRC Press. https://doi.org/10.1201/9781439863855
[62]
J. B. Rosen. 1965. Existence and Uniqueness of Equilibrium Points for Concave N-Person Games. Econometrica, 33, 3 (1965), 520–534. https://doi.org/10.2307/1911749
[63]
Rahul Savani. 2006. Finding Nash equilibria of bimatrix games. Ph. D. Dissertation. London School of Economics and Political Science.
[64]
David Schmeidler. 1973. Equilibrium points of nonatomic games. Journal of statistical Physics, 7 (1973), 295–300. https://doi.org/10.1007/BF01014905
[65]
Reinhard Selten. 1975. Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory, 4 (1975), 25–55. https://doi.org/10.1007/BF01766400
[66]
Troels Bjerre Sørensen. 2012. Computing a proper equilibrium of a bimatrix game. In Proceedings of the 13th ACM Conference on Electronic Commerce (EC). ACM, 916–928. https://doi.org/10.1145/2229012.2229081
[67]
Emanuel Sperner. 1928. Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 6 (1928), 265–272. https://doi.org/10.1007/BF02940617
[68]
Hugo Steinhaus. 1949. Sur la division pragmatique. Econometrica, 315–319. https://doi.org/10.2307/1907319
[69]
Walter Stromquist. 1980. How to Cut a Cake Fairly. The American Mathematical Monthly, 87, 8 (1980), 640–644. https://doi.org/10.1080/00029890.1980.11995109
[70]
Francis Edward Su. 1999. Rental Harmony: Sperner’s Lemma in Fair Division. The American Mathematical Monthly, 106, 10 (1999), 930–942. https://doi.org/10.1080/00029890.1999.12005142
[71]
Michael J. Todd. 1976. Orientation in complementary pivot algorithms. Mathematics of Operations Research, 1, 1 (1976), 54–66. https://doi.org/10.1287/moor.1.1.54
[72]
Vijay V. Vazirani and Mihalis Yannakakis. 2011. Market equilibrium under separable, piecewise-linear, concave utilities. J. ACM, 58, 3 (2011), 1–25. https://doi.org/10.1145/1970392.1970394
[73]
Douglas R. Woodall. 1980. Dividing a cake fairly. J. Math. Anal. Appl., 78, 1 (1980), 233–247. https://doi.org/10.1016/0022-247X(80)90225-5
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DOI:10.1145/3618260
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