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July  2016, 12(3): 931-947. doi: 10.3934/jimo.2016.12.931

## The risk-averse newsvendor game with competition on demand

 1 Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China, China 2 School of Management Science and Engineering, Dongbei University of Finance and Economics, Dalian, 116025, China

Received  July 2014 Revised  April 2015 Published  September 2015

This paper studies the effect of risk-aversion in the competitive newsvendor game. Multiple newsvendors with risk-averse preferences face a random demand and the demand is allocated proportionally to their inventory levels. Each newsvendor aims to maximize his expected utility instead of his expected profit. Assuming a general form of risk-averse utility function, we prove that there exists a pure Nash equilibrium in this game, and it is also unique under certain conditions. We find that the order quantity of each newsvendor is decreasing in the degree of risk-aversion and increasing in the initial wealth. Newsvendors with moderate preferences of risk-aversion make more profits compared with the risk-neutral situation. We also discuss the joint effect of risk-aversion and competition. If the effect of risk-aversion is strong enough to dominate the effect of competition, the total inventory level under competition will be lower than that under centralized decision-making.
Citation: Yuwei Shen, Jinxing Xie, Tingting Li. The risk-averse newsvendor game with competition on demand. Journal of Industrial and Management Optimization, 2016, 12 (3) : 931-947. doi: 10.3934/jimo.2016.12.931
##### References:
 [1] K. J. Arrow, The theory of risk aversion, in Essays in the Theory of Risk-Bearing (ed. K. J. Arrow), Markham Publishing Company, 1971, 90-120. [2] V. Agrawal and S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management, 2 (2000), 410-423. doi: 10.1287/msom.2.4.410.12339. [3] A. O. Brown and C. S. Tang, The impact of alternative performance measures on single-period inventory policy, Journal of Industrial and Management Optimization, 2 (2006), 297-318. doi: 10.3934/jimo.2006.2.297. [4] P. L. Brockett and L. L. Golden, A class of utility functions containing all the common utility functions, Management Science, 33 (1987), 955-964. doi: 10.1287/mnsc.33.8.955. [5] G. P. Cachon, Supply chain coordination with contracts, in Handbooks in Operations Research and Management Science, Vol. 11 (eds. S. C. Graves and A. G. de Kok), Elsevier, 2003, 227-339. doi: 10.1016/S0927-0507(03)11006-7. [6] X. Chen, M. Sim, D. S. Levi and P. Sun, Risk aversion in inventory management, Operations Research, 55 (2007), 828-842. doi: 10.1287/opre.1070.0429. [7] L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 41 (1995), 786-794. doi: 10.1287/mnsc.41.5.786. [8] D. Fudenberg and J. Tirole, Game Theory, MIT Press, London, 1991. [9] I. Friend and M. E. Blume, The demand for risky assets, The American Economic Review, 65 (1975), 900-922. [10] A. Gasparro and J. Beckerman, Whole foods again lowers sales projections; specialty supermarket reports 6 Wall Street Journal (Online), 30 Jul 2014. Available from: http://search.proquest.com/docview/1549531450?accountid=14426. [11] K. Girotra and S. Netessine, How to build risk into your business model, Harvard Business Review, 89 (2011), 100-105. [12] C. A. Holt and S. K. Laury, Risk aversion and incentive effects, American Economic Review, 92 (2002), 1644-1655. doi: 10.1257/000282802762024700. [13] J. R. Hagerty, 3M begins untangling its 'hairballs' - making plastic hooks is harder than it seems; streamlining a four-state, 1,300-mile supply chain, Wall Street Journal, 17 May 2012. Available from: http://search.proquest.com/docview/1013910634?accountid=14426. [14] K. B. Hamal and J. R. Anderson, A note on decreasing absolute risk aversion among farmers in Nepal, Australian Journal of Agricultural Economics, 26 (1982), 220-225. doi: 10.1111/j.1467-8489.1982.tb00414.x. [15] B. Keren and J. S. Pliskin, A benchmark solution for the risk-averse newsvendor problem, European Journal of Operational Research, 174 (2006), 1643-1650. doi: 10.1016/j.ejor.2005.03.047. [16] M. Khouja, The single-period (news-vendor) problem: literature review and suggestions for future research, Omega-The International Journal of Management Science, 27 (1999), 537-553. doi: 10.1016/S0305-0483(99)00017-1. [17] S. A. Lippman and K. F. McCardle, The competitive newsboy, Operations Research, 45 (1997), 54-65. doi: 10.1287/opre.45.1.54. [18] W. Liu, S. J. Song and C. Wu, Impact of loss aversion on the newsvendor game with product substitution, International Journal of Production Economics, 141 (2013), 352-359. doi: 10.1016/j.ijpe.2012.08.017. [19] J. W. Pratt, Risk aversion in the small and in the large, Econometrica, 32 (1964), 122-136. [20] M. Parlar, Game theoretic analysis of the substitutable product inventory problem with random demand, Naval Research Logistics, 35 (1988), 397-409. doi: 10.1002/1520-6750(198806)35:3<397::AID-NAV3220350308>3.0.CO;2-Z. [21] Y. Qin, R. X. Wang, A. J. Vakhria, Y. W. Chen and M. M. H. Seref, The newsvendor problem: review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374. doi: 10.1016/j.ejor.2010.11.024. [22] A. Saha, C. R. Shumway and H. Talpaz, Joint estimation of risk preference structure and technology using expo-power utility, American Journal of Agricultural Economics, 76 (1994), 173-184. doi: 10.2307/1243619. [23] M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 404-420. doi: 10.1287/mnsc.46.3.404.12070. [24] F. W. Siegel and J. P. Hoban, Relative risk aversion revisited, The Review of Economics and Statistics, 64 (1982), 481-487. doi: 10.2307/1925947. [25] T. L. Urban, Inventory models with inventory-level-dependent demand: A comprehensive review and unifying theory, European Journal of Operational Research, 162 (2005), 792-804. doi: 10.1016/j.ejor.2003.08.065. [26] H. B. Wolfe, A model for control of style merchandise, Industrial Management Review, 9 (1968), 69-82. [27] C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452. doi: 10.1016/j.ijpe.2009.12.007. [28] C. X. Wang, S. Webster and N. C. Suresh, Would a risk-averse newsvendor order less at a higher selling price?, European Journal of Operational Research, 196 (2009), 544-553. doi: 10.1016/j.ejor.2008.04.002. [29] M. Wik, T. A. Kebede, O. Bergland and S. T. Holden, On the measurement of risk aversion from experimental data, Applied Economics, 36 (2004), 2443-2451. doi: 10.1080/0003684042000280580. [30] M. Wu, S. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, International Journal of Production Economics, 156 (2014), 13-23. doi: 10.1016/j.ijpe.2014.05.009. [31] Y. Z. Wang and Y. Gerchak, Supply chain coordination when demand is shelf-space dependent, Manufacturing & Service Operations Management, 3 (2001), 82-87. doi: 10.1287/msom.3.1.82.9998. [32] G. Xie, W. Y. Yue and S. Y. Wang, Optimal selection of cleaner products in a green supply chain with risk aversion, Journal of Industrial and Management Optimization, 11 (2015), 515-528. doi: 10.3934/jimo.2015.11.515. [33] T. J. Xiao and D. Q. Yang, Price and service competition of supply chains with risk-averse retailers under demand uncertainty, International Journal of Production Economics, 114 (2008), 187-200. doi: 10.1016/j.ijpe.2008.01.006.

show all references

##### References:
 [1] K. J. Arrow, The theory of risk aversion, in Essays in the Theory of Risk-Bearing (ed. K. J. Arrow), Markham Publishing Company, 1971, 90-120. [2] V. Agrawal and S. Seshadri, Impact of uncertainty and risk aversion on price and order quantity in the newsvendor problem, Manufacturing & Service Operations Management, 2 (2000), 410-423. doi: 10.1287/msom.2.4.410.12339. [3] A. O. Brown and C. S. Tang, The impact of alternative performance measures on single-period inventory policy, Journal of Industrial and Management Optimization, 2 (2006), 297-318. doi: 10.3934/jimo.2006.2.297. [4] P. L. Brockett and L. L. Golden, A class of utility functions containing all the common utility functions, Management Science, 33 (1987), 955-964. doi: 10.1287/mnsc.33.8.955. [5] G. P. Cachon, Supply chain coordination with contracts, in Handbooks in Operations Research and Management Science, Vol. 11 (eds. S. C. Graves and A. G. de Kok), Elsevier, 2003, 227-339. doi: 10.1016/S0927-0507(03)11006-7. [6] X. Chen, M. Sim, D. S. Levi and P. Sun, Risk aversion in inventory management, Operations Research, 55 (2007), 828-842. doi: 10.1287/opre.1070.0429. [7] L. Eeckhoudt, C. Gollier and H. Schlesinger, The risk-averse (and prudent) newsboy, Management Science, 41 (1995), 786-794. doi: 10.1287/mnsc.41.5.786. [8] D. Fudenberg and J. Tirole, Game Theory, MIT Press, London, 1991. [9] I. Friend and M. E. Blume, The demand for risky assets, The American Economic Review, 65 (1975), 900-922. [10] A. Gasparro and J. Beckerman, Whole foods again lowers sales projections; specialty supermarket reports 6 Wall Street Journal (Online), 30 Jul 2014. Available from: http://search.proquest.com/docview/1549531450?accountid=14426. [11] K. Girotra and S. Netessine, How to build risk into your business model, Harvard Business Review, 89 (2011), 100-105. [12] C. A. Holt and S. K. Laury, Risk aversion and incentive effects, American Economic Review, 92 (2002), 1644-1655. doi: 10.1257/000282802762024700. [13] J. R. Hagerty, 3M begins untangling its 'hairballs' - making plastic hooks is harder than it seems; streamlining a four-state, 1,300-mile supply chain, Wall Street Journal, 17 May 2012. Available from: http://search.proquest.com/docview/1013910634?accountid=14426. [14] K. B. Hamal and J. R. Anderson, A note on decreasing absolute risk aversion among farmers in Nepal, Australian Journal of Agricultural Economics, 26 (1982), 220-225. doi: 10.1111/j.1467-8489.1982.tb00414.x. [15] B. Keren and J. S. Pliskin, A benchmark solution for the risk-averse newsvendor problem, European Journal of Operational Research, 174 (2006), 1643-1650. doi: 10.1016/j.ejor.2005.03.047. [16] M. Khouja, The single-period (news-vendor) problem: literature review and suggestions for future research, Omega-The International Journal of Management Science, 27 (1999), 537-553. doi: 10.1016/S0305-0483(99)00017-1. [17] S. A. Lippman and K. F. McCardle, The competitive newsboy, Operations Research, 45 (1997), 54-65. doi: 10.1287/opre.45.1.54. [18] W. Liu, S. J. Song and C. Wu, Impact of loss aversion on the newsvendor game with product substitution, International Journal of Production Economics, 141 (2013), 352-359. doi: 10.1016/j.ijpe.2012.08.017. [19] J. W. Pratt, Risk aversion in the small and in the large, Econometrica, 32 (1964), 122-136. [20] M. Parlar, Game theoretic analysis of the substitutable product inventory problem with random demand, Naval Research Logistics, 35 (1988), 397-409. doi: 10.1002/1520-6750(198806)35:3<397::AID-NAV3220350308>3.0.CO;2-Z. [21] Y. Qin, R. X. Wang, A. J. Vakhria, Y. W. Chen and M. M. H. Seref, The newsvendor problem: review and directions for future research, European Journal of Operational Research, 213 (2011), 361-374. doi: 10.1016/j.ejor.2010.11.024. [22] A. Saha, C. R. Shumway and H. Talpaz, Joint estimation of risk preference structure and technology using expo-power utility, American Journal of Agricultural Economics, 76 (1994), 173-184. doi: 10.2307/1243619. [23] M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Management Science, 46 (2000), 404-420. doi: 10.1287/mnsc.46.3.404.12070. [24] F. W. Siegel and J. P. Hoban, Relative risk aversion revisited, The Review of Economics and Statistics, 64 (1982), 481-487. doi: 10.2307/1925947. [25] T. L. Urban, Inventory models with inventory-level-dependent demand: A comprehensive review and unifying theory, European Journal of Operational Research, 162 (2005), 792-804. doi: 10.1016/j.ejor.2003.08.065. [26] H. B. Wolfe, A model for control of style merchandise, Industrial Management Review, 9 (1968), 69-82. [27] C. X. Wang, The loss-averse newsvendor game, International Journal of Production Economics, 124 (2010), 448-452. doi: 10.1016/j.ijpe.2009.12.007. [28] C. X. Wang, S. Webster and N. C. Suresh, Would a risk-averse newsvendor order less at a higher selling price?, European Journal of Operational Research, 196 (2009), 544-553. doi: 10.1016/j.ejor.2008.04.002. [29] M. Wik, T. A. Kebede, O. Bergland and S. T. Holden, On the measurement of risk aversion from experimental data, Applied Economics, 36 (2004), 2443-2451. doi: 10.1080/0003684042000280580. [30] M. Wu, S. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, International Journal of Production Economics, 156 (2014), 13-23. doi: 10.1016/j.ijpe.2014.05.009. [31] Y. Z. Wang and Y. Gerchak, Supply chain coordination when demand is shelf-space dependent, Manufacturing & Service Operations Management, 3 (2001), 82-87. doi: 10.1287/msom.3.1.82.9998. [32] G. Xie, W. Y. Yue and S. Y. Wang, Optimal selection of cleaner products in a green supply chain with risk aversion, Journal of Industrial and Management Optimization, 11 (2015), 515-528. doi: 10.3934/jimo.2015.11.515. [33] T. J. Xiao and D. Q. Yang, Price and service competition of supply chains with risk-averse retailers under demand uncertainty, International Journal of Production Economics, 114 (2008), 187-200. doi: 10.1016/j.ijpe.2008.01.006.
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