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# A kinetic games framework for insurance plans

• *Corresponding author: Daniel Brinkman
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.

Mathematics Subject Classification: Primary: 91A13, 82C40, 91A40; Secondary: 91B30.

 Citation: • • 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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