Multi-stakeholder decision theory

Abstract

Decisions made by people at work often have outcomes for their employer (sales, costs, profits) and for themselves personally (career progression, bonus), which are not perfectly aligned across the choice options. When such misalignment exists, decision-makers at all levels in organizations must evaluate choices in terms of conflicting outcomes across stakeholders. Existing theories of agency and stewardship provide very different answers to this question, and utility theory does not address this question at all: it assumes that the decision maker will act so as to maximize the expected utility of the firm, essentially a pure steward position. Yet we see commonplace practices of incentives and monitoring of people’s work and decisions aiming to overcome agency problems, with mixed success in practice. We also sometimes see ‘super-agent’ choices, of managers lining their own pockets while their organizations lose out. In this study, we propose, develop, mathematically model and illustrate a new theory of decision-making, based on a multiple stakeholder utility function approach, where the argument of that function explicitly includes both the employer’s and the employee’s utilities as stakeholders. This new approach is shown to be more generalizable than both pure agency and stewardship approaches, within a broad continuum of tradeoff-based decisions.

Appendix

In this appendix we show some typical functional forms and results for the value function under various reasonable assumptions of behavior, to illustrate how the joint value function, V, holds.

Case 1 Contours for (a): Mixed and Steady Individual

Assume $m, y\ge 0$.

• $W(m) = 1 - \exp (-\mu m)$

• $\hbox {U(y)} = 1 - \exp (-\nu y)$

• $a(m,y)=c$

• Plots for $0 \le m,y \le 1$.

Note behavior for small c is similar to the behavior for large c when the m and y axes are swapped.

Case 2 Contours for (b): Sensible and Simplistic Agent

Assume $m, y \ge 0$.

• $W(m) = 1 - \exp (-\mu m)$

• $U(y) = 1 - \exp (-\nu y)$

• Piecewise linear $a(m,y): a(m,y)=a(m)$

• Plots for $0 \le m,y \le 1$ and piecewise linear a(m, y).

Note the dip in V(m,y) when m increases. See next set of plots for further observation.

• Square root (concave) $a(m,y)=a(m) = \min \{\hbox {sqrt}({ sm}),1\}$

• Quadratic (convex) $a(m,y)=a(m) = \min \{({ sm})\hat{\,}\ 2,1\}$

Mathematical explanation of the dip in V(m, y) when m increases:

$$\begin{aligned} \frac{\partial V\left( {m,y} \right) }{\partial m}=a\left( m \right) \left( {\mu e^{-\mu m}} \right) +{a}'\left( m \right) \left( {e^{-\nu y}-e^{-\mu m}} \right) \end{aligned}$$

Solving for $\frac{\partial V\left( {m,y} \right) }{\partial m}=a\left( m \right) \left( {\mu e^{-\mu m}} \right) +{a}'\left( m \right) \left( {e^{-\nu y}-e^{-\mu m}} \right) =0$ is not trivial.

Even if the linear version of a(m), where $a(m) = { sm}$ for $0 \le m \le 1/s$, solving

$$\begin{aligned} \frac{\partial V\left( {m,Y} \right) }{\partial m}= & {} a\left( m \right) \left( {\mu e^{-\mu m}} \right) +{a}'\left( m \right) \left( {e^{-\nu Y}-e^{-\mu m}} \right) \\= & {} sm\left( {\mu e^{-\mu m}} \right) +s\left( {e^{-\nu Y}-e^{-\mu m}} \right) =0 \end{aligned}$$

for a fixed value of Y, is not trivial.

The dip is observed for larger y values.

So why is a dip in V(m, y) possible as m increases? Consider $V(m,y)=a(m)(1 - \exp (-m)) + (1- a(m))(1 - \exp (-y))$. Observe that the first term increases with m, but the second term decreases with m. So if $(1 - \exp (-y))$ is large enough, the dip can be observed within the domain plotted.

What does a dip mean in practice? This is a transition from a steward to an agent.

Case 3 Contours for (d): Shy Agent

Assume $m,y \ge 0$.

• $W(m) = 1 - \exp (-\mu m)$

• $U(y) = 1 - \exp (-\nu y)$

• Piecewise linear a(m,y)$:a\left( {m,y} \right) =\left\{ {{\begin{array}{ll} {0,}&{} {\left( {m-y} \right) \le 0} \\ {s(m-y),}&{} {0\le \left( {m-y} \right) \le \frac{1}{s}} \\ {1,}&{} {\left( {m-y} \right) \ge \frac{1}{s}} \\ \end{array} }} \right.$

• Plots for $0 \le m,y \le 1$ and piecewise linear a(m, y).

$$\begin{aligned} \hbox {Square root a(m,y)}{:}\,a\left( {m,y} \right)= & {} \left\{ {{\begin{array}{ll} {0,}&{}\quad {\left( {m-y} \right) \le 0} \\ {\sqrt{s(m-y)},}&{}\quad {0\le \left( {m-y} \right) \le \frac{1}{s}} \\ {1,}&{}\quad {\left( {m-y} \right) \ge \frac{1}{s}} \\ \end{array} }} \right. \\ \hbox {Quadratic a(m,y)}{:}\,a\left( {m,y} \right)= & {} \left\{ {{\begin{array}{ll} {0,}&{}\quad {\left( {m-y} \right) \le 0} \\ {s^{2}(m-y)^{2},}&{}\quad {0\le \left( {m-y} \right) \le \frac{1}{s}} \\ {1,}&{}\quad {\left( {m-y} \right) \ge \frac{1}{s}} \\ \end{array} }} \right. \end{aligned}$$

Case 4 Contours for (e): Shy Steward

Assume $m,y \ge 0$.

• $W(m) = 1 - \exp (-\mu m)$

• $U(y)= 1 - \exp (-\nu y)$

• Piecewise linear $a(m,y){:}a\left( {m,y} \right) =\left\{ {{\begin{array}{ll} {1,}&{}\quad {\left( {m-y} \right) \le -\frac{1}{s}} \\ {-s(m-y),}&{}\quad {-\frac{1}{s}\le \left( {m-y} \right) \le 0} \\ {0,}&{}\quad {\left( {m-y} \right) \ge 0} \\ \end{array} }} \right.$

• Plots for $0 \le m,y \le 1$ and piecewise linear a(m, y).

$$\begin{aligned} \hbox {Square root a(m,y)}{:}a\left( {m,y} \right)= & {} \left\{ {{\begin{array}{ll} {1,}&{}\quad {\left( {m-y} \right) \le -\frac{1}{s}} \\ {\sqrt{-s(m-y)},}&{}\quad {-\frac{1}{s}\le \left( {m-y} \right) \le 0} \\ {0,}&{}\quad {\left( {m-y} \right) \ge 0} \\ \end{array} }} \right. \\ \hbox {Quadratic a(m,y)}{:}a\left( {m,y} \right)= & {} \left\{ {{\begin{array}{ll} {1,}&{}\quad {\left( {m-y} \right) \le -\frac{1}{s}} \\ {\left( {-s(m-y)} \right) ^{2},}&{}\quad {-\frac{1}{s}\le \left( {m-y} \right) \le 0} \\ {0,}&{}\quad {\left( {m-y} \right) \ge 0} \\ \end{array} }} \right. \end{aligned}$$