A kinetic games framework for insurance plans

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