Perfectly competitive capacity expansion games with risk-averse participants. (English) Zbl 1367.91021
Summary: This paper presents Nash equilibrium models of perfectly competitive capacity expansion involving risk-averse participants in the presence of state uncertainty and pricing mechanisms. Existence of solutions to such models is established based on the nonlinear complementarity formulations of the models to which a general existence result is applicable. This study extends two recent papers [L. Fan et al., J. Environ. Econ. Manage. 60, No. 3, 193–208 (2010; Zbl 1202.91246)] and [J. Zhao et al., Oper. Res. 58, No. 3, 529–548 (2010; Zbl 1228.90172)] pertaining to special cases of our models, complements the extensive work on games with strategic players, and provides an extended treatment of games with price-taking players whose feasible sets may be unbounded. The latter aspect generalizes much of the classical analysis of such models for which price boundedness and feasible region compactness are essential assumptions needed for a fixed-point existence proof.
MSC:
| 91A10 | Noncooperative games |
| 90C15 | Stochastic programming |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
References:
| [1] | Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Progr. A 106, 433-446 (2006) · Zbl 1134.90025 · doi:10.1007/s10107-005-0638-8 |
| [2] | Arrow, K.J., Debreu, G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265-290 (1954) · Zbl 0055.38007 · doi:10.2307/1907353 |
| [3] | Bautista, G., Anjos, M.F., Vannelli, A.: Formulation of oligopolistic competition in AC power networks: an NLP approach. IEEE Trans. Power Syst. 22, 105-115 (2006) · doi:10.1109/TPWRS.2006.888986 |
| [4] | Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997) · Zbl 0892.90142 |
| [5] | Britz, W., Ferris, M.C., Kuhn, A.: Modeling water allocating institutions based on multiple optimization problems with equilibrium constraints. Environ. Model. Softw. 46, 196-207 (2013) · doi:10.1016/j.envsoft.2013.03.010 |
| [6] | California Independent System Operator: Flexible Capacity Procurement Phase 1: Risk of Retirement, Market and Infrastructure Policy, Second Revised Draft Final Proposal (2012) · Zbl 0311.90056 |
| [7] | Cardell, J., Hitt, C.C., Hogan, W.W.: Market power and strategic interaction in electricity networks. Resour. Energy Econ. 19, 109-137 (1997) · doi:10.1016/S0928-7655(97)00006-7 |
| [8] | Centeno, E., Reneses, J., Barquin, J.: Strategic analysis of electricity markets under uncertainty: a conjectured-price-response approach. IEEE Trans. Power Syst. 22, 423-432 (2007) · doi:10.1109/TPWRS.2006.887890 |
| [9] | Chen, Y., Liu, A.L., Hobbs, B.F.: Economic and emissions implications of load-based, source-based, and first-seller emissions trading programs under California AB32. Oper. Res. 59, 696-712 (2011) · Zbl 1238.91107 · doi:10.1287/opre.1110.0917 |
| [10] | Chen, Y., Sijm, J., Hobbs, B.F., Lise, W.: Implications of \[\text{ CO }_2\] CO2 emissions trading for short-run electricity market outcomes in northwest Europe. J. Regul. Econ. 34, 251-281 (2008) · doi:10.1007/s11149-008-9069-9 |
| [11] | Cunningham, L., Baldick, R., Baughman, M.: An empirical study of applied game theory: transmission constrained Cournot behavior. IEEE Trans. Power Syst. 17, 166-172 (2002) · doi:10.1109/59.982209 |
| [12] | Dantzig, G.B.: Linear programming under uncertainty. Manag. Sci. 1, 197-206 (1955) · Zbl 0995.90589 · doi:10.1287/mnsc.1.3-4.197 |
| [13] | Debreu, G.: A social equilibrium existence theorem. Proc. Natl. Acad. Sci. USA 38, 886-893 (1952) · Zbl 0047.38804 · doi:10.1073/pnas.38.10.886 |
| [14] | Dempster, M.A.H.: On stochastic programming I: static linear programming under risk. J. Math. Anal. Appl. 21, 304-343 (1968) · Zbl 0346.90056 · doi:10.1016/0022-247X(68)90215-1 |
| [15] | Dempster, M.A.H.: On stochastic programming II: dynamic problems under risk. Stochastics 25, 15-42 (1988) · Zbl 0653.90054 · doi:10.1080/17442508808833530 |
| [16] | Ehrenmann, A., Smeers, Y.: Generation capacity expansion in a risky environment: a stochastic equilibrium analysis. Oper. Res. 59, 1332-1346 (2011) · Zbl 1241.91062 · doi:10.1287/opre.1110.0992 |
| [17] | Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer, New York (2003) · Zbl 1062.90002 |
| [18] | Facchinei, F., Kanzow, C.: Generalized Nash equilibrium problems. 4OR 5, 173-210 (2007) · Zbl 1211.91162 · doi:10.1007/s10288-007-0054-4 |
| [19] | Facchinei, F.; Pang, JS; Eldar, Y. (ed.); Palomar, D. (ed.), Nash equilibria: the variational approach, 443-493 (2009), Cambridge · Zbl 1210.91008 · doi:10.1017/CBO9780511804458.013 |
| [20] | Facchinei, F., Pang, J.S., Scutari, G., Lampariello, L.: VI-constrained hemivariational inequalities: Distributed algorithms and power control in ad-hoc networks. Math. Progr. A 145, 59-96 (2014) · Zbl 1300.90054 · doi:10.1007/s10107-013-0640-5 |
| [21] | Fan, L., Hobbs, B.F., Norman, C.S.: Risk aversion and \[\text{ CO }_2\] CO2 regulatory uncertainty in power generation investment: Policy and modeling implications. J. Environ. Econ. Manag. 60, 193-208 (2010) · Zbl 1202.91246 · doi:10.1016/j.jeem.2010.08.001 |
| [22] | Ferris, M.C., Wets, R.J.B.: MOPEC: multiple optimization problems with equilibrium constraints. http://www.cs.wisc.edu/ ferris/talks/chicago-mar (2013) · Zbl 0653.90054 |
| [23] | Fukushima, M.: Restricted generalized Nash equilibria and controlled penalty algorithm. Comput. Manag. Sci. 8, 201-218 (2011) · Zbl 1253.91010 · doi:10.1007/s10287-009-0097-4 |
| [24] | Hobbs, B.F., Metzler, C.B., Pang, J.S.: Strategic gaming analysis for electric power systems: an MPEC approach. IEEE Trans. Power Syst. 15, 638-645 (2000) · doi:10.1109/59.867153 |
| [25] | Hobbs, B.F., Hu, M.C., Inon, J., Bhavaraju, M.: A dynamic analysis of a demand curve-based capacity market proposal: the PJM reliability pricing model. IEEE Trans. Power Syst. 22, 3-11 (2007) · doi:10.1109/TPWRS.2006.887954 |
| [26] | Kannan, A., Shanbhag, U.V., Kim, H.M.: Strategic behavior in power markets under uncertainty. Energy Syst. 2, 115-141 (2011) · doi:10.1007/s12667-011-0032-y |
| [27] | Keeney, R.L., Raiffa, H.: Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Wiley, New York (1976) · Zbl 0488.90001 |
| [28] | Krawczyk, J.B.: Coupled constraint Nash equilibria in environmental games. Resour. Energy Econ. 27, 157-181 (2005) · doi:10.1016/j.reseneeco.2004.08.001 |
| [29] | Krawczyk, J.B.: Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems. Comput. Manag. Sci. 4, 183-204 (2007) · Zbl 1134.91303 · doi:10.1007/s10287-006-0033-9 |
| [30] | Kubota, K., Fukushima, M.: Gap function approach to the generalized Nash equilibrium problem. J. Optim. Theory Appl. 144, 511-531 (2010) · Zbl 1188.91021 · doi:10.1007/s10957-009-9614-4 |
| [31] | Kulkarni, A.A., Shanbhag, U.V.: On the variational equilibrium as a refinement of the generalized Nash equilibrium. Automatica 48, 45-55 (2012) · Zbl 1245.91006 · doi:10.1016/j.automatica.2011.09.042 |
| [32] | Kulkarni, A.A., Shanbhag, U.V.: Revisiting generalized Nash games and variational inequalities. J. Optim. Theory Appl. 154, 175-186 (2012) · Zbl 1261.90065 · doi:10.1007/s10957-011-9981-5 |
| [33] | Markowitz, H.: Portfolio selection. J. Finance 7, 77-91 (1952) |
| [34] | McKenzie, L.W.: On the existence of general equilibrium for a competitive market. Econometrica 27, 54-71 (1959) · Zbl 0095.34302 · doi:10.2307/1907777 |
| [35] | Metlzer, C.B., Hobbs, B.F., Pang, J.S.: Nash-Cournot equilibria in power markets on a linearized DC network with arbitrage: formulations and properties. Netw. Spat. Econ. 3, 123-150 (2003) · doi:10.1023/A:1023907818360 |
| [36] | Murphy, F.H., Smeers, Y.: Generation capacity expansion in imperfectly competitive restructured electricity markets. Oper. Res. 53, 646-661 (2005) · Zbl 1165.91342 · doi:10.1287/opre.1050.0211 |
| [37] | Nabetani, K., Tseng, P., Fukushima, M.: Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints. Comput. Optim. Appl. 48, 423-452 (2011) · Zbl 1220.90136 · doi:10.1007/s10589-009-9256-3 |
| [38] | Nash Jr, J.F.: Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 36, 48-49 (1950) · Zbl 0036.01104 · doi:10.1073/pnas.36.1.48 |
| [39] | Nash Jr, J.F.: Non-cooperative games. Ann. Math. 54, 286-295 (1951) · Zbl 0045.08202 · doi:10.2307/1969529 |
| [40] | Neuhoff, K., De Vries, L.: Insufficient incentives for investment in electricity generations. Util. Policy 12, 253-267 (2004) · doi:10.1016/j.jup.2004.06.002 |
| [41] | Ozdemir, O.: Simulation Modeling and Optimization of Competitive Electricity Markets and Stochastic Fluid Systems, Ph.D. thesis. University of Tilburg, Tilburg (2013) |
| [42] | Pang, J.S., Scutari, G.: Nonconvex games with side constraints. SIAM J. Optim. 21, 1491-1522 (2011) · Zbl 1246.91022 · doi:10.1137/100811787 |
| [43] | Reinelt, P.S., Keith, D.W.: Carbon capture retrofits and the cost of regulatory uncertainty. Energy J. 28, 101-127 (2007) · doi:10.5547/ISSN0195-6574-EJ-Vol28-No4-5 |
| [44] | Rosen, J.B.: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33, 520-534 (1965) · Zbl 0142.17603 · doi:10.2307/1911749 |
| [45] | Ruszczyński, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31, 433-452 (2006) · Zbl 1278.90283 · doi:10.1287/moor.1050.0186 |
| [46] | Schiro, D.A., Pang, J.S., Shanbhag, U.V.: On the solution of affine generalized Nash equilibrium problems with shared constraints by Lemke’s method. Math. Progr. A 146, 1-46 (2013) · Zbl 1282.90201 · doi:10.1007/s10107-012-0558-3 |
| [47] | Vázquez, C., Rivier, M., Pérez-Arriaga, I.J.: A market approach to long-term security of supply. IEEE Trans. Power Syst. 17, 349-357 (2002) · doi:10.1109/TPWRS.2002.1007903 |
| [48] | Walras, L.: Elements of Pure Economics, translated by W. Jaffé. Taylor & Francis, New York (2010, original 1874) · JFM 21.0229.03 |
| [49] | Wei, J.Y., Smeers, Y.: Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper. Res. 47, 102-112 (1999) · Zbl 1175.91080 · doi:10.1287/opre.47.1.102 |
| [50] | Wets, R.J.B.: Programming under uncertainty: the equivalent convex program. SIAM J. Appl. Math. 14, 89-105 (1966) · Zbl 0139.13303 · doi:10.1137/0114008 |
| [51] | Wets, R.J.B.: Programming under uncertainty: the solution set. SIAM J. Appl. Math. 14, 1143-1151 (1966) · Zbl 0149.16502 · doi:10.1137/0114091 |
| [52] | Wets, R.J.B.: Stochastic programs with fixed recourse: the equivalent deterministic program. SIAM Rev. 16, 309-339 (1974) · Zbl 0311.90056 · doi:10.1137/1016053 |
| [53] | Wogrin, S., Hobbs, B.F., Ralph, D., Centeno, E., Barquin, J.: Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition. Math. Progr. B 140(2), 295-322 (2013) · Zbl 1273.90113 · doi:10.1007/s10107-013-0696-2 |
| [54] | Yao, J., Oren, S.S., Adler, I.: Cournot equilibria in two-settlement electricity markets with system contingencies. Int. J. Crit. Infrastruct. 3, 142-160 (2007) · doi:10.1504/IJCIS.2007.011549 |
| [55] | Zhao, J., Hobbs, B.F., Pang, J.S.: Long-run equilibrium modeling of emissions allowance allocation systems in electric power markets. Oper. Res. 58, 529-548 (2010) · Zbl 1228.90172 · doi:10.1287/opre.1090.0771 |
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.
