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A game-theoretic framework for estimating a health purchaser’s willingness-to-pay for health and for expansion

Abstract

A health purchaser’s willingness-to-pay (WTP) for health is defined as the amount of money the health purchaser (e.g. a health maximizing public agency or a profit maximizing health insurer) is willing to spend for an additional unit of health. In this paper, we propose a game-theoretic framework for estimating a health purchaser’s WTP for health in markets where the health purchaser offers a menu of medical interventions, and each individual in the population selects the intervention that maximizes her prospect. We discuss how the WTP for health can be employed to determine medical guidelines, and to price new medical technologies, such that the health purchaser is willing to implement them. The framework further introduces a measure for WTP for expansion, defined as the amount of money the health purchaser is willing to pay per person in the population served by the health provider to increase the consumption level of the intervention by one percent without changing the intervention price. This measure can be employed to find how much to invest in expanding a medical program through opening new facilities, advertising, etc. Applying the proposed framework to colorectal cancer screening tests, we estimate the WTP for health and the WTP for expansion of colorectal cancer screening tests for the 2005 US population.

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Notes

  1. These conditions include weak ordering, monotonicity, continuity, tail independence, and constant absolute risk aversion for gains and for losses. The reader is referred to [45] for the exact definition of each axiom.

  2. Remember that for an individual of rank θ, the WTP for a given intervention j, denoted by WTP j , solves u( − e j , − WTP j ) + U j (θ) = U 0 (θ) (See Eq. 1).

  3. Note that “WTP for health” and “WTP for a medical intervention” are totally distinct concepts. The WTP for health is defined as the amount of money that the decision maker (here HP) is willing to spend for one unit of health, while the WTP for a medical intervention is the amount of money that the patient is willing to pay to remain on the same utility curve as not using the intervention at all (refer to footnote 2). While the WTP for health is the direct result of limited medical resources, the WTP for a medical intervention is the result of the patient’s preference over monetary and health-related outcomes.

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Acknowledgements

The authors would like to thank Reha Uzsoy, Michael Pignone, and three anonymous reviewers for their comments and insights which improved this work in substance and style. This work is, in part, supported by a Fitts Graduate Fellowship.

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Correspondence to Reza Yaesoubi.

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Appendices

Appendix A: Glossary of notation

Table 2 summarizes the notation used throughout the paper.

Table 2 Glossary of notation

Appendix B: Proof of Theorem 1

To prove Theorem 1, we need the following lemma.

Lemma 1

Let

$$ \textrm{\bf{A}}_{k + 1} = \left( \begin{array}{ccccc} 0 & a_1 & a_2 & \cdots & a_k \\ a_1 & b_1 & 0 & \cdots & 0 \\ a_2 & 0 & b_2 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_k & 0 & 0 & \cdots & b_k \end{array} \right) $$

then

$$ |{\bf{A}}_{k + 1} |\, = - \sum\nolimits_{i = 1}^k {a_i^2 \prod\nolimits_{j \ne i} {b_j } }. $$
(25)

Proof

Expression (25) is true for k = 1. Now, suppose that it is true for k + 1, k ≥ 2; we now show that it is also true for k + 2.

$$\begin{array}{rll} |{\bf{A}}_{k + 2} |&=& \left| {\begin{array}{*{20}c} 0 & {a_1 } & \cdots & {a_k } & {a_{k + 1} } \\ {a_1 } & {b_1 } & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {a_k } & 0 & \cdots & {b_k } & \vdots \\ {a_{k + 1} } & 0 & \cdots & \cdots & {b_{k + 1} } \\ \end{array}} \right| \\[-3pt] &=& (-1)^{k+1} a_{k + 1} \left| {\begin{array}{*{20}c} {a_1 } & \cdots & {a_k } & {a_{k+1} } \\ {b_1 } & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & {b_k } & 0 \\ \end{array}} \right| \\[-3pt] &&-\, b_{k + 1} |{\bf{A}}_{k + 1} |\, \\[-3pt] &=& (-1)^{k+1}(-1)^k a_{k + 1}^2 b_1 \cdots b_k - b_{k + 1} |{\bf{A}}_{k + 1} |\\[-3pt] & =& - a_{k + 1}^2 b_1 \cdots b_k - \sum\nolimits_{i = 1}^k {a_i^2 \prod\nolimits_{j \ne i} {b_j } } \\[-3pt] &=& - \sum\nolimits_{i = 1}^{k + 1} {a_i^2 \prod\nolimits_{j \ne i} {b_j } }. \end{array}$$

The Lagrangian function of optimization problem (2) is:

$$\begin{array}{rll} L(\theta_1 ,\theta_2 , \ldots ,\theta_M ) &=& \sum\nolimits_{j = 1}^M {V_j (\theta_j )} \nonumber\\ &&+\, \lambda \left( {\sum\nolimits_{j = 1}^M {Q_j (\theta_j ) - Q^0 } } \right). \end{array}$$
(26)

The first-order optimality condition will be:

$$\begin{array}{lll} &&{\kern-6pt} (1-\beta)\big(v_j (\theta_j ) - v_{j - 1} (\theta_j )\big) - (1-\beta_j(\theta_j))\tilde p_j \nonumber\\ &&{\kern6pt} + (1-\beta_{j-1}(\theta_j)) \tilde p_{j-1} + \lambda \big(q_j (\theta_j ) - q_{j - 1} (\theta_j )\nonumber\\ &&{\kern6pt}- e_j +e_{j-1}\big) = 0. \end{array}$$
(27)

For a given λ, define the HP’s net monetary benefit for intervention j as χ j (θ j ) = v j (θ j ) − (1 − β j (θ j ))p j  + λ(q j (θ j ) − e j ). Therefore, the first-order optimality condition (27) becomes equivalent to χ j (θ j ) − χ j − 1 (θ j ) = 0, which implies that the functions χ j (·) and χ j − 1 (·) intersect at point θ j . We claim that at the optimal point \(\theta_j^*\), \(\chi '_j (\theta_j^* ) \leq \chi '_{j - 1} (\theta_j^*)\). To show this, note that if at the optimality intervention j precedes intervention j − 1 (see Fig. 1a), then individuals with rank θ ∈ [θ j ,θ j  + ϵ] use the intervention j − 1 and individuals with rank θ ∈ [θ j ,θ j  − ϵ] uses intervention j; this implies that for θ ∈ [θ j ,θ j  + ϵ], χ j (θ) ≤ χ j − 1 (θ), and for θ ∈ [θ j ,θ j  − ϵ], χ j (θ) ≥ χ j − 1 (θ). Since both function intersect at point \(\theta_j^*\), it results that \(\chi '_j (\theta_j^* ) \leq \chi '_{j - 1} (\theta_j^*)\).

At optimality, the problem (2) has binding constraint and hence the bordered Hessian matrix of Lagrangian function (26) is equal to:

$$ {\bf{H}} \!=\! \left(\! {\begin{array}{*{20}c} 0 & {Q'_1 (\theta_1)} & \cdots & {Q'_M (\theta_M)} \\ {Q'_1 (\theta_1)} & {\chi '_1 (\theta_1 ) \!-\! \chi '_{0} (\theta_1 )} & \cdots & 0 \\ \vdots & \vdots & {} & \vdots \\ {Q'_M (\theta_M)} & 0 & \cdots & {\chi '_M (\theta_M ) \!-\! \chi '_{M - 1} (\theta_M )} \\ \end{array}} \!\right)\!. $$

By Lemma 1 and setting a j  = Q j (θ j ) and b j  = χ j (θ j ) − χj − 1 (θ j ) ≤ 0, for j = 1, 2, ..., M, it is immediate that the last M leading principal minor of matrix H alternate in sign, with the sign of the determinate of H matrix the same as the sign of ( − 1)M. Therefore, the point \((\theta_1^* ,\theta_2^* , \ldots ,\theta_M^* )\) that satisfies the first-order Kuhn-Tucker conditions is the strict maximum [38, Theorem 19.6 and the discussion thereafter]. Hence, there exists a unique Lagrangian multiplier λ * that makes the optimization problem (3) equivalent to problem (2) at the observed solution \((\tilde\theta_1,\tilde\theta_2, \ldots ,\tilde\theta_M)\). That is, assuming that the observed allocation \((\tilde\theta_1,\tilde\theta_2, \ldots ,\tilde\theta_M)\) is optimal for the HP, we can find a unique multiplier λ * such that solving the problem (3) results in the exact same allocation as \((\tilde\theta_1, \tilde\theta_2, \ldots ,\tilde\theta_M)\).

Likewise, it can be shown that when the health purchaser is health maximizer subject to a budget constraint, his optimization problem is equivalent to:

$$\begin{array}{lll} &&{\kern-6pt} \mathop {\max }\limits_{0 = \theta_{M + 1} < \ldots < \theta_1 < \theta_0 = 1} {\rm{ }}K(\theta_1 ,\theta_2 \ldots ,\theta_M ) \nonumber\\ &&\;= \sum\nolimits_{j = 0}^M {Q_j (\theta_j )} + \mu ^* \sum\nolimits_{j = 0}^M {V_j (\theta_j )}, \end{array}$$
(28)

By writing the first-order optimality conditions of problems (3) and (28), it is straight forward to show that μ * = 1/λ *. The proof is then completed by multiplying the objective function (28) by the constant 1/μ *. □

Appendix C: Proof of Theorem 2

We write the equality constraint (9) as two inequality constraints:

$$ U_j(\theta_j) - U_0 (\theta_j ) \geq -u_q(-e_j)u_v(- \beta _j \tilde p_j - c_j ), $$
(29)
$$ U_j(\theta_j) - U_0 (\theta_j ) \leq -u_q(-e_j)u_v(- \beta _j \tilde p_j - c_j ) . $$
(30)

Let τ 1 and τ 2 denote the Lagrangian multiplier of constraints (29) and (30). The first-order conditions for the new optimization problem will be:

$$\begin{array}{rll} \tilde p_j \theta ^*_j &+& (\tau _2 - \tau _1 )\tilde p_j u_q(-e_j) u'_v(-\beta_j^* \tilde p_j-c_j) \nonumber\\ &-& \delta \left\{ \begin{array}{l} = 0,\textrm{ for }\beta _j^* > 0, \\ < 0,\textrm{ for }\beta _j^* = 0. \\ \end{array} \right. \end{array}$$
(31)
$$\begin{array}{rll} \pi _j (\theta_j ^* ,\beta _j^* ) &-& \pi _0 (\theta_j ^* ) - (\tau _2 - \tau _1 ) {C'_j (\theta^*_j) } \nonumber\\ &-& \eta \left\{ \begin{array}{l} = 0, \textrm{ for }\theta_j ^* > 0, \\ < 0, \textrm{ for }\theta_j ^* = 0. \\ \end{array} \right. \end{array}$$
(32)
$$ \eta (\theta_j ^* - 1) = 0 $$
(33)
$$ \delta (\beta _j^* - \beta _j^{\max } ) = 0 $$
(34)

Comparing the first-order conditions (31)–(34) with first-order conditions (13)–(16) reveals that at optimality \( \tau ^* = \tau _2^* - \tau _1^* \). First we show that − τ * > 0 implies a positive WTP for expansion and − τ * < 0 implies a negative WTP for expansion. Suppose − τ * > 0. Therefore we have \(-\tau ^* = -(\tau _2^* - \tau _1^*) > 0\). By constraint (9), to increase the consumption level of intervention j, θ j , we should increase the utility \(u_q(-e_j)u_v(- \beta _j \tilde p_j - c_j )\). By constraint (29), increasing \(u_q(-e_j)u_v(- \beta _j \tilde p_j - c_j )\) by one unit increases the HP’s objective function by \(\tau _1^* \) and by constraint (30), increasing \(u_q(-e_j)u_v(- \beta _j \tilde p_j - c_j )\) by one unit decreases the HP’s objective function by \(\tau _2^* \). Therefore increasing \(u_q(-e_j)u_v(- \beta _j \tilde p_j - c_j )\) by one unit results in increasing the HP’s objective function by \(-(\tau _2^* - \tau _1^*)\) which is positive. Thus − τ * > 0 implies a positive WTP for expansion. The case for − τ * < 0 is similar and hence omitted.

By constraint (9), to increase the consumption level of intervention j, θ j , by one unit, we should increase the utility \(u_q(-e_j)u_v(- \beta _j \tilde p_j - c_j )\) by − (U j (θ) − U0(θ)) unit. Consequently, increasing the consumption level of intervention j by one unit increase the HP’s objective function by \(-\tau ^*\big(-(U'_j(\theta)-U'_0(\theta))\big)\), which is equal to \(\gamma_j^* = \tau^* C_j ^\prime (\theta^*_j)\). □

Appendix D: Proof of Proposition 1

The results are obtained from the first-order conditions (13)–(16). We only show the first part; parts 2 and 3 are proved similarly. By Eq. 16, When \(0<\beta _j^* <\beta_j^{\max }\), we have δ * = 0; hence, by Eq. 13, we have \(\theta_j^* + \tau^* u_q(-e_j) u'_v(-\beta_j^* \tilde p_j-c_j)=0\). Consequently, by Theorem 2, the HP’s WTP for expansion will be \(\gamma_j^* = \tau^* C_j ^\prime (\theta^*_j)\), which results in Eq. 17. To show \(\gamma_j^*>0\), note that by Assumption 1, it is apparent that at the splitting point \(\theta^*_j\), we have U j (θ) < U0(θ), and hence C j (θ) = U j (θ) − U0(θ) < 0.

To show Part 1(b), By Eq. 15, when \(0<\theta_j^*<1\), we have η * = 0, which makes Eq. 14 equivalent to:

$$ \pi_j(\theta_j^*,\beta_j^*)-\pi_0(\theta_j^*)-\tau {C_j'(\theta^*_j)}=0. $$
(35)

Substituting \(\pi_j(\theta ,\beta_j)=(1-\beta)v_j(\theta)-(1-\beta_j)\tilde p_j+\lambda (q_j(\theta) - e_j)\) and π 0 (θ) = (1 − β)v 0 (θ) + λq 0 (θ) in Eq. 35 results in Eq. 18. □

Appendix E: Proof of Proposition 2

Suppose that at optimality, there are M interventions in the optimum sequence s. The contract (β 1, β 2, ...,β M ) divides the population into at most M + 1 sections, each of which uses only one of the available interventions (see Fig. 1a). Assume that the population is split at points \(\left( {\theta _1 , \theta_2 \ldots ,\theta _{M-1} ,\theta _M } \right)\), where 0 = θ M + 1 ≤ θ M  ≤ θ M − 1 ≤ ... ≤ θ 1 ≤ θ 0 = 1. The HP solves the following problem to determine the coinsurance rates (β 1, β 2, ...,β M ) and the consumption level \(\left( {\theta _1 , \theta_2, \ldots ,\theta _M } \right)\) (recall that an individual of type θ ∈ (θ j + 1, θ j ) uses intervention j):

$$ \mathop {\max }\limits_{\scriptstyle \beta _1 , \beta_2, \ldots ,\beta _M \hfill \atop \scriptstyle \theta _1, \theta_2, \ldots ,\theta _M \hfill} \Pi = \sum\limits_{j = 0}^M {\int_{\theta _{j + 1} }^{\theta _j } {\pi_j (t,\beta _j )} \,dt} $$
(36)
$$\begin{array}{lll} &&{\kern-6pt} \textrm{s.t. }u(-e_j, - \beta _j \tilde p_j - c_j ) + U_j(\theta_j) \nonumber\\ &&\;\,= u(-e_{j-1}, - \beta _{j - 1}^{} \tilde p_{j - 1} ) + U_{j-1}(\theta_j),\nonumber\\ &&{\kern-6pt}\textrm{for } j = 1,2, \ldots ,M, \end{array}$$
(37)
$$ 0 = \theta_{M+1} \le \theta _M \le \theta _{M - 1} \le \ldots \le \theta _1 \le \theta _0 = 1, $$
(38)
$$ {\rm{0}} \le \beta _j \le \beta _j^{\max } ,\textrm{ for } j = 1,2, \ldots ,M. $$
(39)

We solve the optimization problems (36)–(39) in M steps (stages). In stage k, given the optimal allocation of the intervention j k − 1, \(\theta^*_{j_{k-1}}\) and ignoring the interventions (j k + 1, ..., j K ), we determine the portion of population that should use the intervention j k − 1 by solving:

$$ \mathop {\mathop {\max}\limits_{0\leq \beta_k \leq \beta_k^{\max} \hfill \atop 0\leq \theta _k \leq \theta _{k - 1}^* } \int_0^{\theta _k } {\left( {\pi_k (t,\beta_k ) - \pi _{k - 1} (t,\beta _{k - 1}^* )} \right)} \,dt} $$
(40)
$$\begin{array}{lll} &&{\kern-6pt} \textrm{s.t. }u(-e_j, - \beta _j \tilde p_j - c_j ) + U_j(\theta) \nonumber\\ &&\;= u(-e_{j-1}, - \beta^* _{j - 1} \tilde p_{j - 1} ) + U_{j-1}(\theta^*_j). \end{array}$$
(41)

If the optimal allocation of the intervention k turns out to be \(\theta _k^* = \theta _{k - 1}^* \), then the intervention k − 1 is dominated by interventions k and k − 2, and hence should be dropped from the current sequence.

We claim that this approach is in fact solving the optimization problems (36)–(39). To prove this, we show that (1) the objective function being maximized by this method is equal to the objective function (36), and (2) at each stage, the feasibility constraints (37)–(39) are not violated.

To show the first part, the first key observation is that by constraint (37), all β k ’s can be eliminated from the optimization problems (36)–(39), and hence, at each stage, the HP’s problem is to find only the allocation of each intervention, i.e. \(\theta^*_k\)’s. For instance, at stage k = 1, by solving problems (40) and (41) for k = 1, the HP determines the portion of population who uses the intervention j = 0 (use nothing), i.e. \([\theta^*_1,1]\) (see Fig. 1c); at stage k = 2, by solving problems (40) and (41) for k = 2, the HP determines the portion of population who uses the intervention j = 1, i.e. \([\theta^*_2,\theta^*_1]\) (see Fig. 1c), and so on.

The second key observation is that the optimal decision at stage k, i.e. \(\theta^*_k\), does not affect the value of the objective function (40) in problems (40) and (41) for the subsequent stages (stages > k). In other words, the decision at stage k does not impose a new bound on the objective function of the subsequent stages. To see this, recall that if the optimal allocation of intervention k + 1 turns out to be \(\theta _{k+1}^* = \theta _{k}^* \), then since the intervention k is dominated by interventions k + 1 and k − 1, we drop it from the sequence; and therefore, in the optimum sequence, we always have \(\theta _{k+1}^* < \theta _{k}^* \), which means that in the optimum sequence, the decision in each stage, say \(\theta _k^*\), does not add a constraint to the decision in the subsequent stages, i.e. \(\theta _{j}^*, j\geq k + 1\).

And finally, we show that the summation of the objective function (40) is equal to the objective function (36). Subtracting the constant \(\int_0^1 {\pi _0 (t)dt} \) from the objective function (36) does not change the optimal solution; therefore, maximizing Π is equivalent to maximizing:

$$\begin{array}{lll} &&{\kern-6pt} \Pi - \int_0^1 {\pi _0 (t)dt} \nonumber\\ &&\;= \sum\limits_{j = 0}^M {\int_{\theta _{j + 1} }^{\theta _j } {\pi _j (t,\beta _j )} \,dt} - \int_0^1 {\pi _0 (t)dt} \nonumber\\ &&\;= \sum\limits_{j = 0}^M {\left( {\int_0^{\theta _j } {\pi _j (t,\beta _j )} \,dt - \int_0^{\theta _{j + 1} } \!\!{\pi _j (t,\beta _j )} \,dt} \right)} \!-\! \int_0^1 {\pi _0 (t)dt} \nonumber \!\!\!\!\\ &&\;= \sum\limits_{j = 1}^M {\int_0^{\theta _j} {\big( {\pi _j (t,\beta _j ) - \pi _{j - 1} (t,\beta _{j - 1} )} \big)} \,dt}. \end{array}$$
(42)

Eventually, according to the preceding argument, it is clear that the optimal allocation obtained in each stage satisfies the feasibility constraints (37)–(39). □

Appendix F: Proof of Proposition 3

We write the equality constraint (20) as two inequality constraints:

$$\begin{array}{lll} &&{\kern-6pt} U_{j_k}(\theta_{j_k }) - U_{j_{k-1}}(\theta_{j_k }) -u(-e_{j_{k-1}}, - \beta _{j_{k - 1} }^* \tilde p_{j_{k - 1} } ) \nonumber\\ &&\;\geq - u(-e_{j_k}, - \beta _{j_k } \tilde p_{j_k } - c_{j_k } ) , \end{array}$$
(43)
$$\begin{array}{lll} &&{\kern-6pt} U_{j_k}(\theta_{j_k }) - U_{j_{k-1}}(\theta_{j_k }) -u(-e_{j_{k-1}}, - \beta _{j_{k - 1} }^* \tilde p_{j_{k - 1} } ) \nonumber\\ &&\;\leq - u(-e_{j_k}, - \beta _{j_k } \tilde p_{j_k } - c_{j_k } ). \end{array}$$
(44)

We then proceed in the exact same way as in the proof of Theorem 2.

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Yaesoubi, R., Roberts, S.D. A game-theoretic framework for estimating a health purchaser’s willingness-to-pay for health and for expansion. Health Care Manag Sci 13, 358–377 (2010). https://doi.org/10.1007/s10729-010-9135-6

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Keywords

  • Game theory
  • Willingness-to-pay
  • Health care
  • Moral hazard
  • Asymmetric information
  • Resource allocation
  • Implementation
  • Colorectal cancer