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March  2017, 10(1): 93-116. doi: 10.3934/krm.2017004

A kinetic games framework for insurance plans

School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ 85287-1804, USA

*Corresponding author: Daniel Brinkman

Received  January 2016 Revised  May 2016 Published  November 2016

Fund Project: The authors are supported by NSF RNMS grant No.1107291

The dynamics of insurance plans have been under the microscope in recent years due to the controversy surrounding the implementation of the Affordable Care Act (Obamacare) in the United States. In this paper, we introduce a game between an insurance company and an ensemble of customers choosing between several insurance plans. We then derive a kinetic model for the strategies of the insurer and the decisions of the customers and establish the conditions for which a Nash equilibrium exists for some specific customer distributions. Finally, we give some agent-based numerical results for how the plan enrollment evolves over time which show qualitative agreement to "experimental" results in the literature from two plans in the state of Massachusetts.

Citation: Daniel Brinkman, Christian Ringhofer. A kinetic games framework for insurance plans. Kinetic & Related Models, 2017, 10 (1) : 93-116. doi: 10.3934/krm.2017004
References:
[1]

M. BrugerA. Lorz and M.-T. Wolfram, On a boltzmann mean field model for knowledge growth, SIAM J. Appl. Math., 76 (2016), 1799-1818(20 pages). doi: 10.1137/15M1018599. Google Scholar

[2]

M. BurgerL. A. CaffarelliP. A. Markowich and M.-T. Wolfram, On a boltzmann-type price formation model, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20130126. doi: 10.1098/rspa.2013.0126. Google Scholar

[3]

M. BurgerL. A. CaffarelliP. A. Markowich and M.-T. Wolfram, On the asymptotic behavior of a boltzmann-type price formation model, Communications in Mathematical Sciences, 12 (2014), 1353-1361. doi: 10.4310/CMS.2014.v12.n7.a10. Google Scholar

[4]

L. A. Caffarelli, P. A. Markowich and M.-T. Wolfram, On a price formation free boundary model by lasry and lions: The neumann problem, Comptes Rendus Mathematique, 349 (2011), 841–844, URL http://www.sciencedirect.com/science/article/pii/S1631073X11001932. doi: 10.1016/j.crma.2011.07.006. Google Scholar

[5]

L. A. Caffarelli, P. A. Markowich and J.-F. Pietschmann, On a price formation free boundary model by lasry and lions, Comptes Rendus Mathematique, 349 (2011), 621–624, URL http://www.sciencedirect.com/science/article/pii/S1631073X11001488. doi: 10.1016/j.crma.2011.05.011. Google Scholar

[6]

C. Camerer and T. Ho, Experience-weighted attraction learning in normal form games, Econometrica, 67 (1999), 827-874. doi: 10.1111/1468-0262.00054. Google Scholar

[7]

C. F. Camerer, Behavioral Game Theory: Experiments in Strategic Interaction, Princeton University Press, 2003.Google Scholar

[8]

K. Crammer, O. Dekel, J. Keshet, S. Shalev-Shwartz and Y. Singer, Online passive-aggressive algorithms, J. Mach. Learn. Res., 7 (2006), 551–585, URL http://dl.acm.org/citation.cfm?id=1248547.1248566. Google Scholar

[9]

D. M. Cutler and R. J. Zeckhauser, Adverse Selection in Health Insurance, Working Paper 6107, National Bureau of Economic Research, 1998. doi: 10.2202/1558-9544.1056. Google Scholar

[10]

P. Degond, J. -G. Liu and C. Ringhofer, A nash equilibrium macroscopic closure for kinetic models coupled with mean-field games, 2012.Google Scholar

[11]

P. Degond, J.-G. Liu and C. Ringhofer, Evolution of the distribution of wealth in an economic environment driven by local nash equilibria, Journal of Statistical Physics, 154 (2014), 751–780, URL http://dx.doi.org/10.1007/s10955-013-0888-4. doi: 10.1007/s10955-013-0888-4. Google Scholar

[12]

B. DüringP. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, Boltzmann and fokker-planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. Google Scholar

[13]

R. Dusansky and C. Koc, Implications of the interaction between insurance choice and medical care demand, Journal of Risk and Insurance, 77 (2010), 129-144. doi: 10.1111/j.1539-6975.2009.01335.x. Google Scholar

[14]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179. Google Scholar

[15]

S. Hoi, J. Wang and P. Zhao, Libol: A library for online learning algorithms, Journal of Machine Learning Research, 15 (2014), 495–499, URL http://jmlr.org/papers/v15/hoi14a.html.Google Scholar

[16]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229–260, URL http://dx.doi.org/10.1007/s11537-007-0657-8. doi: 10.1007/s11537-007-0657-8. Google Scholar

[17]

J. F. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America, 36 (1950), 48-89. doi: 10.1073/pnas.36.1.48. Google Scholar

[18]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A boltzmann approach Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130396, 15 pp. doi: 10.1098/rsta.2013.0396. Google Scholar

[19]

D. Schmeidler, Equilibrium points of nonatomic games, Journal of Statistical Physics, 7 (1973), 295-300. doi: 10.1007/BF01014905. Google Scholar

[20]

N. Slonim, E. Yom-Tov and K. Crammer, Active online classification via information maximization, in Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence -Volume Volume Two, IJCAI'11, AAAI Press, 2011,1498-1504, URL http://dx.doi.org/10.5591/978-1-57735-516-8/IJCAI11-252Google Scholar

[21]

J. M. Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0. Google Scholar

[22]

R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction, IEEE Transactions on Neural Networks, 9 (1998), p1054. doi: 10.1109/TNN.1998.712192. Google Scholar

[23]

G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[24]

G. Toscani and C. Brugna, Wealth redistribution in boltzmann-like models of conservative economies, in Econophysics and Economics of Games, Social Choices and Quantitative Techniques (eds. B. Basu, S. R. Chakravarty, B. K. Chakrabarti and K. Gangopadhyay), New Economic Windows, Springer Milan, 2010, 71-82, URL http://dx.doi.org/10.1007/978-88-470-1501-2_9. doi: 10.1007/978-88-470-1501-2_9. Google Scholar

[25]

C. Wilson, A model of insurance markets with incomplete information, Journal of Economic Theory, 16 (1977), 167–207, URL http://www.sciencedirect.com/science/article/pii/0022053177900047. doi: 10.1016/0022-0531(77)90004-7. Google Scholar

show all references

References:
[1]

M. BrugerA. Lorz and M.-T. Wolfram, On a boltzmann mean field model for knowledge growth, SIAM J. Appl. Math., 76 (2016), 1799-1818(20 pages). doi: 10.1137/15M1018599. Google Scholar

[2]

M. BurgerL. A. CaffarelliP. A. Markowich and M.-T. Wolfram, On a boltzmann-type price formation model, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 469 (2013), 20130126. doi: 10.1098/rspa.2013.0126. Google Scholar

[3]

M. BurgerL. A. CaffarelliP. A. Markowich and M.-T. Wolfram, On the asymptotic behavior of a boltzmann-type price formation model, Communications in Mathematical Sciences, 12 (2014), 1353-1361. doi: 10.4310/CMS.2014.v12.n7.a10. Google Scholar

[4]

L. A. Caffarelli, P. A. Markowich and M.-T. Wolfram, On a price formation free boundary model by lasry and lions: The neumann problem, Comptes Rendus Mathematique, 349 (2011), 841–844, URL http://www.sciencedirect.com/science/article/pii/S1631073X11001932. doi: 10.1016/j.crma.2011.07.006. Google Scholar

[5]

L. A. Caffarelli, P. A. Markowich and J.-F. Pietschmann, On a price formation free boundary model by lasry and lions, Comptes Rendus Mathematique, 349 (2011), 621–624, URL http://www.sciencedirect.com/science/article/pii/S1631073X11001488. doi: 10.1016/j.crma.2011.05.011. Google Scholar

[6]

C. Camerer and T. Ho, Experience-weighted attraction learning in normal form games, Econometrica, 67 (1999), 827-874. doi: 10.1111/1468-0262.00054. Google Scholar

[7]

C. F. Camerer, Behavioral Game Theory: Experiments in Strategic Interaction, Princeton University Press, 2003.Google Scholar

[8]

K. Crammer, O. Dekel, J. Keshet, S. Shalev-Shwartz and Y. Singer, Online passive-aggressive algorithms, J. Mach. Learn. Res., 7 (2006), 551–585, URL http://dl.acm.org/citation.cfm?id=1248547.1248566. Google Scholar

[9]

D. M. Cutler and R. J. Zeckhauser, Adverse Selection in Health Insurance, Working Paper 6107, National Bureau of Economic Research, 1998. doi: 10.2202/1558-9544.1056. Google Scholar

[10]

P. Degond, J. -G. Liu and C. Ringhofer, A nash equilibrium macroscopic closure for kinetic models coupled with mean-field games, 2012.Google Scholar

[11]

P. Degond, J.-G. Liu and C. Ringhofer, Evolution of the distribution of wealth in an economic environment driven by local nash equilibria, Journal of Statistical Physics, 154 (2014), 751–780, URL http://dx.doi.org/10.1007/s10955-013-0888-4. doi: 10.1007/s10955-013-0888-4. Google Scholar

[12]

B. DüringP. MarkowichJ.-F. Pietschmann and M.-T. Wolfram, Boltzmann and fokker-planck equations modelling opinion formation in the presence of strong leaders, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. Google Scholar

[13]

R. Dusansky and C. Koc, Implications of the interaction between insurance choice and medical care demand, Journal of Risk and Insurance, 77 (2010), 129-144. doi: 10.1111/j.1539-6975.2009.01335.x. Google Scholar

[14]

J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998. doi: 10.1017/CBO9781139173179. Google Scholar

[15]

S. Hoi, J. Wang and P. Zhao, Libol: A library for online learning algorithms, Journal of Machine Learning Research, 15 (2014), 495–499, URL http://jmlr.org/papers/v15/hoi14a.html.Google Scholar

[16]

J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007), 229–260, URL http://dx.doi.org/10.1007/s11537-007-0657-8. doi: 10.1007/s11537-007-0657-8. Google Scholar

[17]

J. F. Nash, Equilibrium points in n-person games, Proceedings of the National Academy of Sciences of the United States of America, 36 (1950), 48-89. doi: 10.1073/pnas.36.1.48. Google Scholar

[18]

L. Pareschi and G. Toscani, Wealth distribution and collective knowledge: A boltzmann approach Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 372 (2014), 20130396, 15 pp. doi: 10.1098/rsta.2013.0396. Google Scholar

[19]

D. Schmeidler, Equilibrium points of nonatomic games, Journal of Statistical Physics, 7 (1973), 295-300. doi: 10.1007/BF01014905. Google Scholar

[20]

N. Slonim, E. Yom-Tov and K. Crammer, Active online classification via information maximization, in Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence -Volume Volume Two, IJCAI'11, AAAI Press, 2011,1498-1504, URL http://dx.doi.org/10.5591/978-1-57735-516-8/IJCAI11-252Google Scholar

[21]

J. M. Smith and G. R. Price, The logic of animal conflict, Nature, 246 (1973), 15-18. doi: 10.1038/246015a0. Google Scholar

[22]

R. S. Sutton and A. G. Barto, Reinforcement learning: An introduction, IEEE Transactions on Neural Networks, 9 (1998), p1054. doi: 10.1109/TNN.1998.712192. Google Scholar

[23]

G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496. doi: 10.4310/CMS.2006.v4.n3.a1. Google Scholar

[24]

G. Toscani and C. Brugna, Wealth redistribution in boltzmann-like models of conservative economies, in Econophysics and Economics of Games, Social Choices and Quantitative Techniques (eds. B. Basu, S. R. Chakravarty, B. K. Chakrabarti and K. Gangopadhyay), New Economic Windows, Springer Milan, 2010, 71-82, URL http://dx.doi.org/10.1007/978-88-470-1501-2_9. doi: 10.1007/978-88-470-1501-2_9. Google Scholar

[25]

C. Wilson, A model of insurance markets with incomplete information, Journal of Economic Theory, 16 (1977), 167–207, URL http://www.sciencedirect.com/science/article/pii/0022053177900047. doi: 10.1016/0022-0531(77)90004-7. Google Scholar

Figure 1.  Numerical examples for the first case of Theorem 3.2. All initial plan prices approach zero price trivial steady-states
Figure 2.  Numerical examples for the second case of Theorem 3.2. Initial plan prices below $r = \alpha M = 5$ are essentially stationary steady-states. Initial plan prices above $r = \alpha M = 5$ decay to a steady-state near $r = 5$
Figure 3.  Numerical examples for the third case of Theorem 3.2. All initial plan prices $0 < r < 10$ converge to a steady state with $r \approx 7.14$. The first case had a steady state price of $r=7.165$ and the second steady-state occurred for $r=7.160$
Figure 4.  Numerical examples for the fourth case of Theorem 3.2. Every initial plan price increases to the maximum $M$. Note that according to Equation (24), a plan with an enrollment of zero has an undefined price. The simulation responds by setting the price to zero
Figure 5.  For $\beta = 1.3$, all plans go extinct and profits go to zero. The insurer occasionally realizes negative profits
Figure 6.  For $\beta = 4/3$, the theorem is unclear, but numerically we see relative stability of multiple plans. The insurer profits are bounded away from zero
Figure 7.  For $\beta = 5/3$, we expect a unique steady state plan price. Although the random accidents eliminate a true steady state, we quickly reach a probabalistic steady state. The long-term profits are higher than the other cases
Figure 8.  For $\beta = 7/3$, all plans go extinct and profits go to zero. Unlike the first case, the profits are always positive, but go to zero as the plan prices increase above the costs of the most expensive individual
Figure 9.  Normalized profit as a function of the profit parameter $\beta$. Note that a local maximum occurs for the lowest possible surviving profit, suggesting that greediness on the part of the insurer may actually reduce profits
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