{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T19:52:27Z","timestamp":1771703547822,"version":"3.50.1"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,12,25]],"date-time":"2018-12-25T00:00:00Z","timestamp":1545696000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci. China Inf. Sci."],"published-print":{"date-parts":[[2019,2]]},"DOI":"10.1007\/s11432-018-9414-5","type":"journal-article","created":{"date-parts":[[2018,12,27]],"date-time":"2018-12-27T10:56:10Z","timestamp":1545908170000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Matrix expression of Shapley values and its application to distributed resource allocation"],"prefix":"10.1007","volume":"62","author":[{"given":"Yuanhua","family":"Wang","sequence":"first","affiliation":[]},{"given":"Daizhan","family":"Cheng","sequence":"additional","affiliation":[]},{"given":"Xiyu","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,12,25]]},"reference":[{"key":"9414_CR1","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1109\/TSTE.2012.2192299","volume":"3","author":"S W Mei","year":"2012","unstructured":"Mei S W, Wang Y Y, Liu F, et al. Game approaches for hybrid power system planning. IEEE Trans Sustain Energ, 2012, 3: 506\u2013517","journal-title":"IEEE Trans Sustain Energ"},{"key":"9414_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/100784163","volume":"51","author":"M H Zhu","year":"2013","unstructured":"Zhu M H, Mart\u00ednez S. Distributed coverage games for energy-aware mobile sensor networks. SIAM J Control Optim, 2013, 51: 1\u201327","journal-title":"SIAM J Control Optim"},{"key":"9414_CR3","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1145\/1925019.1925026","volume":"38","author":"R Gopalakrishnan","year":"2011","unstructured":"Gopalakrishnan R, Marden J R, Wierman A. An architectural view of game theoretic control. SIGMETRICS Perform Eval Rev, 2011, 38: 31\u201336","journal-title":"SIGMETRICS Perform Eval Rev"},{"key":"9414_CR4","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1287\/opre.1120.1137","volume":"61","author":"J R Marden","year":"2013","unstructured":"Marden J R, Wierman A. Distributed welfare games. Oper Res, 2013, 61: 155\u2013168","journal-title":"Oper Res"},{"key":"9414_CR5","first-page":"307","volume":"2","author":"L S Shapley","year":"1953","unstructured":"Shapley L S. A value for n-person games. Contrib Theory Games, 1953, 2: 307\u2013317","journal-title":"Contrib Theory Games"},{"key":"9414_CR6","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1006\/jeth.2000.2704","volume":"100","author":"D P\u00e9rez-Castrillo","year":"2001","unstructured":"P\u00e9rez-Castrillo D, Wettstein D. Bidding for the surplus: a non-cooperative approach to the Shapley value. J Econ Theory, 2001, 100: 274\u2013294","journal-title":"J Econ Theory"},{"key":"9414_CR7","volume-title":"Additive and non-additioe set functions","author":"L S Shapley","year":"1953","unstructured":"Shapley L S. Additive and non-additioe set functions. Dissertation for Ph.D. Degree. Princeton: Princeton University, 1953"},{"key":"9414_CR8","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/BF01756292","volume":"16","author":"E Kalai","year":"1987","unstructured":"Kalai E, Samet D. On weighted Shapley values. Int J Game Theory, 1987, 16: 205\u2013222","journal-title":"Int J Game Theory"},{"key":"9414_CR9","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/BF01240278","volume":"20","author":"Y Chun","year":"1991","unstructured":"Chun Y. On the symmetric and weighted shapley values. Int J Game Theory, 1991, 20: 183\u2013190","journal-title":"Int J Game Theory"},{"key":"9414_CR10","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1016\/S0899-8256(05)80008-9","volume":"8","author":"A S Nowak","year":"1995","unstructured":"Nowak A S, Radzik T. On axiomatizations of the weighted Shapley values. Games Econ Behav, 1995, 8: 389\u2013405","journal-title":"Games Econ Behav"},{"key":"9414_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-85729-097-7","volume-title":"Analysis and Control of Boolean Networks: A Semi-tensor Product Approach","author":"D Z Cheng","year":"2011","unstructured":"Cheng D Z, Qi H S, Li Z Q. Analysis and Control of Boolean Networks: A Semi-tensor Product Approach. Berlin: Springer, 2011"},{"key":"9414_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.sysconle.2011.09.019","volume":"61","author":"F F Li","year":"2012","unstructured":"Li F F, Sun J T. Stability and stabilization of Boolean networks with impulsive effects. Syst Control Lett, 2012, 61: 1\u20135","journal-title":"Syst Control Lett"},{"key":"9414_CR13","doi-asserted-by":"publisher","first-page":"010202","DOI":"10.1007\/s11432-017-9238-1","volume":"61","author":"H T Li","year":"2018","unstructured":"Li H T, Zhao G D, Meng M, et al. A survey on applications of semi-tensor product method in engineering. Sci China Inf Sci, 2018, 61: 010202","journal-title":"Sci China Inf Sci"},{"key":"9414_CR14","doi-asserted-by":"publisher","first-page":"4222","DOI":"10.1109\/TAC.2017.2679903","volume":"62","author":"M Meng","year":"2017","unstructured":"Meng M, Liu L, Feng G. Stability and l1 gain analysis of Boolean networks with Markovian jump parameters. IEEE Trans Autom Control, 2017, 62: 4222\u20134228","journal-title":"IEEE Trans Autom Control"},{"key":"9414_CR15","doi-asserted-by":"publisher","first-page":"1658","DOI":"10.1109\/TAC.2015.2478123","volume":"61","author":"J Q Lu","year":"2016","unstructured":"Lu J Q, Zhong J, Huang C, et al. On pinning controllability of Boolean control networks. IEEE Trans Autom Control, 2016, 61: 1658\u20131663","journal-title":"IEEE Trans Autom Control"},{"key":"9414_CR16","doi-asserted-by":"publisher","first-page":"2110","DOI":"10.1049\/iet-cta.2016.1252","volume":"11","author":"G D Zhao","year":"2017","unstructured":"Zhao G D, Fu S H. Matrix approach to trajectory control of higher-order k-valued logical control networks. IET Control Theory Appl, 2017, 11: 2110\u20132115","journal-title":"IET Control Theory Appl"},{"key":"9414_CR17","first-page":"052201","volume":"57","author":"Z B Liu","year":"2014","unstructured":"Liu Z B, Wang Y Z, Li H T. Two kinds of optimal controls for probabilistic mix-valued logical dynamic networks. Sci China Inf Sci, 2014, 57: 052201","journal-title":"Sci China Inf Sci"},{"key":"9414_CR18","doi-asserted-by":"publisher","first-page":"7812","DOI":"10.1016\/j.jfranklin.2017.09.024","volume":"354","author":"Y T Zheng","year":"2017","unstructured":"Zheng Y T, Li H T, Ding X Y, et al. Stabilization and set stabilization of delayed Boolean control networks based on trajectory stabilization. J Franklin Inst, 2017, 354: 7812\u20137827","journal-title":"J Franklin Inst"},{"key":"9414_CR19","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.1016\/j.automatica.2012.03.024","volume":"48","author":"Y Z Wang","year":"2012","unstructured":"Wang Y Z, Zhang C H, Liu Z B. A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems. Automatica, 2012, 48: 1227\u20131236","journal-title":"Automatica"},{"key":"9414_CR20","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1016\/j.neucom.2015.09.073","volume":"174","author":"J Zhong","year":"2016","unstructured":"Zhong J, Lu J Q, Huang C, et al. Finding graph minimum stable set and core via semi-tensor product approach. Neurocomputing, 2016, 174: 588\u2013596","journal-title":"Neurocomputing"},{"key":"9414_CR21","doi-asserted-by":"publisher","first-page":"010205","DOI":"10.1007\/s11432-017-9283-7","volume":"61","author":"J T Zhao","year":"2018","unstructured":"Zhao J T, Chen Z Q, Liu Z X. Modeling and analysis of colored petri net based on the semi-tensor product of matrices. Sci China Inf Sci, 2018, 61: 010205","journal-title":"Sci China Inf Sci"},{"key":"9414_CR22","first-page":"1680","volume-title":"Proceedings of 10th IEEE International Conference on Control and Automation (ICCA)","author":"D Z Cheng","year":"2013","unstructured":"Cheng D Z, Xu T T. Application of STP to cooperative games. In: Proceedings of 10th IEEE International Conference on Control and Automation (ICCA), Hangzhou, 2013. 1680\u20131685"},{"key":"9414_CR23","doi-asserted-by":"publisher","first-page":"1793","DOI":"10.1016\/j.automatica.2014.05.005","volume":"50","author":"D Z Cheng","year":"2014","unstructured":"Cheng D Z. On finite potential games. Automatica, 2014, 50: 1793\u20131801","journal-title":"Automatica"},{"key":"9414_CR24","doi-asserted-by":"publisher","first-page":"2161","DOI":"10.1049\/iet-cta.2016.1454","volume":"11","author":"Y H Wang","year":"2017","unstructured":"Wang Y H, Liu T, Cheng D Z. From weighted potential game to weighted harmonic game. IET Control Theory Appl, 2017, 11: 2161\u20132169","journal-title":"IET Control Theory Appl"},{"key":"9414_CR25","doi-asserted-by":"publisher","first-page":"2402","DOI":"10.1109\/TAC.2015.2404471","volume":"60","author":"D Z Cheng","year":"2015","unstructured":"Cheng D Z, He F H, Qi H S, et al. Modeling, analysis and control of networked evolutionary games. IEEE Trans Autom Control, 2015, 60: 2402\u20132415","journal-title":"IEEE Trans Autom Control"},{"key":"9414_CR26","doi-asserted-by":"publisher","first-page":"092209","DOI":"10.1007\/s11432-016-5532-x","volume":"59","author":"Y H Wang","year":"2016","unstructured":"Wang Y H, Cheng D Z. Dynamics and stability for a class of evolutionary games with time delays in strategies. Sci China Inf Sci, 2016, 59: 092209","journal-title":"Sci China Inf Sci"},{"key":"9414_CR27","doi-asserted-by":"publisher","first-page":"2140","DOI":"10.1049\/iet-cta.2017.0014","volume":"11","author":"P L Guo","year":"2017","unstructured":"Guo P L, Zhang H X, Alsaadi F E, et al. Semi-tensor product method to a class of event-triggered control for finite evolutionary networked games. IET Control Theory Appl, 2017, 11: 2140\u20132145","journal-title":"IET Control Theory Appl"},{"key":"9414_CR28","volume-title":"Models in Cooperative Game Theory","author":"R Branzei","year":"2008","unstructured":"Branzei R, Dimitrov D, Tijs S. Models in Cooperative Game Theory. Berlin: Springer, 2008"},{"key":"9414_CR29","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1145\/356689.356692","volume":"9","author":"R Sedgewick","year":"1977","unstructured":"Sedgewick R. Permutation generation methods. ACM Comput Surv, 1977, 9: 137\u2013164","journal-title":"ACM Comput Surv"},{"key":"9414_CR30","doi-asserted-by":"publisher","DOI":"10.1201\/9781439832905","volume-title":"Handbook of Discrete and Combinatorial Mathematics","author":"K H Rosen","year":"1999","unstructured":"Rosen K H, Michaels J G, et al. Handbook of Discrete and Combinatorial Mathematics. Boca Raton: CRC Press, 1999"}],"container-title":["Science China Information Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11432-018-9414-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11432-018-9414-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11432-018-9414-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,3,22]],"date-time":"2020-03-22T16:14:42Z","timestamp":1584893682000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11432-018-9414-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,25]]},"references-count":30,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,2]]}},"alternative-id":["9414"],"URL":"https:\/\/doi.org\/10.1007\/s11432-018-9414-5","relation":{},"ISSN":["1674-733X","1869-1919"],"issn-type":[{"value":"1674-733X","type":"print"},{"value":"1869-1919","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,12,25]]},"assertion":[{"value":"27 January 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 December 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"22201"}}