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Potential games with aggregation in non-cooperative general insurance markets. (English) Zbl 1390.91216

Summary: In the global insurance market, the number of product-specific policies from different companies has increased significantly, and strong market competition has boosted the demand for a competitive premium. Thus, in the present paper, by considering the competition between each pair of insurers, an N-player game is formulated to investigate the optimal pricing strategy by calculating the Nash equilibrium in an insurance market. Under that framework, each insurer is assumed to maximise its utility of wealth over the unit time interval. With the purpose of solving a game of N-players, the best-response potential game with non-linear aggregation is implemented. The existence of a Nash equilibrium is proved by finding a potential function of all insurers’ payoff functions. A 12-player insurance game illustrates the theoretical findings under the framework in which the best-response selection premium strategies always provide the global maximum value of the corresponding payoff function.

MSC:

91B30 Risk theory, insurance (MSC2010)
91A10 Noncooperative games
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[1] AaseK.K. (1993) Equilibrium in a reinsurance syndicate; existence, uniqueness and characterization. ASTIN Bulletin, 23(2), 185-211.10.2143/AST.23.2.2005091 · doi:10.2143/AST.23.2.2005091
[2] Alos-FerrerC. and AniaA.B. (2005) The evolutionary stability of perfectly competitive behavior. Economic Theory, 26(3), 497-516.10.1007/s00199-004-0474-8 · Zbl 1106.91002 · doi:10.1007/s00199-004-0474-8
[3] BoonenT.J. (2015) Competitive equilibria with distortion risk measures. ASTIN Bulletin, 45(3), 703-728.10.1017/asb.2015.11 · Zbl 1390.91331 · doi:10.1017/asb.2015.11
[4] BoonenT.J. (2016) Nash equilibria of over-the-counter bargaining for insurance risk redistributions: The role of a regulator. European Journal of Operational Research, 250(3), 955-965.10.1016/j.ejor.2015.09.062 · Zbl 1346.91027 · doi:10.1016/j.ejor.2015.09.062
[5] BorchK. H. (1962) Application of game theory to some problems in automobile insurance. ASTIN Bulletin, 2(2), 208-221.10.1017/S051503610000996X · doi:10.1017/S051503610000996X
[6] BorchK.H. (1974) The Mathematical Theory of Insurance: An Annotated Selection of Papers on Insurance Published 1960-1972. Mass.: D.C. Heath and Co Lexington.
[7] BrockettP. and XiaX. (1995) Operations research in insurance: a review. Transactions of the Society of Actuaries, XLVII, 7-88.
[8] BühlmannH. (1980) An economic premium principle. ASTIN Bulletin, 11(1), 52-60.10.1017/S0515036100006619 · doi:10.1017/S0515036100006619
[9] BühlmannH. (1984) The general economic premium principle. ASTIN Bulletin, 14(1), 13-21.10.1017/S0515036100004773 · doi:10.1017/S0515036100004773
[10] ClappJ.M. (1985) Quantity competition in spatial markets with incomplete information. The Quarterly Journal of Economics, 100(2), 519-528.10.2307/1885394 · doi:10.2307/1885394
[11] DaykinC.D., PentikainenT. and PesonenM. (1994) Practical Risk Theory for Actuaries. Suffolk UK: Chapman & Hall/CRC. · Zbl 1140.62345
[12] DubeyP., HaimankoO. and ZapechelnyukA. (2006) Strategic complements and substitutes, and potential games. Games and Economic Behavior, 54(1), 77-94.10.1016/j.geb.2004.10.007 · Zbl 1129.91004 · doi:10.1016/j.geb.2004.10.007
[13] DutangC., AlbrecherH. and LoiselS. (2013) Competition among non-life insurers under solvency constraints: A game-theoretic approach. European Journal of Operational Research, 231(3), 702-711.10.1016/j.ejor.2013.06.029 · Zbl 1317.91042 · doi:10.1016/j.ejor.2013.06.029
[14] EmmsP. (2007a) Dynamic pricing of general insurance in a competitive market. ASTIN Bulletin, 37(01), 1-34.10.2143/AST.37.1.2020796 · Zbl 1162.91409 · doi:10.2143/AST.37.1.2020796
[15] EmmsP. (2007b) Pricing general insurance with constraints. Insurance: Mathematics and Economics, 40(2), 335-355. · Zbl 1141.91504
[16] EmmsP. (2011) Pricing general insurance in a reactive and competitive market. Journal of Computational and Applied Mathematics, 236(6), 1314-1332.10.1016/j.cam.2011.08.014 · Zbl 1228.91033 · doi:10.1016/j.cam.2011.08.014
[17] EmmsP. (2012) Equilibrium pricing of general insurance policies. North American Actuarial Journal, 16(3), 323-349.10.1080/10920277.2012.10590645 · Zbl 1291.91104 · doi:10.1080/10920277.2012.10590645
[18] EmmsP. and HabermanS. (2005) Pricing general insurance using optimal control theory. ASTIN Bulletin, 35(02), 427-453.10.1017/S051503610001432X · Zbl 1155.91401 · doi:10.1017/S051503610001432X
[19] EmmsP. and HabermanS. (2009) Optimal management of an insurers exposure in a competitive general insurance market. North American Actuarial Journal, 13(1), 77-105.10.1080/10920277.2009.10597541 · doi:10.1080/10920277.2009.10597541
[20] EmmsP., HabermanS. and SavoulliI. (2007) Optimal strategies for pricing general insurance. Insurance: Mathematics and Economics, 40(1), 15-34. · Zbl 1273.91236
[21] FudenbergD. and TiroleT. (1991) Game Theory. Cambridge, USA: Massachusetts Institute of Technology Press.
[22] JensenM.K. (2010) Aggregative games and best-reply potentials. Economic Theory, 43(1), 45-66.10.1007/s00199-008-0419-8 · Zbl 1185.91053 · doi:10.1007/s00199-008-0419-8
[23] KukushkinN.S. (2004) Best response dynamics in finite games with additive aggregation. Games and Economic Behavior, 48(1), 94-110.10.1016/j.geb.2003.06.007 · Zbl 1117.91305 · doi:10.1016/j.geb.2003.06.007
[24] LemaireJ. (1984) An application of game theory: Cost allocation. ASTIN Bulletin, 14(1), 61-81.10.1017/S0515036100004815 · doi:10.1017/S0515036100004815
[25] LemaireJ. (1991) Cooperative game theory and its insurance applications. ASTIN Bulletin, 21(1), 17-40.10.2143/AST.21.1.2005399 · doi:10.2143/AST.21.1.2005399
[26] LernerA.P. (1934) The concept of monopoly and the measurement of monopoly power. The Review of Economic Studies, 1(3), 157-175.10.2307/2967480 · doi:10.2307/2967480
[27] MalinovskiiV.K. (2010) Competition-originated cycles and insurance strategies. ASTIN Bulletin, 40(2), 797-843. · Zbl 1235.91099
[28] MartimortD. and StoleL. (2012) Representing equilibrium aggregates in aggregate games with applications to common agency. Games and Economic Behavior, 76(2), 753-772.10.1016/j.geb.2012.08.005 · Zbl 1250.91060 · doi:10.1016/j.geb.2012.08.005
[29] MondererD. and ShapleyL.S. (1996a) Fictitious play property for games with identical interests. Journal of Economic Theory, 68(1), 258-265.10.1006/jeth.1996.0014 · Zbl 0849.90130 · doi:10.1006/jeth.1996.0014
[30] MondererD. and ShapleyL.S. (1996b) Potential games. Games and Economic Behavior, 14(1), 124-143.10.1006/game.1996.0044 · Zbl 0862.90137 · doi:10.1006/game.1996.0044
[31] PantelousA.A. and PassalidouE. (2013) Optimal premium pricing policy in a competitive insurance market environment. Annals of Actuarial Science, 7(2), 175-191.10.1017/S1748499512000152S1748499512000152 · doi:10.1017/S1748499512000152
[32] PantelousA.A. and PassalidouE. (2015) Optimal premium pricing strategies for competitive general insurance markets. Applied Mathematics and Computation, 259, 858-874.10.1016/j.amc.2015.03.027 · Zbl 1390.91202 · doi:10.1016/j.amc.2015.03.027
[33] PantelousA.A. and PassalidouE. (2016) Optimal strategies for a nonlinear premium-reserve model in a competitive insurance market. Annals of Actuarial Science, forthcoming.
[34] PolbornM.K. (1998) A model of an oligopoly in an insurance market. The Geneva Papers on Risk and Insurance Theory, 23(1), 41-48.10.1023/A:1008677913887 · doi:10.1023/A:1008677913887
[35] PowersM.R. and ShubikM. (1998) On the tradeoff between the law of large numbers and oligopoly in insurance. Insurance: Mathematics and Economics, 23(2), 141-156. · Zbl 0912.62115
[36] PowersM.R., ShubikM. and YaoS.T. (1998) Insurance market games: Scale effects and public policy. Journal of Economics, 67(2), 109-134.10.1007/BF01236065 · Zbl 1035.91038 · doi:10.1007/BF01236065
[37] RantalaJ. (1988) Fluctuations in insurance business results: Some control theoretic aspects. In 23rd International Congress of Actuaries, Vol. R, pp. 43-79.
[38] ReesR., GravelleH. and WambachA. (1999) Regulation of insurance markets. The Geneva Papers on Risk and Insurance Theory, 24(1), 55-68.10.1023/A:1008733315931 · doi:10.1023/A:1008733315931
[39] RolskiT., SchmidliH., SchmidtV. and TeugelsJ. (2009) Stochastic Processes for Insurance and Finance, volume 505. New York, USA: John Wiley & Sons.
[40] RothschildM. and StiglitzJ. (1976) Equilibrium in competitive insurance markets: An essay on the economics of imperfect information. The Quarterly Journal of Economics, 90(4), 629-649.10.2307/1885326 · doi:10.2307/1885326
[41] RothschildM. and StiglitzJ. (1992) Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information. Foundations of Insurance Economics, (eds. G.Dionne and S. E.Harrington), pp. 355-375. Huebner International Series on Risk, Insurance and Economic Security, Netherlands: Springer.
[42] SeltenR. (1970) Preispolitik der Mehrproduktenunternehmung in der statischen Theorie, volume 16. Berlin-Heidelberg, Germany: Springer-Verlag. · Zbl 0195.21801
[43] TaylorG.C. (1986) Underwriting strategy in a competitive insurance environment. Insurance: Mathematics and Economics, 5(1), 59-77. · Zbl 0585.62175
[44] TaylorG.C. (1987) Expenses and underwriting strategy in competition. Insurance: Mathematics and Economics, 6(4), 275-287. · Zbl 0638.90020
[45] TaylorG.C. (2008) A simple model of insurance market dynamics. North American Actuarial Journal, 12(3), 242-262.10.1080/10920277.2008.10597520 · doi:10.1080/10920277.2008.10597520
[46] TeugelsJ. and SundtB. (2004) Encyclopedia of Actuarial Science. New Jersey, USA: John Wiley & Sons. · Zbl 1114.62112
[47] TopkisD.M. (1998) Supermodularity and Complementarity. New Jersey, USA: Princeton University Press.
[48] TsanakasA. and ChristofidesN. (2006) Risk exchange with distorted probabilities. ASTIN Bulletin, 36(1), 219-243.10.1017/S051503610001446X · Zbl 1162.91439 · doi:10.1017/S051503610001446X
[49] VoorneveldM. (2000) Best-response potential games. Economics Letters, 66(3), 289-295.10.1016/S0165-1765(99)00196-2 · Zbl 0951.91008 · doi:10.1016/S0165-1765(99)00196-2
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