{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,17]],"date-time":"2026-01-17T04:05:04Z","timestamp":1768622704145,"version":"3.49.0"},"reference-count":30,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T00:00:00Z","timestamp":1618185600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T00:00:00Z","timestamp":1618185600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann Oper Res"],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1007\/s10479-021-04056-6","type":"journal-article","created":{"date-parts":[[2021,4,12]],"date-time":"2021-04-12T20:03:09Z","timestamp":1618257789000},"page":"35-62","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Technical efficiency in firm games with constant returns to scale and $$\\alpha $$-returns to scale"],"prefix":"10.1007","volume":"304","author":[{"given":"Walter","family":"Briec","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7695-8157","authenticated-orcid":false,"given":"Marc","family":"Dubois","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2331-630X","authenticated-orcid":false,"given":"St\u00e9phane","family":"Mussard","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,4,12]]},"reference":[{"key":"4056_CR1","volume-title":"Lectures on functional equations and their applications","author":"J Acz\u00e9l","year":"1966","unstructured":"Acz\u00e9l, J. (1966). Lectures on functional equations and their applications. New York: Academic Press."},{"issue":"4","key":"4056_CR2","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1080\/02331934.2017.1279159","volume":"66","author":"L Andriamasy","year":"2017","unstructured":"Andriamasy, L., Briec, W., & Mussard, S. (2017). On some relations between several generalized convex DEA models. Optimization, 66(4), 547\u2013570.","journal-title":"Optimization"},{"key":"4056_CR3","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1016\/0022-0531(70)90039-6","volume":"2","author":"AB Atkinson","year":"1970","unstructured":"Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2, 244\u2013263.","journal-title":"Journal of Economic Theory"},{"issue":"1","key":"4056_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF00932539","volume":"21","author":"A Ben-Tal","year":"1977","unstructured":"Ben-Tal, A. (1977). On generalized means and generalized convex functions. Journal of Optimization Theory and Applications, 21(1), 1\u201313.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"4056_CR5","doi-asserted-by":"publisher","first-page":"665","DOI":"10.2307\/134822","volume":"14","author":"C Blackorby","year":"1981","unstructured":"Blackorby, C., Donaldson, D., & Auersperg, M. (1981). A new procedure for the measurement of inequality within and among population subgroups. Canadian Journal of Economics, 14, 665\u2013685.","journal-title":"Canadian Journal of Economics"},{"issue":"1","key":"4056_CR6","doi-asserted-by":"publisher","first-page":"332","DOI":"10.1016\/j.ejor.2008.05.028","volume":"197","author":"J-P Boussemart","year":"2009","unstructured":"Boussemart, J.-P., Briec, W., Peypoch, N., & Tav\u00e9ra, C. (2009). $$\\alpha $$-Returns to scale and multi-output production technologies. European Journal of Operational Research, 197(1), 332\u2013339.","journal-title":"European Journal of Operational Research"},{"key":"4056_CR7","first-page":"259","volume":"22","author":"W Briec","year":"2015","unstructured":"Briec, W. (2015). Some remarks on an idempotent and non-associative convex structure. Journal of Convex Analysis, 22, 259\u2013289.","journal-title":"Journal of Convex Analysis"},{"key":"4056_CR8","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/s00712-002-0606-1","volume":"79","author":"W Briec","year":"2003","unstructured":"Briec, W., Dervaux, B., & Leleu, H. (2003). Aggregation of directional distance functions and industrial efficiency. Journal of Economics, 79, 237\u2013261.","journal-title":"Journal of Economics"},{"key":"4056_CR9","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/j.jmaa.2009.01.048","volume":"355","author":"W Briec","year":"2009","unstructured":"Briec, W., & Horvath, C. D. (2009). A B-convex production model for evaluating performance of firms. Journal of Mathematical Analysis and Applications, 355, 131\u2013144.","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"4056_CR10","first-page":"740","volume":"215","author":"W Briec","year":"2011","unstructured":"Briec, W., & Liang, Q. B. (2011). On some Semilattice Structures for Production Technologies. European Journal of Operational Research, 215, 740\u2013749.","journal-title":"European Journal of Operational Research"},{"key":"4056_CR11","doi-asserted-by":"publisher","first-page":"286","DOI":"10.1016\/j.ejor.2014.04.049","volume":"239","author":"W Briec","year":"2014","unstructured":"Briec, W., & Mussard, S. (2014). Efficient firm groups: Allocative efficiency in cooperative games. European Journal of Operational Research, 239, 286\u2013296.","journal-title":"European Journal of Operational Research"},{"issue":"3","key":"4056_CR12","doi-asserted-by":"publisher","first-page":"991","DOI":"10.1016\/j.ejor.2019.11.048","volume":"283","author":"W Briec","year":"2020","unstructured":"Briec, W., & Mussard, S. (2020). Improvement of technical efficiency of firm groups. European Journal of Operational Research, 283(3), 991\u20131001.","journal-title":"European Journal of Operational Research"},{"key":"4056_CR13","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1006\/jeth.1996.0096","volume":"70","author":"RG Chambers","year":"1996","unstructured":"Chambers, R. G., Chung, Y., & F\u00e4re, R. (1996). Benefit and distance functions. Journal of Economic Theory, 70, 407\u2013419.","journal-title":"Journal of Economic Theory"},{"key":"4056_CR14","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1023\/A:1022637501082","volume":"98","author":"RG Chambers","year":"1998","unstructured":"Chambers, R. G., Chung, Y., & F\u00e4re, R. (1998). Profit, directional distance functions, and Nerlovian efficiency. Journal of Optimization Theory and Applications, 98, 351\u2013364.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"4056_CR15","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1007\/s001990050205","volume":"11","author":"RG Chambers","year":"1998","unstructured":"Chambers, R. G., & F\u00e4re, R. (1998). Translation Homotheticity. Economic Theory, 11, 629\u2013641.","journal-title":"Economic Theory"},{"issue":"12","key":"4056_CR16","doi-asserted-by":"publisher","first-page":"1507","DOI":"10.1287\/mnsc.34.12.1507","volume":"34","author":"R F\u00e4re","year":"1988","unstructured":"F\u00e4re, R., Grosskopf, S., & Njinkeu, D. (1988). Note\u2014On piecewise reference technologies. Management Science, 34(12), 1507\u20131511.","journal-title":"Management Science"},{"key":"4056_CR17","unstructured":"F\u00e4re, R., Grosskopf, S., & Zelenyuk, V. (2001). Aggregation of Nerlovian Profit Indicator, Mimeo."},{"key":"4056_CR18","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1080\/13504850600825584","volume":"15","author":"R F\u00e4re","year":"2008","unstructured":"F\u00e4re, R., Grosskopf, S., & Zelenyuk, V. (2008). Aggregation of Nerlovian profit indicator. Applied Economics Letters, 15, 845\u2013847.","journal-title":"Applied Economics Letters"},{"key":"4056_CR19","doi-asserted-by":"publisher","unstructured":"F\u00e4re, R., Mizobuchi, H., & Zelenyuk, V. (2021). Hicks neutrality and homotheticity in technologies with multiple inputs and multiple outputs, Omega, 101, 102240. https:\/\/doi.org\/10.1016\/j.omega.2020.102240.","DOI":"10.1016\/j.omega.2020.102240"},{"issue":"3","key":"4056_CR20","doi-asserted-by":"publisher","first-page":"253","DOI":"10.2307\/2343100","volume":"120","author":"MJ Farrell","year":"1957","unstructured":"Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society, 120(3), 253\u2013290.","journal-title":"Journal of the Royal Statistical Society"},{"key":"4056_CR21","unstructured":"Hardy, G. H., Littlewood, J. E., & P\u00f3lya, G. (1934). Inequalities. Cambridge University Press."},{"key":"4056_CR22","doi-asserted-by":"publisher","first-page":"209","DOI":"10.2307\/2296966","volume":"50","author":"U Kohli","year":"1983","unstructured":"Kohli, U. (1983). Nonjoint technologies. Review of Economic Studies, 50, 209\u201319.","journal-title":"Review of Economic Studies"},{"issue":"4","key":"4056_CR23","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/0304-4068(94)00695-7","volume":"24","author":"S-K Li","year":"1995","unstructured":"Li, S.-K. (1995). Relations between convexity and homogeneity in multioutput technologies. Journal of Mathematical Economics, 24(4), 311\u2013318.","journal-title":"Journal of Mathematical Economics"},{"key":"4056_CR24","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1007\/BF02295790","volume":"1","author":"S-K Li","year":"2001","unstructured":"Li, S.-K., & Ng, Y. C. (2001). Measuring the productive efficiency of a group of firms. International Advances in Economic Research, 1, 377\u2013390.","journal-title":"International Advances in Economic Research"},{"issue":"3","key":"4056_CR25","doi-asserted-by":"publisher","first-page":"558","DOI":"10.1016\/j.ejor.2012.05.014","volume":"222","author":"S Lozano","year":"2012","unstructured":"Lozano, S. (2012). Information aharing in DEA: A cooperative game theory approach. European Journal of Operational Research, 222(3), 558\u2013565.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"4056_CR26","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1016\/j.ejor.2013.06.004","volume":"231","author":"S Lozano","year":"2013","unstructured":"Lozano, S. (2013). DEA production games. European Journal of Operational Research, 231(2), 405\u2013413.","journal-title":"European Journal of Operational Research"},{"key":"4056_CR27","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1016\/j.ejor.2013.04.034","volume":"230","author":"A Peyrache","year":"2013","unstructured":"Peyrache, A. (2013). Industry structural inefficiency and potential gains from mergers and break-ups: A comprehensive approach. European Journal of Operational Research, 230, 422\u2013430.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"4056_CR28","doi-asserted-by":"publisher","first-page":"780","DOI":"10.1016\/j.ejor.2018.07.020","volume":"272","author":"P Ravelojaona","year":"2019","unstructured":"Ravelojaona, P. (2019). On constant elasticity of substitution\u2014Constant elasticity of transformation directional distance functions. European Journal of Operational Research, 272(2), 780\u2013791.","journal-title":"European Journal of Operational Research"},{"key":"4056_CR29","doi-asserted-by":"crossref","unstructured":"Shapley, L. (1953). A value for $$n$$-Person Games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the Theory of Games, 2. Princeton University Press.","DOI":"10.1515\/9781400881970-018"},{"key":"4056_CR30","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BF01753431","volume":"1","author":"L Shapley","year":"1972","unstructured":"Shapley, L. (1972). Cores of convex games. International Journal of Game Theory, 1, 11\u201326.","journal-title":"International Journal of Game Theory"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-021-04056-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10479-021-04056-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-021-04056-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,8,6]],"date-time":"2021-08-06T15:29:15Z","timestamp":1628263755000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10479-021-04056-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,12]]},"references-count":30,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2021,9]]}},"alternative-id":["4056"],"URL":"https:\/\/doi.org\/10.1007\/s10479-021-04056-6","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,12]]},"assertion":[{"value":"22 March 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 April 2021","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}