{"type":"ScholarlyArticle","title":"Trembling-Hand Perfection in Extensive-Form Games with Commitment","authors":[{"name":"Gabriele Farina","slug":"gabriele-farina","given":"Gabriele","family":"Farina","affiliation":"Carnegie Mellon University , 5000 Forbes Avenue , Pittsburgh , PA 15213 , USA"},{"name":"Alberto Marchesi","slug":"alberto-marchesi","given":"Alberto","family":"Marchesi","affiliation":"Politecnico di Milano , Piazza Leonardo da Vinci 32 , I-20133 , Milan , Italy"},{"name":"Christian Kroer","slug":"christian-kroer","given":"Christian","family":"Kroer","affiliation":"Carnegie Mellon University , 5000 Forbes Avenue , Pittsburgh , PA 15213 , USA"},{"name":"Nicola Gatti","slug":"nicola-gatti","given":"Nicola","family":"Gatti","affiliation":"Politecnico di Milano , Piazza Leonardo da Vinci 32 , I-20133 , Milan , Italy"},{"name":"Tuomas Sandholm","slug":"tuomas-sandholm","given":"Tuomas","family":"Sandholm","affiliation":"Carnegie Mellon University , 5000 Forbes Avenue , Pittsburgh , PA 15213 , USA"}],"abstract":"We initiate the study of equilibrium refinements based on trembling-hand perfection in extensive-form games with commitment strategies, that is, where one player commits to a strategy first. We show that the standard strong (and weak) Stackelberg equilibria are not suitable for trembling-hand perfection, because the limit of a sequence of such strong (weak) Stackelberg commitment strategies of a perturbed game may not be a strong (weak) Stackelberg equilibrium itself. However, we show that the universal set of all Stackelberg equilibria (i.e., those that are optimal for at least some follower response function) is natural for trembling- hand perfection: it does not suffer from the problem above. We also prove that determining the existence of a Stackelberg equilibrium--refined or not--that gives the leader expected value at least v is NP-hard. This significantly extends prior complexity results that were specific to strong Stackelberg equilibrium.","doi":"10.24963/ijcai.2018/32","url":"https://ijcai.org/proceedings/2018/32","pdf":"https://ijcai.org/proceedings/2018/0032.pdf","markdown":"https://ijcai.org/proceedings/2018/32.md","bibtex":"https://ijcai.org/proceedings/2018/bibtex/32","volume":{"slug":"2018","title":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2018","url":"https://ijcai.org/proceedings/2018/","year":2018,"location":"Stockholm","dates":"13-19 July 2018","editors":"Jérôme Lang","isbn":"978-0-9992411-2-7","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2018"},"section":"Agent-based and Multi-agent Systems","track":"Main track","number":32,"pages":{"first":233,"last":239},"keywords":["Agent-based and Multi-agent Systems: Algorithmic Game Theory","Agent-based and Multi-agent Systems: Economic Paradigms, Auctions and Market-Based Systems","Agent-based and Multi-agent Systems: Noncooperative Games"],"videos":[],"references":[{"position":1,"raw":"[Bertrand, 1883] Joseph Louis Franc ¸ois Bertrand. Theorie mathematique de la richesse sociale. Journal des Savants, pages 499-508, 1883.","title":"Theorie mathematique de la richesse sociale","authors":"Joseph Louis, Franc ¸ois Bertrand","year":"1883","venue":"Journal des Savants","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Bošanskỳ and Cermak, 2015] Branislav Bošanskỳ and Jiri Cermak. Sequence-form algorithm for computing stack- elberg equilibria in extensive-form games. In AAAI Con- ference on Artificial Intelligence (AAAI), 2015.","title":"Sequence-form algorithm for computing stackelberg equilibria in extensive-form games","authors":"Cermak Bošanskỳ, Branislav Bošanskỳ, Jiri Cermak","year":"2015","venue":"AAAI Conference on Artificial Intelligence (AAAI)","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Bošanskỳ et al., 2017] Branislav Bošanskỳ, Simina Brânzei, Kristoffer Arnsfelt Hansen, Troels Bjerre Lund, and Peter Bro Miltersen. Computation of Stackelberg equilibria of finite sequential games. ACM Transaction on Economics and Computation (TEAC), 5(4):23:1-23:24, December 2017.","title":"Computation of Stackelberg equilibria of finite sequential games","authors":"Branislav Bošanskỳ, Simina Brânzei, Arnsfelt Kristoffer, Troels Bjerre Hansen, Peter Lund, Bro Miltersen","year":"2017","venue":"ACM Transaction on Economics and Computation (TEAC)","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Breton et al., 1988] Michèle Breton, Abderrahmane Alj, and Alain Haurie. Sequential Stackelberg equilibria in two-person games. Journal of Optimization Theory and Applications, 59(1):71-97, October 1988.","title":"Sequential Stackelberg equilibria in two-person games","authors":"Michèle Breton, Abderrahmane Alj, Alain Haurie","year":"1988","venue":"Journal of Optimization Theory and Applications","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Cermak et al., 2016] Jiri Cermak, Branislav Bošanskỳ, Karel Durkota, Viliam Lisy, and Christopher Kiekintveld. Using correlated strategies for computing Stackelberg equilibria in extensive-form games. In AAAI Conference on Artificial Intelligence (AAAI), 2016.","title":"Using correlated strategies for computing Stackelberg equilibria in extensive-form games","authors":"Cermak","year":"2016","venue":"AAAI Conference on Artificial Intelligence (AAAI)","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Conitzer and Sandholm, 2006] Vincent Conitzer and Tuo- mas Sandholm. Computing the optimal strategy to commit to. In Proceedings of the ACM Conference on Economics and Computation (EC), 2006.","title":"Computing the optimal strategy to commit to","authors":"Sandholm Conitzer, Vincent Conitzer, Tuomas Sandholm","year":"2006","venue":"Proceedings of the ACM Conference on Economics and Computation (EC)","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Cournot, 1838] Antoine Augustin Cournot. Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the The- ory of Wealth). Hachette, Paris, 1838.","title":"Recherches sur les principes mathématiques de la théorie des richesses (Researches into the Mathematical Principles of the Theory of Wealth","authors":"Cournot","year":"1838","venue":null,"doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[De Nittis et al., 2018] Giuseppe De Nittis, Alberto March- esi, and Nicola Gatti. Computing the strategy to commit to in polymatrix games. In AAAI Conference on Artificial Intelligence (AAAI), 2018.","title":"Computing the strategy to commit to in polymatrix games","authors":"De Nittis","year":"2018","venue":"AAAI Conference on Artificial Intelligence (AAAI)","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Garey and Johnson, 1979] Michael Garey and David John- son. Computers and Intractability. W. H. Freeman and Company, 1979.","title":"Computers and Intractability","authors":"Michael Garey, David Johnson","year":"1979","venue":null,"doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Koller et al., 1996] Daphne Koller, Nimrod Megiddo, and Bernhard von Stengel. Efficient computation of equilib- ria for extensive two-person games. Games and Economic Behavior, 14(2):247-259, 1996.","title":"Efficient computation of equilibria for extensive two-person games","authors":"Koller","year":"1996","venue":"Games and Economic Behavior","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Kroer and Sandholm, 2015] Christian Kroer and Tuomas Sandholm. Limited lookahead in imperfect-information games. In Proceedings of the International Joint Confer- ence on Artificial Intelligence (IJCAI), 2015.","title":"Christian Kroer and Tuomas Sandholm. Limited lookahead in imperfect-information games","authors":"Sandholm Kroer","year":"2015","venue":"Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI)","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Kroer et al., 2018] Christian Kroer, Gabriele Farina, and Tuomas Sandholm. Robust Stackelberg equilibria in extensive-form games and extension to limited lookahead. In AAAI Conference on Artificial Intelligence (AAAI), 2018.","title":"Robust Stackelberg equilibria in extensive-form games and extension to limited lookahead","authors":"Kroer","year":"2018","venue":"AAAI Conference on Artificial Intelligence (AAAI)","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Kuhn, 1953] Harold W. Kuhn. Extensive games and the problem of information. In H. W. Kuhn and A. W. Tucker, editors, Contributions to the Theory of Games, volume 2 of Annals of Mathematics Studies, 28, pages 193-216.","title":"Extensive games and the problem of information","authors":"Harold W Kuhn","year":null,"venue":"Contributions to the Theory of Games","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Letchford and Conitzer, 2010] Joshua Letchford and Vin- cent Conitzer. Computing optimal strategies to commit to in extensive-form games. In Proceedings of the ACM Conference on Electronic Commerce (EC), 2010.","title":"Joshua Letchford and Vincent Conitzer. Computing optimal strategies to commit to in extensive-form games","authors":"Conitzer Letchford","year":"2010","venue":"Proceedings of the ACM Conference on Electronic Commerce (EC)","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Romanovskii, 1962] I. Romanovskii. Reduction of a game with complete memory to a matrix game. Soviet Mathe- matics, 3:678-681, 1962.","title":"Reduction of a game with complete memory to a matrix game","authors":"I Romanovskii","year":"1962","venue":"Soviet Mathematics","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Selten, 1975] Reinhard Selten. Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory, 4(1):25- 55, March 1975.","title":"Reexamination of the perfectness concept for equilibrium points in extensive games","authors":"Reinhard Selten, Selten","year":"1975","venue":"International Journal of Game Theory","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Shoham and Leyton-Brown, 2008] Yoav Shoham and Kevin Leyton-Brown. Multiagent systems: Algorithmic, game-theoretic, and logical foundations. Cambridge University Press, 2008.","title":"Multiagent systems: Algorithmic, game-theoretic, and logical foundations","authors":"Yoav Shoham, Kevin Leyton-Brown","year":"2008","venue":null,"doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Tambe, 2011] Milind Tambe. Security and game theory: al- gorithms, deployed systems, lessons learned. Cambridge University Press, 2011.","title":"Security and game theory: algorithms, deployed systems, lessons learned","authors":"Milind Tambe","year":"2011","venue":null,"doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[van Damme, 1984] Eric van Damme. A relation between perfect equilibria in extensive form games and proper equilibria in normal form games. International Journal of Game Theory, 1984.","title":"A relation between perfect equilibria in extensive form games and proper equilibria in normal form games","authors":"; Van Damme, Eric Van Damme","year":"1984","venue":"International Journal of Game Theory","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[van Damme, 1991] Eric van Damme. Stability and perfec- tion of Nash equilibria, volume 339. Springer, 1991.","title":"Stability and perfection of Nash equilibria","authors":"; Van Damme, Eric Van Damme","year":"1991","venue":null,"doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[von Stackelberg, 1934] Heinrich von Stackelberg. Markt- form und Gleichgewicht. Springer, Vienna, 1934.","title":"Heinrich von Stackelberg. Marktform und Gleichgewicht","authors":"Von Stackelberg","year":"1934","venue":null,"doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[von Stengel and Zamir, 2010] Bernhard von Stengel and Shmuel Zamir. Leadership games with convex strategy sets. Games and Economic Behavior, 69(2):446-457, 2010.","title":"Leadership games with convex strategy sets","authors":"Von Stengel, ; Zamir, Von Bernhard, Shmuel Stengel, Zamir","year":"2010","venue":"Games and Economic Behavior","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[von Stengel, 1996] Bernhard von Stengel. Efficient compu- tation of behavior strategies. Games and Economic Behav- ior, 14(2):220-246, 1996.","title":"Efficient computation of behavior strategies","authors":"Von Stengel, ; Bernhard, Von Stengel","year":"1996","venue":"Games and Economic Behavior","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Yin et al., 2012] Zhengyu Yin, Albert Xin Jiang, Milind Tambe, Christopher Kiekintveld, Kevin Leyton-Brown, Tuomas Sandholm, and John P. Sullivan. TRUSTS: Scheduling randomized patrols for fare inspection in tran- sit systems using game theory. AI Magazine, 33(4):59-72, 2012.","title":"TRUSTS: Scheduling randomized patrols for fare inspection in transit systems using game theory","authors":"Zhengyu Yin, Albert Xin Jiang, Milind Tambe, Christopher Kiekintveld, Kevin Leyton-Brown, Tuomas Sandholm, John P Sullivan","year":"2012","venue":"AI Magazine","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}