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A generalized linear production model: A unifying model.(English)Zbl 0604.90142

The paper deals with linear production games in the tradition of G. Owens. However, the author does not assume that the resources of a coalition S of producers is necessarily obtained by adding up the resources of its members (additivity assumption). It is shown that these generalized production games are balanced if the associated resource games are balanced. Moreoer, a procedure for generating a vector in the core of the generalized production game is given. These two results are easily obtained and then applied to explain the non-emptiness of the core of various cooperative games, which are derived from optimization problems but do not satisfy Owen’s additivity assumption.
Reviewer: D.Hinrichsen

MSC:

91A12 Cooperative games
91B38 Production theory, theory of the firm
90C90 Applications of mathematical programming
90C05 Linear programming
90C35 Programming involving graphs or networks
05C05 Trees

Cite

References:

[1]L.J. Billera and R.E. Bixby, ”Market representation ofn-person games”,Bulletin of the American Mathematical Society 80 (1974) 522–526. ·Zbl 0288.90092 ·doi:10.1090/S0002-9904-1974-13478-6
[2]C. Bird, ”On cost allocation for a spanning tree: A game theoretic approach”,Networks 6 (1976) 335–350. ·Zbl 0357.90083 ·doi:10.1002/net.3230060404
[3]O.N. Bondareva, ”Some applications of linear programming methods to the theory of cooperative games”, (in Russian)Problemy Kibernetiki 10 (1963) 119–139. ·Zbl 1013.91501
[4]A. Charnes and K.O. Kortanek, ”On balanced sets, cores and linear programming”,Cahiers du Centre d’Etudes de Recherches Opérationnelles 9 (1967) 32–43. ·Zbl 0158.19205
[5]P. Dubey and L.S. Shapley, ”Totally balanced games arising from controlled programming problems”,Mathematical Programming 29 (1984) 245–267. ·Zbl 0557.90109 ·doi:10.1007/BF02591996
[6]J. Edmonds, ”Maximum matching and a polyhedron with 0,1 vertices”,National Bureau of Standards Journal of Research 69B (1965) 125–130. ·Zbl 0141.21802
[7]J. Edmonds, ”Optimum branchings”,National Bureau of Standards Journal of Research 71B (1967) 233–240. ·Zbl 0155.51204
[8]D. Gale and L.S. Shapley, ”College admissions and the stability of marriage”,American Mathematical Monthly 69 (1962) 9–14. ·Zbl 0109.24403 ·doi:10.2307/2312726
[9]R.E. Gomory and T.C. Hu, ”An application of generalized linear programming to network flows”,Journal of the Society for Industrial and Applied Mathematics 10 (1962) 260–283. ·Zbl 0105.12805 ·doi:10.1137/0110020
[10]D. Granot, ”A note on the room-mates problem and a related revenue allocation problem”,Management Science 30 (1984) 633–643. ·Zbl 0551.90106 ·doi:10.1287/mnsc.30.5.633
[11]D. Granot and G. Huberman, ”Minimum cost spanning tree games”,Mathematical Programming 21 (1981) 1–18. ·Zbl 0461.90099 ·doi:10.1007/BF01584227
[12]D. Granot and G. Huberman, ”On the core and nucleolus of minimum cost spanning tree games”,Mathematical Programming 29 (1984) 323–347. ·Zbl 0541.90099 ·doi:10.1007/BF02592000
[13]D. Granot and F. Granot, ”On some network flow games”, Working Paper No. 1056, Faculty of Commerce and Business Administration, University of British Columbia, Vancouver (1984). ·Zbl 0555.90077
[14]E. Kalai and E. Zemel, ”Generalized network problems yielding totally balanced games”,Operations Research 30 (1982) 998–1008. ·Zbl 0493.90032 ·doi:10.1287/opre.30.5.998
[15]G. Owen, ”On the core of linear production games”,Mathematical Programming 9 (1975) 358–370. ·Zbl 0318.90060 ·doi:10.1007/BF01681356
[16]L.S. Shaplley, ”On balanced sets and cores”,Naval Research Logistics Quarterly 4 (1967) 453–460. ·doi:10.1002/nav.3800140404
[17]L.S. Shapley and M. Shubik, ”The assignment game I: The core”,International Journal of Game Theory 1 (1972) 111–130. ·Zbl 0236.90078 ·doi:10.1007/BF01753437
[18]M.J. Todd, Private Communication, June 1985.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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