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doi: 10.3934/jimo.2021086

The impacts of retailers' regret aversion on a random multi-period supply chain network

1. 

Department of Management Science and Engineering, Qingdao University, Qingdao, Shandong 266071, China

2. 

Hong Kong Statistical Society, Hong Kong, China

3. 

School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

* Corresponding author

Received  August 2020 Revised  March 2021 Published  April 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China under grant 71901129 71771129

Most current studies on the equilibrium decision-makings of a supply chain network (SCN) consider only completely rational retailers who always try to maximize their expected profits under the situations that demands are random. However, many evidences show that a retailer wants to choose the decision-making involving random demand which provides him with minimum regret. In this paper, we consider the impacts of retailers' anticipated regret aversion and experiential regret aversion on the equilibrium decision-makings of a random SCN with multiple production periods. Due to the random demand, the decision-makings of retailers are influenced not only by their anticipated regret, but also by their experiential regret after they have encountered bad or good experiences during past periods. The equilibrium conditions of the model are established. A numerical example is solved to illustrate the benefit of retailers obtained by considering their regret-averse behaviors. Moreover, it is found that retailers should consider their anticipated regret aversion or experiential regret aversion according to different situations of the demand markets.

Citation: Yan Zhou, Chi Kin Chan, Kar Hung Wong. The impacts of retailers' regret aversion on a random multi-period supply chain network. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021086
References:
[1]

M. T. Ahmed and C. Kwon, Optimal contract-sizing in online display advertising for publishers with regret considerations, Omega, 42 (2014), 201-212.  doi: 10.1016/j.omega.2013.06.001.  Google Scholar

[2]

N. Ayvaz-CavdarogluS. Kachani and C. Maglaras, Revenue management with minimax regret negotiations, Omega, 63 (2016), 12-22.  doi: 10.1016/j.omega.2015.09.009.  Google Scholar

[3]

Q. Bai and F. Meng, Impact of risk aversion on two-echelon supply chain systems with carbon emission reduction constraints, J. Ind. Manag. Optim., 16 (2020), 1943-1965.  doi: 10.3934/jimo.2019037.  Google Scholar

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D. Bell, Regret in decision making under uncertainty, Operations Research, 30 (1982), 961-981.   Google Scholar

[5]

N. CamilleG. CoricelliJ. SalletP. Pradat-DiehlJ.-R. Duhamel and A. Sirigu, The involvement of the orbitofrontal cortex in the experience of regret, Science, 304 (2004), 1167-1170.  doi: 10.1126/science.1094550.  Google Scholar

[6]

C. K. ChanY. Zhou and K. H. Wong, A dynamic equilibrium model of the oligopolistic closed-loop supply chain network under uncertain and time-dependent demands, Transportation Research Part E, 118 (2018), 325-354.  doi: 10.1016/j.tre.2018.07.008.  Google Scholar

[7]

C. K. ChanY. Zhou and K. H. Wong, An equilibrium model of the supply chain network under multi-attribute behaviors analysis, European J. Oper. Res., 275 (2019), 514-535.  doi: 10.1016/j.ejor.2018.11.068.  Google Scholar

[8]

A. Chassein and M. Goerigk, Minmax regret combinatorial optimization problems with ellipsoidal uncertainty sets, European J. Oper. Res., 258 (2017), 58-69.  doi: 10.1016/j.ejor.2016.10.055.  Google Scholar

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C. G. Chorus, Regret theory-based route choices and traffic equilibria, Transportmetrica, 8 (2012), 291-305.  doi: 10.1080/18128602.2010.498391.  Google Scholar

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E. CondeM. Leal and J. Puerto, A minmax regret version of the time-dependent shortest path problem, European J. Oper. Res., 270 (2018), 968-981.  doi: 10.1016/j.ejor.2018.04.030.  Google Scholar

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J. M. Cruz and Z. Liu, Modeling and analysis of the multiperiod effects of social relationship on supply chain networks, European J. Oper. Res., 214 (2011), 39-52.  doi: 10.1016/j.ejor.2011.03.044.  Google Scholar

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H. Deng, Y. Li, Z. Wan and Z. Wan, Partially smoothing and gradient-based algorithm for optimizing the VMI system with competitive retailers under random demands, Math. Probl. Eng., (2020), 3687471, 18 pp. doi: 10.1155/2020/3687471.  Google Scholar

[14]

J. DongD. Zhang and A. Nagurney, A supply chain network equilibrium model with random demand, European J. Oper. Res., 156 (2004), 194-212.  doi: 10.1016/S0377-2217(03)00023-7.  Google Scholar

[15]

R. Engelbrecht-Wiggans and E. Katok, Regret and feedback information in first-price sealed-bid auctions, Management Science, 54 (2008), 808-819.  doi: 10.1287/mnsc.1070.0806.  Google Scholar

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Y. Hamdouch, Multi-period supply chain network equilibrium with capacity constraints and purchasing strategies, Transportation Research Part C, 19 (2011), 803-820.  doi: 10.1016/j.trc.2011.02.006.  Google Scholar

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T.-H. Ho and J. Zhang, Designing pricing contracts for boundedly rational customers: Does the framing of the fixed fee matter?, Manageament Science, 54 (2008), 686-700.  doi: 10.1287/mnsc.1070.0788.  Google Scholar

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A. JabbarzadehB. Fahimnia and J.-B. Sheu, An enhanced robustness approach for managing supply and demand uncertainties, International Journal of Production Economics, 183 (2017), 620-631.  doi: 10.1016/j.ijpe.2015.06.009.  Google Scholar

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D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.  Google Scholar

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G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, Ékonom. i Mat. Metody, 12 (1976), 747-756.   Google Scholar

[23]

Y. Kuang and C. T. Ng, Pricing substitutable products under consumer regrets, International Journal of Production Economics, 203 (2018), 286-300.  doi: 10.1016/j.ijpe.2018.07.006.  Google Scholar

[24]

H. LiT. LuoY. Xu and J. Xu, Minimax regret vertex centdian location problem in general dynamic networks, Omega, 75 (2018), 87-96.  doi: 10.1016/j.omega.2017.02.004.  Google Scholar

[25]

D. LiA. Nagurney and M. Yu, Consumer learning of product quality with time delay: Insights from spatial price equilibrium models with differentiated products, Omega, 81 (2018), 150-168.  doi: 10.1016/j.omega.2017.10.007.  Google Scholar

[26]

G. Loomes and R. Sugden, Regret theory: An alternative theory of rational choice, The Economic Journal, 92 (1982), 805-824.   Google Scholar

[27]

A. Nagurney, Network Economics: A Variational Inequality Approach, Advances in Computational Economics, 1. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[28]

A. NagurneyJ. Dong and D. Zhang, A supply chain network equilibrium model, Transportation Research Part E, 38 (2002), 281-303.  doi: 10.1016/S1366-5545(01)00020-5.  Google Scholar

[29]

A. NagurneyM. Salarpour and P. Daniele, An integrated financial and logistical game theory model for humanitarian organizations with purchasing costs, multiple freight service providers, and budget, capacity, and demand constraints, International Journal of Production Economics, 212 (2019), 212-226.  doi: 10.1016/j.ijpe.2019.02.006.  Google Scholar

[30]

J. F. Nash Jr., Equilibrium points in $n$-person games, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.  Google Scholar

[31]

J. Nash, Non-cooperative games, Ann. of Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[32]

G. Perakis and G. Roels, Regret in the newsvendor model with partial information, Oper. Res., 56 (2008), 188-203.  doi: 10.1287/opre.1070.0486.  Google Scholar

[33]

G. de. O. RamosA. L. C. Bazzan and B. C. da Silva, Analysing the impact of travel information for minimising the regret of route choice, Transportation Research Part C, 88 (2018), 257-271.  doi: 10.1016/j.trc.2017.11.011.  Google Scholar

[34]

S. SaberiJ. M. CruzJ. Sarkis and A. Nagurney, A competitive multiperiod supply chain network model with freight carriers and green technology investment option, European J. Oper. Res., 266 (2018), 934-949.  doi: 10.1016/j.ejor.2017.10.043.  Google Scholar

[35]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Manageament Science, 46 (2000), 404-420.  doi: 10.1287/mnsc.46.3.404.12070.  Google Scholar

[36]

Z. WanH. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Appl. Math. Model., 58 (2018), 281-299.  doi: 10.1016/j.apm.2017.06.028.  Google Scholar

[37]

J. Wang and B. Xiao, A minmax regret price control model for managing perishable products with uncertain parameters, European J. Oper. Res., 258 (2017), 652-663.  doi: 10.1016/j.ejor.2016.09.024.  Google Scholar

[38]

W. WangP. ZhangJ. DingJ. LiH. Sun and L. He, Closed-loop supply chain network equilibrium model with retailer-collection under legislation, J. Ind. Manag. Optim., 15 (2019), 199-219.  doi: 10.3934/jimo.2018039.  Google Scholar

[39]

P. XidonasG. MavrotasC. Hassapis and C. Zopounidis, Robust multiobjective portfolio optimization: A minimax regret approach, European J. Oper. Res., 262 (2017), 299-305.  doi: 10.1016/j.ejor.2017.03.041.  Google Scholar

[40]

X. YanH.-Y. ChongJ. ZhouZ. Sheng and F. Xu, Fairness preference based decision-making model for concession period in PPP projects, J. Ind. Manag. Optim., 16 (2020), 11-23.  doi: 10.3934/jimo.2018137.  Google Scholar

[41]

M. Zeelenberg, Anticipated regret, expected feedback and behavioral decision making, Journal of Behavioral Decision Making, 12 (1999), 93-106.  doi: 10.1002/(SICI)1099-0771(199906)12:2<93::AID-BDM311>3.0.CO;2-S.  Google Scholar

[42]

Y. ZhouC. Chan and K. Wong, A multi-period supply chain network equilibrium model considering retailers' uncertain demands and dynamic loss-averse behaviors, Transportation Research Part E, 118 (2018), 51-76.  doi: 10.1016/j.tre.2018.06.006.  Google Scholar

[43]

Y. ZhouZ. ShenR. Ying and X. Xu, A loss-averse two-product odering model with information updating in two-echelon inventory system, J. Ind. Manag. Optim., 14 (2018), 687-705.  doi: 10.3934/jimo.2017069.  Google Scholar

show all references

References:
[1]

M. T. Ahmed and C. Kwon, Optimal contract-sizing in online display advertising for publishers with regret considerations, Omega, 42 (2014), 201-212.  doi: 10.1016/j.omega.2013.06.001.  Google Scholar

[2]

N. Ayvaz-CavdarogluS. Kachani and C. Maglaras, Revenue management with minimax regret negotiations, Omega, 63 (2016), 12-22.  doi: 10.1016/j.omega.2015.09.009.  Google Scholar

[3]

Q. Bai and F. Meng, Impact of risk aversion on two-echelon supply chain systems with carbon emission reduction constraints, J. Ind. Manag. Optim., 16 (2020), 1943-1965.  doi: 10.3934/jimo.2019037.  Google Scholar

[4]

D. Bell, Regret in decision making under uncertainty, Operations Research, 30 (1982), 961-981.   Google Scholar

[5]

N. CamilleG. CoricelliJ. SalletP. Pradat-DiehlJ.-R. Duhamel and A. Sirigu, The involvement of the orbitofrontal cortex in the experience of regret, Science, 304 (2004), 1167-1170.  doi: 10.1126/science.1094550.  Google Scholar

[6]

C. K. ChanY. Zhou and K. H. Wong, A dynamic equilibrium model of the oligopolistic closed-loop supply chain network under uncertain and time-dependent demands, Transportation Research Part E, 118 (2018), 325-354.  doi: 10.1016/j.tre.2018.07.008.  Google Scholar

[7]

C. K. ChanY. Zhou and K. H. Wong, An equilibrium model of the supply chain network under multi-attribute behaviors analysis, European J. Oper. Res., 275 (2019), 514-535.  doi: 10.1016/j.ejor.2018.11.068.  Google Scholar

[8]

A. Chassein and M. Goerigk, Minmax regret combinatorial optimization problems with ellipsoidal uncertainty sets, European J. Oper. Res., 258 (2017), 58-69.  doi: 10.1016/j.ejor.2016.10.055.  Google Scholar

[9]

C. G. Chorus, Regret theory-based route choices and traffic equilibria, Transportmetrica, 8 (2012), 291-305.  doi: 10.1080/18128602.2010.498391.  Google Scholar

[10]

E. CondeM. Leal and J. Puerto, A minmax regret version of the time-dependent shortest path problem, European J. Oper. Res., 270 (2018), 968-981.  doi: 10.1016/j.ejor.2018.04.030.  Google Scholar

[11]

G. CoricelliH. D. CritchleyM. JoffilyJ. P. O'DohertyA. Sirigu and R. J. Dolan, Regret and its avoidance: A neuroimaging study of choice behavior, Nature Neuroscience, 8 (2005), 1255-1262.  doi: 10.1038/nn1514.  Google Scholar

[12]

J. M. Cruz and Z. Liu, Modeling and analysis of the multiperiod effects of social relationship on supply chain networks, European J. Oper. Res., 214 (2011), 39-52.  doi: 10.1016/j.ejor.2011.03.044.  Google Scholar

[13]

H. Deng, Y. Li, Z. Wan and Z. Wan, Partially smoothing and gradient-based algorithm for optimizing the VMI system with competitive retailers under random demands, Math. Probl. Eng., (2020), 3687471, 18 pp. doi: 10.1155/2020/3687471.  Google Scholar

[14]

J. DongD. Zhang and A. Nagurney, A supply chain network equilibrium model with random demand, European J. Oper. Res., 156 (2004), 194-212.  doi: 10.1016/S0377-2217(03)00023-7.  Google Scholar

[15]

R. Engelbrecht-Wiggans and E. Katok, Regret and feedback information in first-price sealed-bid auctions, Management Science, 54 (2008), 808-819.  doi: 10.1287/mnsc.1070.0806.  Google Scholar

[16]

M. Fisher and A. Raman, Reducing the cost of demand uncertainty through accurate response to early sales, Operations Research, 44 (1996), 87-99.   Google Scholar

[17]

H. Gilbert and O. Spanjaard, A double oracle approach to minmax regret optimization problems with interval data, European J. Oper. Res., 262 (2017), 929-943.  doi: 10.1016/j.ejor.2017.04.058.  Google Scholar

[18]

Y. Hamdouch, Multi-period supply chain network equilibrium with capacity constraints and purchasing strategies, Transportation Research Part C, 19 (2011), 803-820.  doi: 10.1016/j.trc.2011.02.006.  Google Scholar

[19]

T.-H. Ho and J. Zhang, Designing pricing contracts for boundedly rational customers: Does the framing of the fixed fee matter?, Manageament Science, 54 (2008), 686-700.  doi: 10.1287/mnsc.1070.0788.  Google Scholar

[20]

A. JabbarzadehB. Fahimnia and J.-B. Sheu, An enhanced robustness approach for managing supply and demand uncertainties, International Journal of Production Economics, 183 (2017), 620-631.  doi: 10.1016/j.ijpe.2015.06.009.  Google Scholar

[21]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. doi: 10.1137/1.9780898719451.  Google Scholar

[22]

G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, Ékonom. i Mat. Metody, 12 (1976), 747-756.   Google Scholar

[23]

Y. Kuang and C. T. Ng, Pricing substitutable products under consumer regrets, International Journal of Production Economics, 203 (2018), 286-300.  doi: 10.1016/j.ijpe.2018.07.006.  Google Scholar

[24]

H. LiT. LuoY. Xu and J. Xu, Minimax regret vertex centdian location problem in general dynamic networks, Omega, 75 (2018), 87-96.  doi: 10.1016/j.omega.2017.02.004.  Google Scholar

[25]

D. LiA. Nagurney and M. Yu, Consumer learning of product quality with time delay: Insights from spatial price equilibrium models with differentiated products, Omega, 81 (2018), 150-168.  doi: 10.1016/j.omega.2017.10.007.  Google Scholar

[26]

G. Loomes and R. Sugden, Regret theory: An alternative theory of rational choice, The Economic Journal, 92 (1982), 805-824.   Google Scholar

[27]

A. Nagurney, Network Economics: A Variational Inequality Approach, Advances in Computational Economics, 1. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-011-2178-1.  Google Scholar

[28]

A. NagurneyJ. Dong and D. Zhang, A supply chain network equilibrium model, Transportation Research Part E, 38 (2002), 281-303.  doi: 10.1016/S1366-5545(01)00020-5.  Google Scholar

[29]

A. NagurneyM. Salarpour and P. Daniele, An integrated financial and logistical game theory model for humanitarian organizations with purchasing costs, multiple freight service providers, and budget, capacity, and demand constraints, International Journal of Production Economics, 212 (2019), 212-226.  doi: 10.1016/j.ijpe.2019.02.006.  Google Scholar

[30]

J. F. Nash Jr., Equilibrium points in $n$-person games, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 48-49.  doi: 10.1073/pnas.36.1.48.  Google Scholar

[31]

J. Nash, Non-cooperative games, Ann. of Math., 54 (1951), 286-295.  doi: 10.2307/1969529.  Google Scholar

[32]

G. Perakis and G. Roels, Regret in the newsvendor model with partial information, Oper. Res., 56 (2008), 188-203.  doi: 10.1287/opre.1070.0486.  Google Scholar

[33]

G. de. O. RamosA. L. C. Bazzan and B. C. da Silva, Analysing the impact of travel information for minimising the regret of route choice, Transportation Research Part C, 88 (2018), 257-271.  doi: 10.1016/j.trc.2017.11.011.  Google Scholar

[34]

S. SaberiJ. M. CruzJ. Sarkis and A. Nagurney, A competitive multiperiod supply chain network model with freight carriers and green technology investment option, European J. Oper. Res., 266 (2018), 934-949.  doi: 10.1016/j.ejor.2017.10.043.  Google Scholar

[35]

M. E. Schweitzer and G. P. Cachon, Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence, Manageament Science, 46 (2000), 404-420.  doi: 10.1287/mnsc.46.3.404.12070.  Google Scholar

[36]

Z. WanH. Wu and L. Dai, A polymorphic uncertain equilibrium model and its deterministic equivalent formulation for decentralized supply chain management, Appl. Math. Model., 58 (2018), 281-299.  doi: 10.1016/j.apm.2017.06.028.  Google Scholar

[37]

J. Wang and B. Xiao, A minmax regret price control model for managing perishable products with uncertain parameters, European J. Oper. Res., 258 (2017), 652-663.  doi: 10.1016/j.ejor.2016.09.024.  Google Scholar

[38]

W. WangP. ZhangJ. DingJ. LiH. Sun and L. He, Closed-loop supply chain network equilibrium model with retailer-collection under legislation, J. Ind. Manag. Optim., 15 (2019), 199-219.  doi: 10.3934/jimo.2018039.  Google Scholar

[39]

P. XidonasG. MavrotasC. Hassapis and C. Zopounidis, Robust multiobjective portfolio optimization: A minimax regret approach, European J. Oper. Res., 262 (2017), 299-305.  doi: 10.1016/j.ejor.2017.03.041.  Google Scholar

[40]

X. YanH.-Y. ChongJ. ZhouZ. Sheng and F. Xu, Fairness preference based decision-making model for concession period in PPP projects, J. Ind. Manag. Optim., 16 (2020), 11-23.  doi: 10.3934/jimo.2018137.  Google Scholar

[41]

M. Zeelenberg, Anticipated regret, expected feedback and behavioral decision making, Journal of Behavioral Decision Making, 12 (1999), 93-106.  doi: 10.1002/(SICI)1099-0771(199906)12:2<93::AID-BDM311>3.0.CO;2-S.  Google Scholar

[42]

Y. ZhouC. Chan and K. Wong, A multi-period supply chain network equilibrium model considering retailers' uncertain demands and dynamic loss-averse behaviors, Transportation Research Part E, 118 (2018), 51-76.  doi: 10.1016/j.tre.2018.06.006.  Google Scholar

[43]

Y. ZhouZ. ShenR. Ying and X. Xu, A loss-averse two-product odering model with information updating in two-echelon inventory system, J. Ind. Manag. Optim., 14 (2018), 687-705.  doi: 10.3934/jimo.2017069.  Google Scholar

Figure 1.  The structure of SCN for $ T $ production periods
Figure 2.  The regret function $ R(\Delta v_j(t)) $ with different regret aversion coefficient $ \delta $
Figure 3.  The profit, regret and utility of all the decision makers in the rise situation of Example 1
Figure 4.  The profit, regret and utility of all the decision makers in the depleted situation of Example 1
Table 1.  The equilibrium in the rise situation of Example 1
$ i=1,2 $, $ j=1,2 $ regret-neutral anticipated regret-averse
$ k_j(t) $ $ k_j(t) $
$ 0 $ $ 0.2 $ $ 0.4 $ $ 0.6 $ $ 0.8 $
$ \hat{q}_i^*(t),t=1 $ 1.8633 1.6578 1.4318 1.1823 0.9097
$ \hat{q}_i^*(t),t=2 $ 1.5945 1.4185 1.2248 1.0108 0.7769
$ \hat{q}_i^*(t),t=3 $ 1.4934 1.3291 1.148 0.9476 0.728
$ I_i^*(t),t=1 $ 0.5254 0.4672 0.4026 0.3309 0.2523
$ I_i^*(t),t=2 $ 0.4356 0.3887 0.3364 0.2779 0.213
$ q_{ij}^*(t),t=1 $ 0.6692 0.5938 0.5125 0.4237 0.3272
$ q_{ij}^*(t),t=2 $ 0.8424 0.7469 0.6431 0.5297 0.4064
$ q_{ij}^*(t),t=3 $ 0.9648 0.8571 0.7396 0.6102 0.4686
$ \rho_{j}^*(t),t=1 $ 35.8181 40.1667 46.2534 55.4406 70.7135
$ \rho_{j}^*(t),t=2 $ 40.228 45.183 52.1475 62.7221 80.4573
$ \rho_{j}^*(t),t=3 $ 45.2266 50.7262 58.4607 70.2238 90.025
$ i=1,2 $, $ j=1,2 $ regret-neutral anticipated regret-averse
$ k_j(t) $ $ k_j(t) $
$ 0 $ $ 0.2 $ $ 0.4 $ $ 0.6 $ $ 0.8 $
$ \hat{q}_i^*(t),t=1 $ 1.8633 1.6578 1.4318 1.1823 0.9097
$ \hat{q}_i^*(t),t=2 $ 1.5945 1.4185 1.2248 1.0108 0.7769
$ \hat{q}_i^*(t),t=3 $ 1.4934 1.3291 1.148 0.9476 0.728
$ I_i^*(t),t=1 $ 0.5254 0.4672 0.4026 0.3309 0.2523
$ I_i^*(t),t=2 $ 0.4356 0.3887 0.3364 0.2779 0.213
$ q_{ij}^*(t),t=1 $ 0.6692 0.5938 0.5125 0.4237 0.3272
$ q_{ij}^*(t),t=2 $ 0.8424 0.7469 0.6431 0.5297 0.4064
$ q_{ij}^*(t),t=3 $ 0.9648 0.8571 0.7396 0.6102 0.4686
$ \rho_{j}^*(t),t=1 $ 35.8181 40.1667 46.2534 55.4406 70.7135
$ \rho_{j}^*(t),t=2 $ 40.228 45.183 52.1475 62.7221 80.4573
$ \rho_{j}^*(t),t=3 $ 45.2266 50.7262 58.4607 70.2238 90.025
Table 2.  The equilibrium in the depleted situation of Example 1
$ i=1,2 $, $ j=1,2 $ regret-neutral anticipated regret
$ k_j(t) $ $ k_j(t) $
$ 0 $ $ 0.2 $ $ 0.4 $ $ 0.6 $ $ 0.8 $
$ \hat{q}_i^*(t),t=1 $ 2.1408 1.899 1.634 1.3421 1.0253
$ \hat{q}_i^*(t),t=2 $ 1.6228 1.4407 1.2413 1.0214 0.7833
$ \hat{q}_i^*(t),t=3 $ 1.253 1.1216 0.9755 0.813 0.6293
$ I_i^*(t),t=1 $ 0.0457 0.044 0.0421 0.0395 0.0381
$ I_i^*(t),t=2 $ 0 0 0 0 0.0055
$ q_{ij}^*(t),t=1 $ 1.0476 0.9261 0.7939 0.6492 0.492
$ q_{ij}^*(t),t=2 $ 0.8344 0.7408 0.6395 0.5282 0.4062
$ q_{ij}^*(t),t=3 $ 0.6275 0.559 0.4851 0.404 0.3157
$ \rho_{j}^*(t),t=1 $ 42.1631 47.4737 54.9998 66.5564 86.286
$ \rho_{j}^*(t),t=2 $ 40.6374 45.5225 52.3862 62.818 80.423
$ \rho_{j}^*(t),t=3 $ 37.6092 42.0773 48.2821 57.565 72.7609
$ i=1,2 $, $ j=1,2 $ regret-neutral anticipated regret
$ k_j(t) $ $ k_j(t) $
$ 0 $ $ 0.2 $ $ 0.4 $ $ 0.6 $ $ 0.8 $
$ \hat{q}_i^*(t),t=1 $ 2.1408 1.899 1.634 1.3421 1.0253
$ \hat{q}_i^*(t),t=2 $ 1.6228 1.4407 1.2413 1.0214 0.7833
$ \hat{q}_i^*(t),t=3 $ 1.253 1.1216 0.9755 0.813 0.6293
$ I_i^*(t),t=1 $ 0.0457 0.044 0.0421 0.0395 0.0381
$ I_i^*(t),t=2 $ 0 0 0 0 0.0055
$ q_{ij}^*(t),t=1 $ 1.0476 0.9261 0.7939 0.6492 0.492
$ q_{ij}^*(t),t=2 $ 0.8344 0.7408 0.6395 0.5282 0.4062
$ q_{ij}^*(t),t=3 $ 0.6275 0.559 0.4851 0.404 0.3157
$ \rho_{j}^*(t),t=1 $ 42.1631 47.4737 54.9998 66.5564 86.286
$ \rho_{j}^*(t),t=2 $ 40.6374 45.5225 52.3862 62.818 80.423
$ \rho_{j}^*(t),t=3 $ 37.6092 42.0773 48.2821 57.565 72.7609
Table 3.  The equilibrium decisions, profit, regret and utility in experiential regret-averse model of Example 1
$ i=1,2 $ in the rise situation in the depleted situation
$ j=1,2 $ anticipated experiential regret anticipated experiential regret
$ \tau=0\% $ $ \tau=10\% $ $ \tau=20\% $ $ \tau=0\% $ $ \tau=10\% $ $ \tau=20\% $
$ \hat{q}_i^*(t),t=1 $ 1.1823 1.111 1.0317 1.3421 1.3922 1.4385
$ \hat{q}_i^*(t),t=2 $ 1.0108 0.9324 0.8457 1.0214 1.0765 1.1285
$ \hat{q}_i^*(t),t=3 $ 0.9476 0.8502 0.744 0.813 0.8763 0.9326
$ I_i^*(t),t=1 $ 0.3309 0.2645 0.1918 0.0395 0.0863 0.1325
$ I_i^*(t),t=2 $ 0.2779 0.2087 0.135 0 0.0302 0.0597
$ q_{ij}^*(t),t=1 $ 0.4237 0.4215 0.4187 0.6492 0.6506 0.6505
$ q_{ij}^*(t),t=2 $ 0.5297 0.4921 0.4497 0.5282 0.564 0.5983
$ q_{ij}^*(t),t=3 $ 0.6102 0.5269 0.4374 0.404 0.4509 0.4941
$ \rho_{j}^*(t),t=1 $ 55.4406 56.6548 58.0501 66.5564 65.7988 65.2914
$ \rho_{j}^*(t),t=2 $ 62.7221 67.4657 73.9462 62.818 59.1709 56.155
$ \rho_{j}^*(t),t=3 $ 70.2238 79.7648 93.8077 57.565 52.5551 48.7155
$ \Sigma M_{i}(t) $ 12.1586 10.1711 8.2357 11.8539 13.1645 14.4529
$ \Sigma E(\pi_{j}(t)) $ 105.2413 110.8787 116.4356 105.4868 101.6604 97.9989
$ \Sigma E(R_j) $ 48.8578 48.3116 47.7065 48.959 49.2566 49.5897
$ E(U_{j}) $ 75.9265 78.371 80.7928 76.1115 74.4315 72.8693
Total profit 234.7998 242.0996 249.3426 234.6814 229.6498 224.9036
$ i=1,2 $ in the rise situation in the depleted situation
$ j=1,2 $ anticipated experiential regret anticipated experiential regret
$ \tau=0\% $ $ \tau=10\% $ $ \tau=20\% $ $ \tau=0\% $ $ \tau=10\% $ $ \tau=20\% $
$ \hat{q}_i^*(t),t=1 $ 1.1823 1.111 1.0317 1.3421 1.3922 1.4385
$ \hat{q}_i^*(t),t=2 $ 1.0108 0.9324 0.8457 1.0214 1.0765 1.1285
$ \hat{q}_i^*(t),t=3 $ 0.9476 0.8502 0.744 0.813 0.8763 0.9326
$ I_i^*(t),t=1 $ 0.3309 0.2645 0.1918 0.0395 0.0863 0.1325
$ I_i^*(t),t=2 $ 0.2779 0.2087 0.135 0 0.0302 0.0597
$ q_{ij}^*(t),t=1 $ 0.4237 0.4215 0.4187 0.6492 0.6506 0.6505
$ q_{ij}^*(t),t=2 $ 0.5297 0.4921 0.4497 0.5282 0.564 0.5983
$ q_{ij}^*(t),t=3 $ 0.6102 0.5269 0.4374 0.404 0.4509 0.4941
$ \rho_{j}^*(t),t=1 $ 55.4406 56.6548 58.0501 66.5564 65.7988 65.2914
$ \rho_{j}^*(t),t=2 $ 62.7221 67.4657 73.9462 62.818 59.1709 56.155
$ \rho_{j}^*(t),t=3 $ 70.2238 79.7648 93.8077 57.565 52.5551 48.7155
$ \Sigma M_{i}(t) $ 12.1586 10.1711 8.2357 11.8539 13.1645 14.4529
$ \Sigma E(\pi_{j}(t)) $ 105.2413 110.8787 116.4356 105.4868 101.6604 97.9989
$ \Sigma E(R_j) $ 48.8578 48.3116 47.7065 48.959 49.2566 49.5897
$ E(U_{j}) $ 75.9265 78.371 80.7928 76.1115 74.4315 72.8693
Total profit 234.7998 242.0996 249.3426 234.6814 229.6498 224.9036
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