Optimal incentives for teams: a multiscale decision theory approach

Abstract

We present a novel modeling approach for supervised teams, which can determine optimal incentives when individual team member contributions are unknown. Our approach is based on multiscale decision theory, which models the agents’ decision processes and their mutual influence. To estimate the initially unknown influence of team members on their supervisor’s success, we develop a linear approximation method that estimates model parameters from historic team performance data. In our analysis, we derive the optimal incentives the supervisor should offer to team members accounting for their varying skill levels. In addition, we identify the information and communication requirements between all agents such that the supervisor can calculate the optimal incentives, and such that team members can calculate their optimal effort responses. We illustrate our methods and the results through a systems engineering example.

Appendices

Appendix A

Using equation (12), the first order derivative of $e^{\text {INF}x}_{*}(b^{\text {INF}x})$ is given by

$$\begin{aligned}&\frac{\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}b^{\text {INF}x}} = -\Bigg ( (c^{\text {INF}x}+{\tilde{c}}^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}}) \frac{\mathrm {d}}{\mathrm {d} e^{\text {INF}x}}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \Bigg ) \bigg / \nonumber \\&\quad \Bigg ( \Big ( (h^{\text {INF}x} - l^{\text {INF}x}) + b^{\text {INF}x}({\tilde{c}}^{\text {INF}x} + c^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}}) \Big ) \frac{\mathrm {d}^2}{\mathrm {d} (e^{\text {INF}x})^2}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \nonumber \\&\quad \frac{\mathrm {d}^2}{\mathrm {d} (e^{\text {INF}x})^2}k^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \Bigg ). \end{aligned}$$

(24)

Using Eqs. (12) and (24), the second order derivative of $e^{\text {INF}x}_{*}(b^{\text {INF}x})$ is given by

$$\begin{aligned}&\frac{\mathrm {d}^2e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}(b^{\text {INF}x})^2} \nonumber \\&\quad = -\Bigg ( 2(c^{\text {INF}x}+{\tilde{c}}^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}})\frac{\mathrm {d}^2}{\mathrm {d} (e^{\text {INF}x})^2}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big )\times \frac{\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}b^{\text {INF}x}} \nonumber \\&\qquad + \Big ( b^{\text {INF}x}({\tilde{c}}^{\text {INF}x} + c^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}}) + (h^{\text {INF}x} - l^{\text {INF}x})\Big ) \frac{\mathrm {d}^3}{\mathrm {d} (e^{\text {INF}x})^3}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \nonumber \\&\qquad - \frac{\mathrm {d}^3}{\mathrm {d} (e^{\text {INF}x})^3}k^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big )\Big )\times \Big (\frac{\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}b^{\text {INF}x}}\Big )^2 \Bigg ) \bigg / \nonumber \\&\qquad \Bigg ( \Big ( (h^{\text {INF}x} - l^{\text {INF}x}) + b^{\text {INF}x}({\tilde{c}}^{\text {INF}x} + c^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}}) \Big ) \frac{\mathrm {d}^2}{\mathrm {d} (e^{\text {INF}x})^2}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \nonumber \\&\qquad \frac{\mathrm {d}^2}{\mathrm {d} (e^{\text {INF}x})^2}k^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \Bigg ). \end{aligned}$$

(25)

Appendix B

Proof of Theorem 2

From Eq. (24), we know that $\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x}) \big / \mathrm {d}b^{\text {INF}x} > 0$ for all x and for all $b \in (0,1]^n$. In addition, if $\displaystyle { \frac{ \partial ^2 }{\partial (e^{\text {INF}x})^2} \Big (\frac{\partial }{\partial e^{\text {INF}x}} R^{\mathrm {INF}x}(e,b) \Big ) \le 0 }$, then

$$\begin{aligned}&\Big ( b^{\text {INF}x}(c^{\text {INF}x} + {\tilde{c}}^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}}) + (h^{\text {INF}x} - l^{\text {INF}x}) \Big ) \frac{\mathrm {d}^3}{\mathrm {d} (e^{\text {INF}x})^3}\alpha ^{\text {INF}x}(e^{\text {INF}x})\\&\quad - \frac{\mathrm {d}^3}{\mathrm {d} (e^{\text {INF}x})^3}k^{\text {INF}x}(e^{\text {INF}x}) \le 0 \end{aligned}$$

for all x and for all e. From Eq. (25) it follows that $\mathrm {d}^2e^{\text {INF}x}_{*}(b^{\text {INF}x}) \big / \mathrm {d}(b^{\text {INF}x})^2 < 0$ for all x and for all $b \in (0,1]^n$. The second order partial derivatives of $R^{\mathrm {SUP}}(b)$ are

$$\begin{aligned}&\frac{\partial ^2 R^{\mathrm {SUP}}(b)}{\partial (b^{\text {INF}x})^2} \nonumber \\&\quad = - 2(c^{\text {INF}x} + {\tilde{c}}^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}}) \frac{\mathrm {d}}{\mathrm {d} e^{\text {INF}x}}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big )\times \frac{\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}b^{\text {INF}x}} \nonumber \\&\qquad +\Big ( 1 - \sum _{k=1}^n b^{\text {INF}k} \Big )(c^{\text {INF}x} + {\tilde{c}}^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}}) \nonumber \\&\qquad \times \ldots \bigg ( \frac{\mathrm {d}^2}{\mathrm {d} (e^{\text {INF}x})^2}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \times \Big ( \frac{\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}b^{\text {INF}x}} \Big )^2 \nonumber \\&\qquad +\frac{\mathrm {d}}{\mathrm {d} e^{\text {INF}x}}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \times \frac{\mathrm {d}^2e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}(b^{\text {INF}x})^2} \bigg ) \end{aligned}$$

(26)

and

$$\begin{aligned}&\frac{\partial ^2 R^{\mathrm {SUP}}(b)}{\partial b^{\text {INF}x} \partial b^{\text {INF}w}} = -(c^{\text {INF}x} + {\tilde{c}}^{\text {INF}x})(h^{\text {SUP}} - l^{\text {SUP}})\nonumber \\&\quad \times \ldots \bigg ( \frac{\mathrm {d}}{\mathrm {d} e^{\text {INF}x}}\alpha ^{\text {INF}x}\big (e^{\text {INF}x}_{*}(b^{\text {INF}x})\big ) \times \frac{\mathrm {d}e^{\text {INF}x}_{*}(b^{\text {INF}x})}{\mathrm {d}b^{\text {INF}x}} \nonumber \\&\quad + \frac{\mathrm {d}}{\mathrm {d} e^{\text {INF}w}}\alpha ^{\text {INF}w}\big (e^{\text {INF}w}_{*}(b^{\text {INF}w})\big ) \frac{\mathrm {d}e^{\text {INF}w}_{*}(b^{\text {INF}w})}{\mathrm {d}b^{\text {INF}w}} \bigg ) \end{aligned}$$

(27)

for $w \in \{1,\ldots ,n\}$ and $w \ne x$. Since all the second order partial derivatives of $R^{\mathrm {SUP}}(b)$ are negative, we know that the Hessian matrix of $R^{\mathrm {SUP}}(b)$ is negative definite for all $b \in (0,1]^n$. This implies that $R^{\mathrm {SUP}}(b)$ is strictly concave for all $b \in (0,1]^n$. $\square$