American Institute of Mathematical Sciences

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April  2015, 2(3&4): 207-225. doi: 10.3934/jdg.2015002

Endogenous budget constraints in the assignment game

David Cantala 1, and  Juan Sebastián Pereyra 2,

1. 

Centro de Estudios Económicos, El Colegio de México, Camino al Ajusco 20, Fuentes del Pedregal, 10740 Mexico City, Mexico
2. 

ECARES - Solvay Brussels School of Economics and Management, Université libre de Bruxelles and F.R.S.-FNRS, Ave. F.D. Roosevelt 42, B1050 - Brussels, Belgium

Received  October 2014 Revised  April 2015 Published  November 2015

This paper studies economies with indivisible goods and budget-constrained agents with unit-demand who act as both sellers and buyers. In prior literature on the existence of competitive equilibrium, it is assumed the indispensability of money, which in turn implies that budgets constraints are irrelevant. We introduce a new condition, Money Scarcity (MS), that considers agents' budget constraints and ensures the existence of equilibrium. Moreover, an extended version of Gale's top trading cycles algorithm is presented, and it is shown that under MS this mechanism is strategy-proof. Finally, we prove that this mechanism is the unique mechanism that minimizes money transactions at equilibrium.
Keywords: auction mechanism., indivisibility, endogenous budget constraint, top trading cycles, Assignment game, competitive equilibrium.
Mathematics Subject Classification: Primary: 91B68, 91A80; Secondary: 91B2.
Citation: David Cantala, Juan Sebastián Pereyra. Endogenous budget constraints in the assignment game. Journal of Dynamics and Games, 2015, 2 (3&4) : 207-225. doi: 10.3934/jdg.2015002
References:
[1]

A. Abdulkadiroǧlu and T. Sönmez, House allocation with existing tenants, Journal of Economic Theory, 88 (1999), 233-260.

[2]

A. Abdulkadiroǧlu and T. Sönmez, School choice: A mechanism design approach, The American Economic Review, 93 (2003), 729-747.

[3]

T. Andersson, C. Andersson and A. J. J. Talman, Sets in excess demand in ascending auctions with unit-demand bidders, CentER Discussion Paper, 51 (2010), 1-17.

[4]

C. Beviá, M. Quinzii and J. A. Silva, Buying several indivisible goods, Mathematical Social Sciences, 37 (1999), 1-23. doi: 10.1016/S0165-4896(98)00015-8.

[5]

A. Caplin and J. Leahy, A graph theoretic approach to markets for indivisible goods, Journal of Mathematical Economics, 52 (2014), 112-122. doi: 10.1016/j.jmateco.2014.03.011.

[6]

G. Demange and D. Gale, The strategy structure of two-sided matching markets, Econometrica, 53 (1985), 873-883. doi: 10.2307/1912658.

[7]

G. Demange, D. Gale and M. Sotomayor, Multi-item auctions, Journal of Political Economy, 94 (1986), 863-872. doi: 10.1086/261411.

[8]

S. Fujishige and Z. Yang, Existence of an equilibrium in a general competitive exchange economy with indivisible goods and money, Annals of Economics and Finance, 3 (2002), 135-147.

[9]

P. Hall, On representatives of subsets, Journal of London Mathematical Society, 10 (1935), 26-30.

[10]

Y. Hwang and M. Shih, Equilibrium in a market game, Economic Theory, 31 (2007), 387-392. doi: 10.1007/s00199-006-0098-2.

[11]

O. Kesten, Coalitional strategy-proofness and resource monotonicity for house allocation problems, International Journal of Game Theory, 38 (2009), 17-21. doi: 10.1007/s00182-008-0136-3.

[12]

F. Kojima and P. Pathak, Incentives and stability in large two-sided matching markets, American Economic Review, 99 (2009), 608-627. doi: 10.1257/aer.99.3.608.

[13]

S. Lars-Gunnar, Nash implementation of competitive equilibria in a model with indivisible goods, Econometrica, 59 (1991), 869-877. doi: 10.2307/2938231.

[14]

S. Morimoto and S. Serizawa, Strategy-proofness and efficiency with nonquasi-linear preferences: A characterization of minimum price walrasian rule, Theoretical Economics, 10 (2015), 445-487. doi: 10.3982/TE1470.

[15]

E. Miyagawa, House allocation with transfers, Journal of Economic Theory, 100 (2001), 329-355. doi: 10.1006/jeth.2000.2703.

[16]

M. Quinzii, Core and competitive equilibria with indivisibilities, International Journal of Game Theory, 13 (1984), 41-60. doi: 10.1007/BF01769864.

[17]

H. Saitoh, Existence of positive equilibrium price vectors in indivisible goods markets: A note, Mathematical Social Sciences, 48 (2004), 109-112. doi: 10.1016/j.mathsocsci.2003.12.003.

[18]

L. S. Shapley and H. E. Scarf, On cores and indivisibility, Journal of Mathematical Economics, 1 (1974), 23-37. doi: 10.1016/0304-4068(74)90033-0.

[19]

L. S. Shapley and M. Shubik, The assignment game I: The core, International Journal of Game Theory, 1 (1972), 111-130.

[20]

T. Sönmez and U. Ünver, House allocation with existing tenants: A characterization, Games and Economic Behavior, 69 (2010), 425-445. doi: 10.1016/j.geb.2009.10.010.

[21]

M. Sotomayor, A simultaneous descending bid auction for multiple items and unitary demand, Rev. Bras. Econ., 56 (2002), 497-510. doi: 10.1590/S0034-71402002000300006.

[22]

G. van der Laan, D. Talman and Z. Yang, Existence of an equilibrium in a competitive economy with indivisibilities and money, Journal of Mathematical Economics, 28 (1997), 101-109. doi: 10.1016/S0304-4068(97)83316-2.

[23]

J. Wako, Strong core and competitive equilibria of an exchange market with indivisible goods, International Economic Review, 32 (1991), 843-852. doi: 10.2307/2527037.

[24]

Z. Yang, A competitive market model for indivisible commodities, Economics Letters, 78 (2003), 41-47. doi: 10.1016/S0165-1765(02)00206-9.

show all references

References:
[1]

A. Abdulkadiroǧlu and T. Sönmez, House allocation with existing tenants, Journal of Economic Theory, 88 (1999), 233-260.

[2]

A. Abdulkadiroǧlu and T. Sönmez, School choice: A mechanism design approach, The American Economic Review, 93 (2003), 729-747.

[3]

T. Andersson, C. Andersson and A. J. J. Talman, Sets in excess demand in ascending auctions with unit-demand bidders, CentER Discussion Paper, 51 (2010), 1-17.

[4]

C. Beviá, M. Quinzii and J. A. Silva, Buying several indivisible goods, Mathematical Social Sciences, 37 (1999), 1-23. doi: 10.1016/S0165-4896(98)00015-8.

[5]

A. Caplin and J. Leahy, A graph theoretic approach to markets for indivisible goods, Journal of Mathematical Economics, 52 (2014), 112-122. doi: 10.1016/j.jmateco.2014.03.011.

[6]

G. Demange and D. Gale, The strategy structure of two-sided matching markets, Econometrica, 53 (1985), 873-883. doi: 10.2307/1912658.

[7]

G. Demange, D. Gale and M. Sotomayor, Multi-item auctions, Journal of Political Economy, 94 (1986), 863-872. doi: 10.1086/261411.

[8]

S. Fujishige and Z. Yang, Existence of an equilibrium in a general competitive exchange economy with indivisible goods and money, Annals of Economics and Finance, 3 (2002), 135-147.

[9]

P. Hall, On representatives of subsets, Journal of London Mathematical Society, 10 (1935), 26-30.

[10]

Y. Hwang and M. Shih, Equilibrium in a market game, Economic Theory, 31 (2007), 387-392. doi: 10.1007/s00199-006-0098-2.

[11]

O. Kesten, Coalitional strategy-proofness and resource monotonicity for house allocation problems, International Journal of Game Theory, 38 (2009), 17-21. doi: 10.1007/s00182-008-0136-3.

[12]

F. Kojima and P. Pathak, Incentives and stability in large two-sided matching markets, American Economic Review, 99 (2009), 608-627. doi: 10.1257/aer.99.3.608.

[13]

S. Lars-Gunnar, Nash implementation of competitive equilibria in a model with indivisible goods, Econometrica, 59 (1991), 869-877. doi: 10.2307/2938231.

[14]

S. Morimoto and S. Serizawa, Strategy-proofness and efficiency with nonquasi-linear preferences: A characterization of minimum price walrasian rule, Theoretical Economics, 10 (2015), 445-487. doi: 10.3982/TE1470.

[15]

E. Miyagawa, House allocation with transfers, Journal of Economic Theory, 100 (2001), 329-355. doi: 10.1006/jeth.2000.2703.

[16]

M. Quinzii, Core and competitive equilibria with indivisibilities, International Journal of Game Theory, 13 (1984), 41-60. doi: 10.1007/BF01769864.

[17]

H. Saitoh, Existence of positive equilibrium price vectors in indivisible goods markets: A note, Mathematical Social Sciences, 48 (2004), 109-112. doi: 10.1016/j.mathsocsci.2003.12.003.

[18]

L. S. Shapley and H. E. Scarf, On cores and indivisibility, Journal of Mathematical Economics, 1 (1974), 23-37. doi: 10.1016/0304-4068(74)90033-0.

[19]

L. S. Shapley and M. Shubik, The assignment game I: The core, International Journal of Game Theory, 1 (1972), 111-130.

[20]

T. Sönmez and U. Ünver, House allocation with existing tenants: A characterization, Games and Economic Behavior, 69 (2010), 425-445. doi: 10.1016/j.geb.2009.10.010.

[21]

M. Sotomayor, A simultaneous descending bid auction for multiple items and unitary demand, Rev. Bras. Econ., 56 (2002), 497-510. doi: 10.1590/S0034-71402002000300006.

[22]

G. van der Laan, D. Talman and Z. Yang, Existence of an equilibrium in a competitive economy with indivisibilities and money, Journal of Mathematical Economics, 28 (1997), 101-109. doi: 10.1016/S0304-4068(97)83316-2.

[23]

J. Wako, Strong core and competitive equilibria of an exchange market with indivisible goods, International Economic Review, 32 (1991), 843-852. doi: 10.2307/2527037.

[24]

Z. Yang, A competitive market model for indivisible commodities, Economics Letters, 78 (2003), 41-47. doi: 10.1016/S0165-1765(02)00206-9.

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