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The Agent priority model

Contents

The concept of priorities
Modelling priority change
Drivers of priorities
Combining the drivers

The concept of priorities

The idea of priorities is one of the key pillars of the framework. The priority of a given Agent state represents its relative importance in a rank order from the perspective of the Agent. Priorities drive behaviour at the Agent level. Different states have different priorities given an Agent’s context, or current state.

As discussed previously, we model each Agent as a stochastic finite state automaton. The state transition matrix of each Agent represents their immediate priorities at any moment in time: which state, or task, they are most likely to proceed to allocate their time and resources to next.

Based on this model, we established the probability of state transitions $P_{\alpha_{(S_{ \sigma})}}$ to be:

(1) $\begin{equation*} P_{\alpha t_{(S_{\tau})}}=\sum_{\sigma=1}^{|\textbf{S}|}{P_{\alpha t_{(S_{\tau}|S_{\sigma})}}P_{\alpha (t-1)_{(S_{\sigma})}}}\end{equation*}$

We call the likelihood of state transition from $S_{\sigma}$ to $S_{\tau}$ , $P_{\alpha t_{(S_{\tau}|S_{\sigma})}$ , $A_{\alpha}$ ‘s priority for $S_{\tau}$ over $S_{\sigma}$ at time $t$ , because the probabilities can be interpreted as the relative importance of each state over the other, given the agent’s context, i.e. the current state.

We simplify notation by using $P_{\alpha\sigma\tau}$ for $P_{\alpha_{(S_{\tau}|S_{\sigma})}}$ , and refer to $A_{\alpha}$ ‘s state transition matrix as $A_{\alpha}$ ‘s priority matrix, $\textbf{P}_a$ :

(2) $\begin{equation*}\textbf{P}_{\alpha}=\begin{bmatrix}P_{\alpha11} & P_{\alpha12} & \cdots & P_{\alpha1\lvert S \rvert} \\P_{\alpha21} & P_{\alpha22} & \cdots & P_{\alpha2\lvert S \rvert} \\\vdots & \vdots & \ddots & \vdots \\P_{\alpha \lvert S \rvert 1} & P_{\alpha \lvert S \rvert 2} & \cdots & P_{\alpha \lvert S \rvert \lvert S \rvert }\end{bmatrix}\end{equation*}$

Since the total probability of a state transition to any state is one, and probabilities must be non-negative and less than or equal to one, it is clear that each agent’s priority matrix must satisfy the following set of conditions at all times:

(3) $\begin{equation*} \sum_{\tau=1}^{|\textbf{S}|}{P_{\alpha\sigma\tau}}=1 \quad \forall\:\sigma\in [1,|\textbf{S}|]\subset \mathbb{N} \\\end{equation*}$

(4) $\begin{equation*} 0 \leqslant P_{\alpha\sigma\tau} \leqslant 1\end{equation*}$

The organisation’s priority tensor $\textbf{P}$ is a tensor of dimension $\lvert\textbf{A}\rvert\times\lvert\textbf{S}\rvert\times\lvert\textbf{S}\rvert$ composed of the priority matrices of each agent, such that $P_{\alpha\sigma\tau}$ is the probability of $A_{\alpha}$ transitioning from state $S_{\sigma}$ to $S_{\tau}$ .

Modelling priority change

We argue that the mechanism through which Agents set their day-to-day priorities is fundamental to the behaviour of the system.

We wish to establish a sensible set of rules on how the priorities of each agent should vary to reflect in some ways human near-term priority-setting processes. Noting the constraints in equations (3) and (4), we use a softmax function to model the elements of the priority tensor P, $P_{\alpha\sigma\tau}$ , with a latent variable input scalar metric $w_{\alpha\sigma\tau}$ measuring priority weight, with units of logistic units, or logits.

(5) $\begin{equation*} P_{\alpha\sigma\tau}=\frac{e^{w_{\alpha\sigma\tau}}}{\sum_{\tau=1}^{|\textbf{S}|}{e^{w_{\alpha\sigma\tau}}}}\end{equation*}$

With this model, the priority matrix P will always satisfy the constraints in (4) for any real values of $w_{\alpha\sigma\tau}$ .

It follows that:

(6) $\begin{equation*} \frac{dP_{\alpha\sigma\tau}}{dt}=\sum_{\lambda=1}^{|\textbf{S}|}{\frac{\partial{P_{\alpha\sigma\tau}}}{\partial{w_{\alpha\sigma\lambda}}}\frac{dw_{\alpha\sigma\lambda}}{dt}}\end{equation*}$

From (5), using the Jacobian of the softmax function and where $\delta_{\tau\lambda}$ is the Kroeneker delta function:

(7) $\begin{equation*}\frac{\partial{P_{\alpha\sigma\tau}}}{\partial{w_{\alpha\sigma\lambda}}}=P_{\alpha\sigma\tau}(\delta_{\tau\lambda}-P_{\alpha\sigma\lambda})\end{equation*}$

Therefore, following from (6) the rate of change of each priority matrix entry is given by:

(8) $\begin{equation*}\frac{dP_{\alpha\sigma\tau}}{dt}=P_{\alpha\sigma\tau}(\delta_{\tau\lambda}-P_{\alpha\sigma\lambda})\frac{dw_{\alpha\sigma\lambda}}{dt}\end{equation*}$

Drivers of priorities

Based on the input signals into the Agent set from the Plant and reporting system, as well as cross-agent interactions, we define three attributes shaping priorities: influence, judgment and incentives.

With equation (8), we now have the rate of change of each priority matrix entry in a way that is consistent with (3), (4) and based on the evolution of the priority weights $w_{\alpha\sigma\tau}$ .

What drives $w_{\alpha\sigma\tau}$ and, consequently, the priority matrix of each agent? In the real world, individuals perform a complex assessment of their context at any time to establish appropriate near-term priorities on time allocation. In alignment with our model of intra-organisational signal flows, we decompose each $A_{\alpha}$ ‘s context into three sets of input signals at $A_{\alpha}$ ‘s disposal:

Influence

Agent priorities are partly driven through organisational influence, primarily the communication of priorities between agents, including, but not limited to, along reporting lines. Broadly, this can be summarised as the effect of political power on individuals and is a direct consequence of human emotional responses to relationships with others, whether persuasive, voluntary, compliant or coercive.

If your peers are prioritising time spent on sales over customer service, you are more likely to do so too, all other things being equal. If you have been invited to a meeting by your line manager, you are likely to attend that instead of completing other tasks, all things being equal.

The toolchain or “plumbing” of influence is that of power. It includes interpersonal relationships, employee communication, org charts, reporting lines and town hall meetings. It is the domain of charisma, coercion and all “soft”, and sometimes less soft, skills often associated with leadership – and, in unhealthy organisations, oppression.

Judgment

Agent priorities are also partially driven by their own training and experience, the cumulative effect of which we summarise under the term judgment. Professionals are expected to be trained for judgment. Engineers, doctors, lawyers, are expected to study facts and form an opinion of what is a “correct”, or at least optimal, set of priorities on next steps. Importantly, judgment is informed by facts on the ground and, specifically, the physical state of the plant.

If a manufacturing system is worked so hard that there is a risk of an accident occurring, you are likely to slow down production, all other things being equal. If the warehouse is full, you are likely to turn away shipments and escalate to your manager.

The toolchain of judgment is reasoning, experience and training. Our educational institutions – schools, polytechnics, universities – are meant to train individuals on the appropriate heuristics for judgment in their future professions. Experience on the job builds on this training. Judgment is also affected by the cultures we grow up in. Organisations are meant to be hiring for judgment when assessing the skills and profile of an applicant against a job description.

Incentives

Agent priorities are further influenced by financial incentives linked to reported performance. Whether a bonus, or benefits, or fixed salary, we classify all monetary or monetary-equivalent rewards (and punishments) offered by the organisation under this term. In certain, unwanted, circumstances, employee incentives may also include “shadow” ones, for example, opportunities to misappropriate inventory or to take bribes.

As a salesperson, if you have met your maximum sales bonus quota for the year, you might choose to sandbag the rest of your sales pipeline lead for the next bonus cycle instead of continuing to push hard for this year. An opportunity for substantial equity ownership is likely to incentivise employees to align their interests with their majority shareholders’, all other things being equal.

The toolchain of incentives is compensation structures, including fixed salary, bonus/ malus structures and benefits – anything that can be easily thought of as convertible to cash.

Combining the drivers

We use a simple, linear model of Agent priority-setting that gives a weight to each of influence, judgment and incentives.

We wish to define an interaction mechanism for the three driver inputs of Agent priorities.

We will use a linear model for simplicity, with the rate of change of the priority weights of each agent be governed by the following differential equation:

(9) $\begin{equation*}\frac{dw_{\alpha\sigma\tau}}{dt}=F_{\beta\alpha\sigma\tau}(P_{\beta\sigma\tau}-z_{\beta}P_{\alpha\sigma\tau})+G_{\alpha\sigma\tau\chi}(x_{\chi}-\hat{x}_{\chi})+E_{\alpha\sigma\tau\psi}(y_{\psi}-\hat{y}_{\psi})\end{equation*}$

Where $z_{\beta}=1~\forall~\beta~\in~\mathbb{N}$ , $\hat{y}_{\psi}$ are elements of the Board target vector $\hat{\textbf{y}}$ , and $\hat{x}_{\chi}$ elements of the plant target state vector ${\hat{\textbf{x}}_{\chi}}$ .

${F_{\beta\alpha\sigma\tau}}$ , ${G_{\alpha\sigma\tau\chi}}$ and ${E_{\alpha\sigma\tau\psi}}$ are elements in tensors F, G and E, respectively, holding the following meaning:

F links the rate of change in $A_{\alpha}$ ‘s priorities to other Agents’ priorities, thus representing other Agents’ influence on $A_{\alpha}$ ‘s actions. F is therefore called the influence tensor, with term $F_{\beta\alpha\sigma\tau}$ quantifying the influence agent $A_{\alpha}$ has on agent $A_{\beta}$ on the probability of a state transition from $S_{\sigma}$ to $S_{\tau}$ .
G models the impact of $A_{\alpha}$ ‘s assessment of the system state, thus the weight of their judgment of their observed environment, G is therefore called the judgment tensor.
E captures $A_{\alpha}$ ‘s responsiveness to the organisation’s reported results, modelling the alignment of financial performance incentives to $A_{\alpha}$ ‘s action priorities. $\textbf{E}$ is therefore called the incentive tensor.

Substituting (9) into (8), we have:

(10) $\begin{equation*}\frac{dP_{\alpha\sigma\tau}}{dt}=P_{\alpha\sigma\tau}(\delta_{\tau\lambda}-P_{\alpha\sigma\lambda})\left[F_{\beta\alpha\sigma\lambda}(P_{\beta\sigma\lambda}-z_{\beta}P_{\alpha\sigma\lambda})+G_{\alpha\sigma\lambda\chi}(x_{\chi}-\hat{x}_{\chi})+E_{\alpha\sigma\lambda\psi}(y_{\psi}-\hat{y}_{\psi})\right]\end{equation*}$

With (10), we now have a set of $|\textbf{A}|\times|\textbf{S}|\times|\textbf{T}|$ nonlinear differential equations governing the dynamics of the organisation’s priorities across all $\alpha$ , $\sigma$ , $\tau$ that models the interactions across the relationship graph of agents and with the dynamics of the Plant and Reporting system.