Modeling and designing health care payment innovations for medical imaging

Abstract

Payment innovations that better align incentives in health care are a promising approach to reduce health care costs and improve quality of care. Designing effective payment systems, however, is challenging due to the complexity of the health care system with its many stakeholders and their often conflicting objectives. There is a lack of mathematical models that can comprehensively capture and efficiently analyze the complex, multi-level interactions and thereby predict the effect of new payment systems on stakeholder decisions and system-wide outcomes. To address the need for multi-level health care models, we apply multiscale decision theory (MSDT) and build upon its recent advances. In this paper, we specifically study the Medicare Shared Savings Program (MSSP) for Accountable Care Organizations (ACOs) and determine how this incentive program affects computed tomography (CT) use, and how it could be redesigned to minimize unnecessary CT scans. The model captures the multi-level interactions, decisions and outcomes for the key stakeholders, i.e., the payer, ACO, hospital, primary care physicians, radiologists and patients. Their interdependent decisions are analyzed game theoretically, and equilibrium solutions - which represent stakeholders’ normative decision responses - are derived. Our results provide decision-making insights for the payer on how to improve MSSP, for ACOs on how to distribute MSSP incentives among their members, and for hospitals on whether to invest in new CT imaging systems.

Appendix

1.1 A1. Representation of agents’ UMPs

In agent P’s UMP, its monetary payoff is

$$\begin{array}{@{}rcl@{}} &&{{\Pi}^{P1}}(\theta_{P}|a_{h})={{N}_{1}}\cdot \{\frac{1}{2}({{c}_{H,A,N}}+{{c}_{S,A,N}})\\ &&-\theta_{P}\cdot [{{c}_{S,A,N}}-\tilde{q}{{c}_{S,I,T}}-(1-\tilde{q}){{c}_{S,I,N}}]\\ &&+{\theta_{P}^{2}}\cdot \frac{1}{2}[({{c}_{S,A,N}}-{{c}_{H,A,N}})+(1-\tilde{q})({{c}_{H,I,T}}-{{c}_{S,I,N}})\\ &&+\tilde{q}({{c}_{H,I,N}}-{{c}_{S,I,T}})]\}. \end{array} $$

Similarly, agent P’s health benefit is

$$\begin{array}{@{}rcl@{}} &&{{B}^{P1}}(\theta_{P}|a_{h})={{N}_{1}}\cdot \{\frac{1}{2}({{\mu}_{H,A,N}}+{{\mu}_{S,A,N}})\\ &&-\theta_{P}\cdot [{{\mu}_{S,A,N}}-\tilde{q}{{\mu}_{S,I,T}}-(1-\tilde{q}){{\mu}_{S,I,N}}] \\ &&+{\theta_{P}^{2}}\cdot \frac{1}{2}[({{\mu}_{S,A,N}}-{{\mu}_{H,A,N}})+(1\!-\tilde{q})({{\mu}_{H,I,T}}\!-{{\mu}_{S,I,N}})\\ &&+\tilde{q}({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}})]\}. \end{array} $$

In agent R’s UMP, its monetary payoff is

$${{\Pi}^{R1}}(\theta_{P})+{{\Pi}^{R2}}(\theta_{R})={{N}_{1}}\cdot \theta_{P}\cdot {{c}_{I}}+{{N}_{2}}\cdot \theta_{R}\cdot {{c}_{I}}. $$

Agent R’s health benefit is

$$\begin{array}{@{}rcl@{}} &&{{B}^{R2}}(\theta_{R}|a_{h})={{N}_{2}}\cdot \{\frac{1}{2}{{\mu}_{H,A,N}}(2-r)+\frac{1}{2}{{\mu}_{S,A,N}}\cdot r \\ &&-\theta_{R}\cdot \{r\cdot [{{\mu}_{S,A,N}}-\tilde{q}{{\mu}_{S,I,T}}-(1-\tilde{q}){{\mu}_{S,I,N}}]\\ &&+(1-r)[{{\mu}_{H,A,N}}-\tilde{q}{{\mu}_{H,I,N}}-(1-\tilde{q}){{\mu}_{H,I,T}}]\} \\ &&+{\theta_{R}^{2}}\cdot \frac{1}{2}r\cdot [({{\mu}_{S,A,N}}-{{\mu}_{H,A,N}})\\ &&+(1-\tilde{q})({{\mu}_{H,I,T}}-{{\mu}_{S,I,N}})+\tilde{q}({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}})]. \end{array} $$

1.2 A2 . Proof of Theorem 1

It is easy to obtain the expressions of $\theta _{P}^{*h}$ and $\theta _{R}^{*h}$ via taking derivatives of B ^P1(𝜃 _P |a _h ) and B ^R2(𝜃 _R |a _h ). Additionally, it is easy to check that $\theta _{P}^{*m}=\theta _{R}^{*m}=1$.

Next, we show that $\theta _{P}^{*}$ satisfies $\theta _{P}^{*h}<\theta _{P}^{*}\le 1$. Denote

$$\begin{array}{@{}rcl@{}} {{X}_{1}}&=&{{\mu}_{S,A,N}}-\tilde{q}{{\mu}_{S,I,T}}-(1-\tilde{q}){{\mu}_{S,I,N}},\text{} \\ {{X}_{2}}&=&{{\mu}_{S,A,N}}-{{\mu}_{H,A,N}}+(1-\tilde{q})({{\mu}_{H,I,T}}-{{\mu}_{S,I,N}})\\ &&+\tilde{q}({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}}). \end{array} $$

We have $0<\theta _{P}^{*h}=\frac {{{X}_{1}}}{{{X}_{2}}}<1,\text {} {{X}_{1}}<0,\text {} {{X}_{2}}<0$. The first derivative of U ^P(𝜃 _P |a _h ) with respect to 𝜃 _P is

$$\begin{array}{@{}rcl@{}} &\frac{\partial {{U}^{P}}(\theta_{P}|a_{h})}{\partial \theta_{P}}=-{{N}_{1}}[{{c}_{S,A,N}}-\tilde{q}{{c}_{S,I,T}}-(1-\tilde{q}){{c}_{S,I,N}}\\ &+{{\lambda}^{P}}{{X}_{1}}]+{{N}_{1}}\theta_{P}[(1-2\tilde{q})({{c}_{H,I,T}}-{{c}_{H,I,N}})+{{\lambda} ^{P}}{{X}_{2}}]. \end{array} $$

Given the inequalities ${{c}_{S,A,N}}-\tilde {q}{{c}_{S,I,T}}-(1-\tilde {q}){{c}_{S,I,N}}<0,\text {} (1-2\tilde {q})({{c}_{H,I,T}}-{{c}_{H,I,N}})<0\text {,} {{X}_{1}}<0,\text {} {{X}_{2}}<0$, we have

$$\theta_{P}^{*}=\left\{\begin{array}{l} \frac{{{c}_{S,A,N}}-\tilde{q}{{c}_{H,I,T}}-(1-\tilde{q}){{c}_{H,I,N}}+{{\lambda}^{P}}{{X}_{1}}}{(1-2\tilde{q})({{c}_{H,I,T}}-{{c}_{H,I,N}})+{{\lambda}^{P}}{{X}_{2}}}, \\ \text{} \text{} \text{} \text{ if} \frac{{{c}_{S,A,N}}-\tilde{q}{{c}_{H,I,T}}-(1-\tilde{q}){{c}_{H,I,N}}+{{\lambda}^{P}}{{X}_{1}}}{(1-2\tilde{q})({{c}_{H,I,T}}-{{c}_{H,I,N}})+{{\lambda}^{P}}{{X}_{2}}}<1 \\ 1, \text{ if} \frac{{{c}_{S,A,N}}-\tilde{q}{{c}_{H,I,T}}-(1-\tilde{q}){{c}_{H,I,N}}+{{\lambda} ^{P}}{{X}_{1}}}{(1-2\tilde{q})({{c}_{H,I,T}}-{{c}_{H,I,N}})+{{\lambda}^{P}}{{X}_{2}}}\ge 1 \end{array}\right.. $$

Because ${{c}_{S,A,N}}-\tilde {q}{{c}_{S,I,T}}-(1-\tilde {q}){{c}_{S,I,N}}-(1-2\tilde {q})({{c}_{H,I,T}}-{{c}_{H,I,N}})<0$, the result $\theta _{P}^{*h}<\theta _{P}^{*}\le 1$ is then immediate.

Following the similar reasoning process as above, we have $\theta _{R}^{*h}<\theta _{R}^{*}\le 1$. $\square$

1.3 A3. Proof of Theorem 2

For agent P: denote

$$\begin{array}{@{}rcl@{}} {{X}_{3}}&=&{{\mu}_{S,A,N}}-q{{\mu}_{S,I,T}}-(1-q){{\mu}_{S,I,N}},\text{} \\ {{X}_{4}}&=&{{\mu}_{S,A,N}}-{{\mu}_{H,A,N}}+(1-q)({{\mu}_{H,I,T}}-{{\mu}_{S,I,N}})\\ &&+q({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}}),\\ {{X}_{5}}&=&-\delta {{\mu}_{S,I,T}}+\delta {{\mu}_{S,I,N}},\text{} \\ {{X}_{6}}&=&-\delta ({{\mu}_{H,I,T}}-{{\mu}_{S,I,N}})+\delta ({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}}). \end{array} $$

We have $\theta _{P}^{*h}(a_{2})=\frac {{{X}_{3}}}{{{X}_{4}}}$, $\theta _{P}^{*h}(a_{1})=\frac {{{X}_{3}}+{{X}_{5}}}{{{X}_{4}}+{{X}_{6}}}$. Notice that X ₃<0,X ₄<0,X ₃>X ₄,X ₅−X ₆<0,X ₅<0,X ₆<0, we have

$$\theta_{P}^{*h}(a_{1})-\theta_{P}^{*h}(a_{2})=\frac{{{X}_{4}}{{X}_{5}}-{{X}_{3}}{{X}_{6}}}{({{X}_{4}}+{{X}_{6}}){{X}_{4}}}>0. $$

Therefore, when agent H switches from a ₂=1 to a ₁=1, $\theta _{P}^{*h}$ increases.

Next we consider the changes in $\theta _{P}^{*}$ when agent H switches from a ₂=1 to a ₁=1. Denote

$$\begin{array}{@{}rcl@{}} &&X_{7}=\\ &&\frac{\left( \begin{array}{l} {c}_{S,A,N}-(q+{\delta}){c_{S,I,T}}-(1- q- \delta){c_{S,I,N}}\\ +{\lambda}^{P}[{{\mu}_{S,A,N}}-(q+ \delta){{\mu}_{S,I,T}}-(1- q- \delta){\mu_{S,I,N}}] \end{array}\right)}{\left( \begin{array}{l} [1-2(q +\delta)](c_{H,I,T}-c_{H,I,T})+{\lambda}^{P}[(\mu_{S,A,N}-\mu_{H,A,N})]\\ \quad +(1- q- \delta)({{\mu}_{H,I,T}}-{{\mu}_{S,I,N}})+(q+ \delta)({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}})\left.\right] \end{array}\right)}. \end{array} $$

When $\theta _{P}^{*}(a_{2})<1$, $\theta _{P}^{*}(a_{1})-\theta _{P}^{*}(a_{2})>0$ is equivalent to

$$\left\{\begin{array}{l} {{X}_{7}}\ge 1>\theta_{P}^{*}(a_{2}), \text{ if} \theta_{P}^{*}(a_{1})=1 \\ {{X}_{7}}>\theta_{P}^{*}(a_{2}), \text{ if} \theta_{P}^{*}(a_{1})<1 \end{array}\right.. $$

Hence, $\theta _{P}^{*}(a_{1})-\theta _{P}^{*}(a_{2})>0$ is equivalent to: ${{X}_{7}}-\theta _{P}^{*}(a_{2})>0$. Compute this inequality, we obtain the condition Eq. 14 in Theorem 2.

For agent R: denote $\theta _{R}^{*h}(a_{2})=\frac {{{X}_{8}}}{{{X}_{9}}}$, X ₈<0,X ₉<0,X ₈>X ₉; denote $\theta _{R}^{*h}(a_{1})=\frac {{{X}_{8}}+{{X}_{10}}}{{{X}_{9}}+{{X}_{11}}}$, where

$$\begin{array}{@{}rcl@{}} {{X}_{10}}&=&\delta \cdot r\cdot (-{{\mu}_{S,I,T}}+{{\mu}_{S,I,N}})\\ &&+\delta (1-r)(-{{\mu}_{H,I,N}}+{{\mu}_{H,I,T}}),\text{} \\ {{X}_{11}}&=&\delta \cdot r\cdot (-{{\mu}_{H,I,T}}+{{\mu}_{S,I,N}}+{{\mu}_{H,I,N}}-{{\mu}_{S,I,T}}). \end{array} $$

Because X ₁₀−X ₁₁<0,X ₁₀<0,X ₁₁<0, we have

$$\theta_{R}^{*h}(a_{1})-\theta_{R}^{*h}(a_{2})=\frac{{{X}_{9}}{{X}_{10}}-{{X}_{8}}{{X}_{11}}}{({{X}_{9}}+{{X}_{11}}){{X}_{9}}}>0. $$

Therefore, when agent H switches from a ₂=1 to a ₁=1, $\theta _{R}^{*h}$ increases.

Next we consider the changes in $\theta _{R}^{*}$ when agent H switches from a ₂=1 to a ₁=1. Denote

$$\begin{array}{@{}rcl@{}} &&{X}_{12} =\\ &&\frac{\left( \begin{array}{l} -{{c}_{I}}+\lambda^{R}\left\{r\cdot[\mu_{S,A,N}-(q+\delta)\mu_{S,I,T}\right.\\ \quad -(1-q-\delta)\mu_{S,I.N}]+ (1-r)\left[\mu_{H,A,N}\right.\\ \qquad \left.- (q+ \delta)\mu_{H,I,N}-(1-q-\delta)\mu_{H,I,T}\right\}\end{array}\right)}{\left( \begin{array}{l} \lambda^{R}\cdot r \cdot\left[\mu_{S,A,N}-\mu_{H,A,N}+ (1-q-\delta)\left( \mu_{H,I,T}\right.\right.\\ \qquad\quad \left.\left.-\mu_{S,I,N}\right)+(q+ \delta)(\mu_{H,I,N}-{\mu_{S,I,T}})\right] \end{array}\right)}. \end{array} $$

Similarly, we have $\theta _{R}^{*}(a_{1})-\theta _{R}^{*}(a_{2})>0$ equivalent to ${{X}_{12}}-\theta _{R}^{*}(a_{2})>0$. Next, following the same reasoning process in the proof for the change in $\theta _{R}^{*h}$, it is easy to check that when agent H switches from a ₂=1 to a ₁=1 and when $\theta _{R}^{*}(a_{2})<1$, $\theta _{R}^{*}$ increases. $\square$

1.4 A4. Proof of Theorem 3

First, we provide the mathematical expressions for $\theta _{P}^{**}$ and $\theta _{R}^{**}$.

$$\theta_{P}^{**}=\left\{\begin{array}{l} \widetilde{\theta}_{P}, \text{ if} 0<\widetilde{\theta}_{P}<1 \\ 0, \text{ if} \widetilde{\theta}_{P}\le 0 \\ 1, \text{ if} \widetilde{\theta}_{P}\ge 1 \end{array}\right.,\text{} where $$

$$\widetilde{\theta}_{P}= \frac{\left( \begin{array}{r}\eta \alpha (1+{{\gamma}_{p}}){{c}_{I}}+(1-\eta \alpha)[c_{S,A,N}-\tilde{q}{c}_{S,I,T}\\-(1-\tilde{q})c_{S,I,N}]+\lambda^{P}[\mu_{S,A,N}\\-\tilde{q}\mu_{S,I,T}- (1-\tilde{q})\mu_{S,I,N}]\end{array} \right)}{\left( \begin{array}{l}(1-\eta \alpha)(1-2\tilde{q})({{c}_{H,I,T}}-{{c}_{H,I,N}})\\\quad+{{\lambda}^{P}}[({{\mu}_{S,A,N}}-{{\mu} _{H,A,N}})\\ \quad\quad+(1-\tilde{q})({{\mu}_{H,I,T}}-{{\mu}_{S,I,N}})+\tilde{q}({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}})]\end{array} \right)}; $$

$$\theta_{R}^{**}=\left\{\begin{array}{ll} \widetilde{\theta}_{R}, \text{ if} 0<\widetilde{\theta}_{R}<1\\ 0, \text{ if} \widetilde{\theta}_{R}\le 0\\ 1, \text{ if} \widetilde{\theta}_{R}\ge 1 \end{array}\right.,\text{} where $$

$$\widetilde{\theta}_{R}\,=\, \frac{\left( \begin{array}{l} {-{{c}_{I}}+\eta \beta (1+{{\gamma}_{p}}){{c}_{I}}} \\ \quad+{{\lambda}^{R}}\{r\cdot [{{\mu}_{S,A,N}}-\tilde{q}{{\mu}_{S,I,T}}-(1-\tilde{q}){{\mu}_{S,I,N}}]\\\qquad++(1-r)[{{\mu} _{H,A,N}}-\tilde{q}{{\mu}_{H,I,N}}\,-\,(1-\tilde{q}){{\mu}_{H,I,T}}]\} \end{array}\right)} {\left( \begin{array}{l}{{\lambda} ^{R}}\cdot r\cdot [{{\mu}_{S,A,N}}-{{\mu}_{H,A,N}}\\\quad\,+\,(1-\tilde{q})({{\mu}_{H,I,T}}-{{\mu} _{S,I,N}})\,+\,\tilde{q}({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}})]\end{array} \right)}. $$

Next, we prove Theorem 3(a). For agent P: denote

$$\begin{array}{@{}rcl@{}} {{X}_{13}}&=&{{c}_{S,A,N}}\,-\,\tilde{q}{{c}_{S,I,T}}\,-\,(1-\tilde{q}){{c}_{S,I,N}},\text{}\\ {{X}_{14}}&=&(1-2\tilde{q})({{c}_{H,I,T}}-{{c}_{H,I,N}}),\\ {{X}_{15}}&=&{{\lambda}^{P}}[{{\mu}_{S,A,N}}-\tilde{q}{{\mu}_{S,I,T}}\,-\,(1-\tilde{q}){{\mu}_{S,I,N}}],\\ {{X}_{16}}&=&{{\lambda}^{P}}[({{\mu}_{S,A,N}}\,-\,{{\mu}_{H,A,N}})\,+\,(1-\tilde{q})({{\mu}_{H,I,T}}-{{\mu}_{S,I,N}})\\ &&+\tilde{q}({{\mu}_{H,I,N}}-{{\mu}_{S,I,T}})]. \end{array} $$

By Theorem 1, we have X ₁₃<X ₁₄<0,X ₁₆<X ₁₅<0. When $\theta _{P}^{**}\in (0,1)$, we have

$$\theta_{P}^{**}=\frac{\eta \alpha (1+{{\gamma}_{p}}){{c}_{I}}+(1-\eta \alpha ){{X}_{13}}+{{X}_{15}}}{(1-\eta \alpha ){{X}_{14}}+{{X}_{16}}}. $$

Take the first derivative of $\theta _{P}^{**}$ with respect to α, and we have the inequality

$$\frac{\partial \theta_{P}^{**}}{\partial \alpha} \,=\,\frac{[{{X}_{14}}{{X}_{15}}\,-\,{{X}_{13}}{{X}_{16}}\,+\,({{X}_{14}}\,+\,{{X}_{16}})(1\,+\,{{\gamma}_{p}}){{c}_{I}}]\eta} {{{({{X}_{14}}\,+\,{{X}_{16}}\,-\,{{X}_{14}}\eta \alpha )}^{2}}}<0, $$

$$\eta \in (0,1],\text{} \alpha \in [0,1],\text{} {{\gamma}_{p}}\in {{\mathbb{R}}^{+}}. $$

Hence $\theta _{P}^{**}$ is a strict decreasing function of α when $\theta _{P}^{**}\in (0,1)$.

For agent R: similarly, when $\theta _{R}^{**}\in (0,1)$, denote $\theta _{R}^{**}=\frac {-{{c}_{I}}+\eta \beta (1+{{\gamma }_{p}}){{c}_{I}}+{{X}_{17}}}{{{X}_{18}}}$. We have X ₁₇<0,X₁₈<0. Take the first derivative of $\theta _{R}^{**}$ with respect to β, we have the inequality

$$\frac{\partial \theta_{R}^{**}}{\partial \beta} =\frac{\eta (1+{{\gamma}_{p}}){{c}_{I}}}{{{X}_{18}}}<0, \eta \in (0,1],\text{} \beta \in [0,1],\text{} {{\gamma}_{p}}\in {{\mathbb{R}}^{+}}. $$

Hence $\theta _{R}^{**}$ is a strict decreasing function of β when $\theta _{R}^{**}\in (0,1).$ $\square$

Lastly, we prove Theorem 3(b).

For agent P: using previous notations, we have

$$\frac{\eta \alpha (1+{{\gamma}_{p}}){{c}_{I}}+(1-\eta \alpha ){{X}_{13}}+{{X}_{15}}}{(1-\eta \alpha ){{X}_{14}}+{{X}_{16}}}\le \frac{{{X}_{13}}+{{X}_{15}}}{{{X}_{14}}+{{X}_{16}}}, $$

and the equality is reached when α=0 (by monotonicity).

Recall that $\theta _{P}^{*}=Min\{1,\text {} \frac {{{X}_{13}}+{{X}_{15}}}{{{X}_{14}}+{{X}_{16}}}\}$.

When $\frac {{{X}_{13}}+{{X}_{15}}}{{{X}_{14}}+{{X}_{16}}}\le 1$, $\theta _{P}^{**}\le \frac {{{X}_{13}}+{{X}_{15}}}{{{X}_{14}}+{{X}_{16}}}=\theta _{P}^{*}$.

When $\frac {{{X}_{13}}+{{X}_{15}}}{{{X}_{14}}+{{X}_{16}}}>1$, $\theta _{P}^{**}\le 1=\theta _{P}^{*}$.

Hence we always have $\theta _{P}^{**}\le \theta _{P}^{*}$.

For agent R: following previous notations, we have

$$\frac{-{{c}_{I}}+\eta \beta (1+{{\gamma}_{p}}){{c}_{I}}+{{X}_{17}}}{{{X}_{18}}}\le \frac{-{{c}_{I}}+{{X}_{17}}}{{{X}_{18}}}, $$

and the equality is reached when β=0 (by monotonicity).

Recall that $\theta _{R}^{*}=Min\{1,\text {} \frac {-{{c}_{I}}+{{X}_{17}}}{{{X}_{18}}}\}$.

When $\frac {-{{c}_{I}}+{{X}_{17}}}{{{X}_{18}}}\le 1$, $\theta _{R}^{**}\le \frac {-{{c}_{I}}+{{X}_{17}}}{{{X}_{18}}}=\theta _{R}^{*}$.

When $\frac {-{{c}_{I}}+{{X}_{17}}}{{{X}_{18}}}>1$, $\theta _{R}^{**}\le 1=\theta _{R}^{*}$.

Hence we always have $\theta _{R}^{**}\le \theta _{R}^{*}$. $\square$

1.5 A5. Parameter values for numerical analysis

The parameter values used in Section 5 Numerical Analysis are listed in Table 5.

Table 5 Parameter Values for Numerical Analysis