{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:17:45Z","timestamp":1773656265121,"version":"3.50.1"},"reference-count":38,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T00:00:00Z","timestamp":1498089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Games"],"abstract":"<jats:p>We study a non-cooperative game with aggregative structure, namely when the payoffs depend on the strategies of the opponent players through an aggregator function. We assume that a subset of players behave as leaders in a Stackelberg model. The leaders, as well the followers, act non-cooperatively between themselves and solve a Nash equilibrium problem. We assume an exogenous uncertainty affecting the aggregator and we obtain existence results for the stochastic resulting game. Some examples are illustrated.<\/jats:p>","DOI":"10.3390\/g8030025","type":"journal-article","created":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T10:11:35Z","timestamp":1498126295000},"page":"25","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Multi-Leader Multi-Follower Model with Aggregative Uncertainty"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7326-8735","authenticated-orcid":false,"given":"Lina","family":"Mallozzi","sequence":"first","affiliation":[{"name":"Department of Mathematics and Applications, University of Naples Federico II, Via Claudio 21, 80125 Naples, Italy"}]},{"given":"Roberta","family":"Messalli","sequence":"additional","affiliation":[{"name":"Department of Economic and Statistic Science, University of Naples Federico II, Complesso Monte Sant\u2019Angelo 21, 80125 Naples, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1287\/opre.31.2.253","article-title":"Stackelberg-Nash-Cournot Equilibria: Characterizations and computations","volume":"31","author":"Sheraly","year":"1983","journal-title":"Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/BFb0006224","article-title":"Stackelberg-Nash model for the design of deregulated transit system","volume":"Volume 157","author":"Hamalainen","year":"1991","journal-title":"Dynamic Games in Economic Analysis"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1109\/59.867153","article-title":"Strategic gaming analysis for electric power systems: An MPEC approach","volume":"15","author":"Hobbs","year":"2000","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.compchemeng.2016.01.005","article-title":"Water integration in eco-industrial parks using a multi-leader-follower approach","volume":"87","author":"Ramos","year":"2016","journal-title":"Comput. Chem. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3682","DOI":"10.1016\/j.comnet.2012.08.004","article-title":"Multi-leader multi-follower Stackelberg model for cognitive radio spectrum sharing scheme","volume":"56","author":"Kim","year":"2012","journal-title":"Comput. Netw."},{"key":"ref_6","first-page":"1","article-title":"Multi-Leader-Follower Games: Models, Methods and Applications","volume":"58","author":"Hu","year":"2015","journal-title":"J. Oper. Res. Soc. Jpn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1287\/opre.1080.0686","article-title":"A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application","volume":"57","author":"Xu","year":"2009","journal-title":"Oper. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1007\/s00199-004-0566-5","article-title":"Asymmetric contests with general technologies","volume":"3","author":"Cornes","year":"2005","journal-title":"Econ. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s00199-008-0419-8","article-title":"Aggregative Games and Best-Reply Potentials","volume":"43","author":"Jensen","year":"2010","journal-title":"Econ. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0304-4068(90)90005-T","article-title":"Nash equilibrium with strategic complementaries","volume":"19","author":"Vives","year":"1990","journal-title":"J. Math. Econ."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Selten, R. (1970). Preispolitik der Mehrproduktenunternehmung in der Statichen Theorie, Springer.","DOI":"10.1007\/978-3-642-48888-7"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.geb.2013.03.009","article-title":"Aggregate comparative statics","volume":"81","author":"Acemoglu","year":"2013","journal-title":"Games Econ. Behav."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Grammatico, S. (2017). Dynamic Control of Agents playing Aggregative Games with Coupling Constraints. IEEE Trans. Autom. Control.","DOI":"10.1109\/TAC.2017.2672902"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"680","DOI":"10.1287\/opre.2016.1501","article-title":"Distributed Algorithms for Aggregative Games on Graphs","volume":"64","author":"Koshal","year":"2016","journal-title":"Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.econlet.2014.12.010","article-title":"One-leader and multiple-follower Stackelberg games with private information","volume":"127","author":"Nakamura","year":"2015","journal-title":"Econ. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1137\/S1052623499363220","article-title":"The Sample Average Approximation Method for Stochastic Discrete Optimization","volume":"12","author":"Kleywegt","year":"2001","journal-title":"SIAM J. Optim."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s00199-004-0474-8","article-title":"The evolutionary stability of perfectly competitive behavior","volume":"26","author":"Ania","year":"2005","journal-title":"Econ. Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"367","DOI":"10.2307\/2296766","article-title":"The structure of utility functions","volume":"35","author":"Gorman","year":"1968","journal-title":"Rev. Econ. Stud."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.2307\/2938316","article-title":"Rationalizability, Learning and Equilibrium in Games with Strategic Complementarities","volume":"58","author":"Milgrom","year":"1990","journal-title":"Econometrica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1287\/opre.26.2.305","article-title":"Minimizing a Submodular Function on a Lattice","volume":"26","author":"Topkis","year":"1978","journal-title":"Oper. Res."},{"key":"ref_21","unstructured":"Topkis, D.M. (1998). Supermodularity and Complementarity, Princeton University Press."},{"key":"ref_22","unstructured":"Vives, X. (2000). Oligopoly Pricing: Old Ideas and New Tools, MIT Press."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1287\/moor.10.3.471","article-title":"Substitutes Complements and Ripples in Network Flows","volume":"10","author":"Granot","year":"1985","journal-title":"Math. Oper. Res."},{"key":"ref_24","first-page":"281","article-title":"Pseudo-convex functions","volume":"3","author":"Mangasarian","year":"1965","journal-title":"SIAM J. Control"},{"key":"#cr-split#-ref_25.1","unstructured":"Von Stackelberg, H. (1934). Marktform und Gleichgewicht, Julius Springer. (The Theory of the Market Economy, English Edition"},{"key":"#cr-split#-ref_25.2","unstructured":"Peacock, A., Ed.;William Hodge: London, UK, 1952.)."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"390","DOI":"10.1287\/opre.32.2.390","article-title":"A Multiple Leader Stackelberg Model and Analysis","volume":"32","author":"Sherali","year":"1984","journal-title":"Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Luo, Z.Q., Pang, J.S., and Ralph, D. (1996). Mathematical Programs with Equilibrium Constraints, Cambridge University Press.","DOI":"10.1017\/CBO9780511983658"},{"key":"ref_28","unstructured":"Cornes, R., and Hartley, R. (2011, January 6\u20137). Well-behaved Aggregative Games. Proceedings of the 2011 Workshop on Aggregative Games at Strathclyde University, Glasgow, UK."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1051\/ro\/2015054","article-title":"Strategic Investments in R & D and Efficiency in the presence of free riders","volume":"50","author":"Brahim","year":"2015","journal-title":"RAIRO-Oper. Res."},{"key":"ref_30","unstructured":"Finus, M. (2000). Game Theory and International Environmental Cooperation: A Survey with an Application to the Kyoto-Protocol, Fondazione Eni Enrico Mattei."},{"key":"ref_31","first-page":"23","article-title":"Partial Cooperation and Non-Signatories Multiple Decision","volume":"2","author":"Mallozzi","year":"2008","journal-title":"AUCO Czech Econ. Rev."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/BF00939867","article-title":"Sequential Stackelberg equilibria in two-person games","volume":"59","author":"Breton","year":"1988","journal-title":"J. Optim. Theory Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3379","DOI":"10.1109\/TAC.2015.2423891","article-title":"An Existence Result for Hierarchical Stackelberg v\/s Stackelberg Games","volume":"60","author":"Kulkarni","year":"2015","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/BF00933155","article-title":"On generalized Stackelberg strategies","volume":"26","author":"Leitmann","year":"1978","journal-title":"J. Optim. Theory Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1023\/A:1021729618112","article-title":"Stackelberg solutions to multi objective two-level linear programming problems","volume":"103","author":"Nishizaki","year":"1999","journal-title":"J. Optim. Theory Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.geb.2004.10.007","article-title":"Strategic complements and substitutes, and potential games","volume":"54","author":"Dubey","year":"2006","journal-title":"Games Econ. Behav."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/s10957-004-1856-6","article-title":"Oligopolistic markets with leadership and demand functions possibly discontinuous","volume":"125","author":"Mallozzi","year":"2005","journal-title":"J. Optim. Theory Appl."}],"container-title":["Games"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/3\/25\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:40:05Z","timestamp":1760208005000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4336\/8\/3\/25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,22]]},"references-count":38,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,9]]}},"alternative-id":["g8030025"],"URL":"https:\/\/doi.org\/10.3390\/g8030025","relation":{},"ISSN":["2073-4336"],"issn-type":[{"value":"2073-4336","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,6,22]]}}}