March & April  2015, 2(3&4): 227-256. doi: 10.3934/jdg.2015003

A survey on assignment markets

1. 

Departament de Matemàtica Econòmica, Financera i Actuarial, Universitat de Barcelona, Av. Diagonal, 690, 08034 Barcelona, Spain, Spain

Received  December 2014 Revised  January 2015 Published  November 2015

The assignment game is a two-sided market, say buyers and sellers, where demand and supply are unitary and utility is transferable by means of prices. This survey is structured in three parts: a first part, from the introduction of the assignment game by Shapley and Shubik (1972) until the publication of the book of Roth and Sotomayor (1990), focused on the notion of core; the subsequent investigations that broaden the scope to other notions of solution for these markets; and its extensions to assignment markets with multiple sides or multiple partnership. These extended two-sided assignment markets, that allow for multiple partnership, better represent the situation in a labour market or an auction.
Citation: Marina Núñez, Carles Rafels. A survey on assignment markets. Journal of Dynamics & Games, 2015, 2 (3&4) : 227-256. doi: 10.3934/jdg.2015003
References:
[1]

R. P. Arribillaga, J. Massó and A. Neme, On the structure of cooperative and competitive solutions of a generalized assignment game,, Journal of Applied Mathematics, (2014). doi: 10.1155/2014/190614. Google Scholar

[2]

L. M. Ausubel and P. Milgrom, Ascending auctions with package bidding,, Frontiers of Theoretical Economics, 1 (2002). doi: 10.2202/1534-5955.1019. Google Scholar

[3]

L. M. Ausubel and P. Milgrom, The lovely but lonely Vickrey auction,, In Combinatorial Auctions (ed. P. Cramton, (2005). Google Scholar

[4]

L. M. Ausubel and P. Milgrom, Ascending Proxy Auctions,, In Combinatorial Auctions (ed. P. Cramton, (2005). Google Scholar

[5]

M. L. Balinski and D. Gale, On the core of the assignment game,, In Functional Analysis, (1990), 274. Google Scholar

[6]

S. Bikhchandani and J. M. Ostroy, The package assignment model,, Journal of Economic Theory, 107 (2002), 377. doi: 10.1006/jeth.2001.2957. Google Scholar

[7]

G. Birkhoff, Tres observaciones sobre el álgebra lineal,, Revista Universidad Nacional de Tucuman, 5 (1946), 147. Google Scholar

[8]

R. van den Brink and M. Pintér, On axiomatizations of the Shapley value for assignment games,, Journal of Mathematical Economics, 60 (2015), 110. doi: 10.1016/j.jmateco.2015.06.016. Google Scholar

[9]

E. Camiña, A generalized assignment game,, Mathematical Social Sciences, 52 (2006), 152. doi: 10.1016/j.mathsocsci.2006.06.003. Google Scholar

[10]

V. P. Crawford and E. M. Knoer, Job matching with heterogeneous firms and workers,, Econometrica, 49 (1981), 437. doi: 10.2307/1913320. Google Scholar

[11]

M. Davis and M. Maschler, The kernel of a cooperative game,, Naval Research Logistics Quarterly 12 (1965), 12 (1965), 223. doi: 10.1002/nav.3800120303. Google Scholar

[12]

G. Demange, Strategyproofness in the Assignment Market Game,, Laboratoire d'Econometrie de l'Ecole Polytechnique, (1982). Google Scholar

[13]

G. Demange and D. Gale, The strategy structure of two-sided matching markets,, Econometrica, 53 (1985), 873. doi: 10.2307/1912658. Google Scholar

[14]

G. Demange, D. Gale and M. Sotomayor, Multi-item auctions,, Journal of Political Economy, 94 (1986), 863. doi: 10.1086/261411. Google Scholar

[15]

T. S. H. Driessen, A note on the inclusion of the kernel in the core of the bilateral assignment game,, International Journal of Game Theory, 27 (1998), 301. doi: 10.1007/s001820050073. Google Scholar

[16]

A. Fagebaume, D. Gale and M. Sotomayor, A note on the multiple-partners assignment game,, Journal of Mathematical Economics, 46 (2010), 388. doi: 10.1016/j.jmateco.2009.06.014. Google Scholar

[17]

D. Gale, The Theory of Linear Economic Models,, McGraw-Hill, (1960). Google Scholar

[18]

D. Gale and L. S. Shapley, College Admission and the Stability of Marriage,, American Mathematical Monthly, 69 (1962), 9. doi: 10.2307/2312726. Google Scholar

[19]

H. Hamers, F. Klijn, T. Solymosi, S. Tijs and J. P. Villar, Assignment games satisfy the CoMa-property,, Games and Economic Behavior, 38 (2002), 231. Google Scholar

[20]

M. Hoffmann and P. Sudhölter, The Shapley value of exact assignment games,, International Journal of Game Theory 35 (2007), 35 (2007), 557. doi: 10.1007/s00182-006-0068-8. Google Scholar

[21]

J. M. Izquierdo, M. Núñez and C. Rafels, A simple procedure to obtain the extreme core allocations of an assignment market,, International Journal of Game Theory, 36 (2007), 17. doi: 10.1007/s00182-007-0091-4. Google Scholar

[22]

D. Jaume, J. Massó and A. Neme, The multiple-partners assignment game with heterogeneous sales and multi-unit demands: competitive equilibria,, Mathematical Methods of Operations Research, 76 (2012), 161. doi: 10.1007/s00186-012-0395-4. Google Scholar

[23]

M. Kaneko, On the core and competitive equilibria of a market with indivisible goods,, Naval Research Logistics Quarterly, 23 (1976), 321. doi: 10.1002/nav.3800230214. Google Scholar

[24]

M. Kaneko, The central assignment game and the assignment markets,, Journal of Mathematical Economics, 10 (1982), 205. doi: 10.1016/0304-4068(82)90038-6. Google Scholar

[25]

M. Kaneko and M. Wooders, Cores of partitioning games,, Mathematical Social Sciences, 3 (1982), 313. doi: 10.1016/0165-4896(82)90015-4. Google Scholar

[26]

T. C. Koopmans and M. Beckmann, Assignment Problems and the Location of Economic Activities,, Econometrica, 25 (1957), 53. doi: 10.2307/1907742. Google Scholar

[27]

H. B. Leonard, Elicitation of honest preferences for the assignment of individuals to positions,, Journal of Political Economy, 91 (1983), 461. doi: 10.1086/261158. Google Scholar

[28]

F. Llerena and M. Núñez, A geometric characterization of the nucleolus of the assignment game,, Economics Bulletin 31 (2011), 31 (2011), 3275. Google Scholar

[29]

F. Llerena, M. Núñez and C. Rafels, An axiomatization of the nucleolus of the assignment game,, International Journal of Game Theory, 44 (2015), 1. doi: 10.1007/s00182-014-0416-z. Google Scholar

[30]

W. F. Lucas, A game with no solution,, Bulletin of the American Mathematical Society, 74 (1968), 237. doi: 10.1090/S0002-9904-1968-11901-9. Google Scholar

[31]

W. F. Lucas, Core theory for multiple-sided assignment games,, Duke Mathematical Journal, 81 (1995), 55. doi: 10.1215/S0012-7094-95-08106-X. Google Scholar

[32]

F. J. Martínez de Albéniz, M. Núñez and C. Rafels, Assignment markets with the same core,, Games and Economic Behavior, 73 (2011), 553. doi: 10.1016/j.geb.2011.02.011. Google Scholar

[33]

F. J. Martínez de Albéniz, C. Rafels and N. Ybern, On the nucleolus of 2x2 assignment games,, Economics Bulletin 3 (2013), 3 (2013), 2938. Google Scholar

[34]

F. J. Martínez de Albéniz, C. Rafels and N. Ybern, A procedure to compute the nucleolus of the assignment game,, Operations Research Letters 41 (2013), 41 (2013), 675. Google Scholar

[35]

F. J. Martínez de Albéniz and C. Rafels, Cooperative assignment games with the inverse Monge property,, Discrete Applied Mathematics, 162 (2014), 42. doi: 10.1016/j.dam.2013.08.027. Google Scholar

[36]

M. Maschler, B. Peleg and L. S. Shapley, Geometric properties of the kernel, nucleolus and related solution concepts,, Mathematics of Operations Research, 4 (1979), 303. doi: 10.1287/moor.4.4.303. Google Scholar

[37]

J. Massó and A. Neme, On cooperative solutions of a generalized assignment game: Limit theorems to the set of competitive equilibria,, Journal of Economic Theory, 154 (2014), 187. doi: 10.1016/j.jet.2014.09.016. Google Scholar

[38]

J. P. Mo, Entry and structures of interest groups in assignment games,, Journal of Economic Theory, 46 (1988), 66. doi: 10.1016/0022-0531(88)90150-0. Google Scholar

[39]

M. Núñez, A note on the nucleolus and the kernel of the assignment game,, International Journal of Game Theory, 33 (2004), 55. doi: 10.1007/s001820400184. Google Scholar

[40]

M. Núñez and C. Rafels, Buyer-seller exactness in the assignment game,, International Journal of Game Theory, 31 (2002), 423. doi: 10.1007/s001820300128. Google Scholar

[41]

M. Núñez and C. Rafels, The assignment game: the $\tau$-value,, International Journal of Game Theory, 31 (2002), 411. doi: 10.1007/s001820300127. Google Scholar

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M. Núñez and C. Rafels, Characterization of the extreme core allocations of the assignment game,, Games and Economic Behavior, 44 (2003), 311. doi: 10.1016/S0899-8256(03)00054-X. Google Scholar

[43]

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[44]

M. Núñez and C. Rafels, A glove-market partitioned matrix related to the assignment game,, Games and Economic Behavior, 67 (2009), 598. doi: 10.1016/j.geb.2009.03.014. Google Scholar

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M. Núñez and C. Rafels, Von Neumann-Morgenstern solutions in the assignment market,, Journal of Economic Theory, 148 (2013), 1282. doi: 10.1016/j.jet.2012.10.002. Google Scholar

[46]

M. Núñez and T. Solymosi, Lexicographic allocations and extreme core payoffs: The case of assignment games,, Corvinus Economics Working Papers, (2014). Google Scholar

[47]

G. Owen, The Assignment Game: The Reduced Game,, Annales d'Économie et de Statistique, (1992), 71. Google Scholar

[48]

B. Peleg, On the reduced game property and its converse,, International Journal of Game Theory, 15 (1986), 187. doi: 10.1007/BF01769258. Google Scholar

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D. Pérez-Castrillo and M. Sotomayor, A simple selling and buying procedure,, Journal of Economic Theory, 103 (2002), 461. doi: 10.1006/jeth.2000.2783. Google Scholar

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D. Pérez-Castrillo and M. Sotomayor, On the manipulability of competitive equilibrium rules in many-to-many buyer-seller markets,, Working paper, (2014). Google Scholar

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show all references

References:
[1]

R. P. Arribillaga, J. Massó and A. Neme, On the structure of cooperative and competitive solutions of a generalized assignment game,, Journal of Applied Mathematics, (2014). doi: 10.1155/2014/190614. Google Scholar

[2]

L. M. Ausubel and P. Milgrom, Ascending auctions with package bidding,, Frontiers of Theoretical Economics, 1 (2002). doi: 10.2202/1534-5955.1019. Google Scholar

[3]

L. M. Ausubel and P. Milgrom, The lovely but lonely Vickrey auction,, In Combinatorial Auctions (ed. P. Cramton, (2005). Google Scholar

[4]

L. M. Ausubel and P. Milgrom, Ascending Proxy Auctions,, In Combinatorial Auctions (ed. P. Cramton, (2005). Google Scholar

[5]

M. L. Balinski and D. Gale, On the core of the assignment game,, In Functional Analysis, (1990), 274. Google Scholar

[6]

S. Bikhchandani and J. M. Ostroy, The package assignment model,, Journal of Economic Theory, 107 (2002), 377. doi: 10.1006/jeth.2001.2957. Google Scholar

[7]

G. Birkhoff, Tres observaciones sobre el álgebra lineal,, Revista Universidad Nacional de Tucuman, 5 (1946), 147. Google Scholar

[8]

R. van den Brink and M. Pintér, On axiomatizations of the Shapley value for assignment games,, Journal of Mathematical Economics, 60 (2015), 110. doi: 10.1016/j.jmateco.2015.06.016. Google Scholar

[9]

E. Camiña, A generalized assignment game,, Mathematical Social Sciences, 52 (2006), 152. doi: 10.1016/j.mathsocsci.2006.06.003. Google Scholar

[10]

V. P. Crawford and E. M. Knoer, Job matching with heterogeneous firms and workers,, Econometrica, 49 (1981), 437. doi: 10.2307/1913320. Google Scholar

[11]

M. Davis and M. Maschler, The kernel of a cooperative game,, Naval Research Logistics Quarterly 12 (1965), 12 (1965), 223. doi: 10.1002/nav.3800120303. Google Scholar

[12]

G. Demange, Strategyproofness in the Assignment Market Game,, Laboratoire d'Econometrie de l'Ecole Polytechnique, (1982). Google Scholar

[13]

G. Demange and D. Gale, The strategy structure of two-sided matching markets,, Econometrica, 53 (1985), 873. doi: 10.2307/1912658. Google Scholar

[14]

G. Demange, D. Gale and M. Sotomayor, Multi-item auctions,, Journal of Political Economy, 94 (1986), 863. doi: 10.1086/261411. Google Scholar

[15]

T. S. H. Driessen, A note on the inclusion of the kernel in the core of the bilateral assignment game,, International Journal of Game Theory, 27 (1998), 301. doi: 10.1007/s001820050073. Google Scholar

[16]

A. Fagebaume, D. Gale and M. Sotomayor, A note on the multiple-partners assignment game,, Journal of Mathematical Economics, 46 (2010), 388. doi: 10.1016/j.jmateco.2009.06.014. Google Scholar

[17]

D. Gale, The Theory of Linear Economic Models,, McGraw-Hill, (1960). Google Scholar

[18]

D. Gale and L. S. Shapley, College Admission and the Stability of Marriage,, American Mathematical Monthly, 69 (1962), 9. doi: 10.2307/2312726. Google Scholar

[19]

H. Hamers, F. Klijn, T. Solymosi, S. Tijs and J. P. Villar, Assignment games satisfy the CoMa-property,, Games and Economic Behavior, 38 (2002), 231. Google Scholar

[20]

M. Hoffmann and P. Sudhölter, The Shapley value of exact assignment games,, International Journal of Game Theory 35 (2007), 35 (2007), 557. doi: 10.1007/s00182-006-0068-8. Google Scholar

[21]

J. M. Izquierdo, M. Núñez and C. Rafels, A simple procedure to obtain the extreme core allocations of an assignment market,, International Journal of Game Theory, 36 (2007), 17. doi: 10.1007/s00182-007-0091-4. Google Scholar

[22]

D. Jaume, J. Massó and A. Neme, The multiple-partners assignment game with heterogeneous sales and multi-unit demands: competitive equilibria,, Mathematical Methods of Operations Research, 76 (2012), 161. doi: 10.1007/s00186-012-0395-4. Google Scholar

[23]

M. Kaneko, On the core and competitive equilibria of a market with indivisible goods,, Naval Research Logistics Quarterly, 23 (1976), 321. doi: 10.1002/nav.3800230214. Google Scholar

[24]

M. Kaneko, The central assignment game and the assignment markets,, Journal of Mathematical Economics, 10 (1982), 205. doi: 10.1016/0304-4068(82)90038-6. Google Scholar

[25]

M. Kaneko and M. Wooders, Cores of partitioning games,, Mathematical Social Sciences, 3 (1982), 313. doi: 10.1016/0165-4896(82)90015-4. Google Scholar

[26]

T. C. Koopmans and M. Beckmann, Assignment Problems and the Location of Economic Activities,, Econometrica, 25 (1957), 53. doi: 10.2307/1907742. Google Scholar

[27]

H. B. Leonard, Elicitation of honest preferences for the assignment of individuals to positions,, Journal of Political Economy, 91 (1983), 461. doi: 10.1086/261158. Google Scholar

[28]

F. Llerena and M. Núñez, A geometric characterization of the nucleolus of the assignment game,, Economics Bulletin 31 (2011), 31 (2011), 3275. Google Scholar

[29]

F. Llerena, M. Núñez and C. Rafels, An axiomatization of the nucleolus of the assignment game,, International Journal of Game Theory, 44 (2015), 1. doi: 10.1007/s00182-014-0416-z. Google Scholar

[30]

W. F. Lucas, A game with no solution,, Bulletin of the American Mathematical Society, 74 (1968), 237. doi: 10.1090/S0002-9904-1968-11901-9. Google Scholar

[31]

W. F. Lucas, Core theory for multiple-sided assignment games,, Duke Mathematical Journal, 81 (1995), 55. doi: 10.1215/S0012-7094-95-08106-X. Google Scholar

[32]

F. J. Martínez de Albéniz, M. Núñez and C. Rafels, Assignment markets with the same core,, Games and Economic Behavior, 73 (2011), 553. doi: 10.1016/j.geb.2011.02.011. Google Scholar

[33]

F. J. Martínez de Albéniz, C. Rafels and N. Ybern, On the nucleolus of 2x2 assignment games,, Economics Bulletin 3 (2013), 3 (2013), 2938. Google Scholar

[34]

F. J. Martínez de Albéniz, C. Rafels and N. Ybern, A procedure to compute the nucleolus of the assignment game,, Operations Research Letters 41 (2013), 41 (2013), 675. Google Scholar

[35]

F. J. Martínez de Albéniz and C. Rafels, Cooperative assignment games with the inverse Monge property,, Discrete Applied Mathematics, 162 (2014), 42. doi: 10.1016/j.dam.2013.08.027. Google Scholar

[36]

M. Maschler, B. Peleg and L. S. Shapley, Geometric properties of the kernel, nucleolus and related solution concepts,, Mathematics of Operations Research, 4 (1979), 303. doi: 10.1287/moor.4.4.303. Google Scholar

[37]

J. Massó and A. Neme, On cooperative solutions of a generalized assignment game: Limit theorems to the set of competitive equilibria,, Journal of Economic Theory, 154 (2014), 187. doi: 10.1016/j.jet.2014.09.016. Google Scholar

[38]

J. P. Mo, Entry and structures of interest groups in assignment games,, Journal of Economic Theory, 46 (1988), 66. doi: 10.1016/0022-0531(88)90150-0. Google Scholar

[39]

M. Núñez, A note on the nucleolus and the kernel of the assignment game,, International Journal of Game Theory, 33 (2004), 55. doi: 10.1007/s001820400184. Google Scholar

[40]

M. Núñez and C. Rafels, Buyer-seller exactness in the assignment game,, International Journal of Game Theory, 31 (2002), 423. doi: 10.1007/s001820300128. Google Scholar

[41]

M. Núñez and C. Rafels, The assignment game: the $\tau$-value,, International Journal of Game Theory, 31 (2002), 411. doi: 10.1007/s001820300127. Google Scholar

[42]

M. Núñez and C. Rafels, Characterization of the extreme core allocations of the assignment game,, Games and Economic Behavior, 44 (2003), 311. doi: 10.1016/S0899-8256(03)00054-X. Google Scholar

[43]

M. Núñez and C. Rafels, On the dimension of the core of the assignment game,, Games and Economic Behavior, 64 (2008), 290. doi: 10.1016/j.geb.2008.01.004. Google Scholar

[44]

M. Núñez and C. Rafels, A glove-market partitioned matrix related to the assignment game,, Games and Economic Behavior, 67 (2009), 598. doi: 10.1016/j.geb.2009.03.014. Google Scholar

[45]

M. Núñez and C. Rafels, Von Neumann-Morgenstern solutions in the assignment market,, Journal of Economic Theory, 148 (2013), 1282. doi: 10.1016/j.jet.2012.10.002. Google Scholar

[46]

M. Núñez and T. Solymosi, Lexicographic allocations and extreme core payoffs: The case of assignment games,, Corvinus Economics Working Papers, (2014). Google Scholar

[47]

G. Owen, The Assignment Game: The Reduced Game,, Annales d'Économie et de Statistique, (1992), 71. Google Scholar

[48]

B. Peleg, On the reduced game property and its converse,, International Journal of Game Theory, 15 (1986), 187. doi: 10.1007/BF01769258. Google Scholar

[49]

D. Pérez-Castrillo and M. Sotomayor, A simple selling and buying procedure,, Journal of Economic Theory, 103 (2002), 461. doi: 10.1006/jeth.2000.2783. Google Scholar

[50]

D. Pérez-Castrillo and M. Sotomayor, On the manipulability of competitive equilibrium rules in many-to-many buyer-seller markets,, Working paper, (2014). Google Scholar

[51]

T. Quint, Characterization of cores of assignment games,, International Journal of Game Theory, 19 (1991), 413. doi: 10.1007/BF01766430. Google Scholar

[52]

T. Quint, The core of an m-sided assignment game,, Games and Economic Behavior, 3 (1991), 487. doi: 10.1016/0899-8256(91)90017-9. Google Scholar

[53]

S. C. Rochford, Symmetrically Pairwise-Bargained Allocations in an Assignment Market,, Journal of Economic Theory, 34 (1984), 262. doi: 10.1016/0022-0531(84)90144-3. Google Scholar

[54]

A. E. Roth and M. Sotomayor, Interior points in the core of two-sided matching problems,, Journal of Economic Theory, 45 (1988), 85. doi: 10.1016/0022-0531(88)90255-4. Google Scholar

[55]

A. E. Roth and M. Sotomayor, Two-sided Matching,, Econometric Society Monograph 18, (1990). doi: 10.1017/CCOL052139015X. Google Scholar

[56]

J. Sánchez-Soriano, M. A. López and I. García-Jurado, On the core of transportation games,, Mathematical Social Sciences, 41 (2001), 215. doi: 10.1016/S0165-4896(00)00057-3. Google Scholar

[57]

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