B THE AUTOMATION MODEL

Imagine a job whose performance requires the execution of $N$ tasks, and let $n$ of these tasks be executed by humans, $(N - n)$ by machines. The effectiveness with which the job gets performed can be measured by the product of two factors: the measure of job performance that would obtain if all participants, both humans and machines, were able to operate correctly under the prevailing circumstance $γ$ , and the probability that the creator’s vision $γ^{*}$ of the circumstances under which his machine will have to operate included the actual circumstance $γ$ ,

μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{n}; ϕ_{n + 1}; . . .; ϕ_{N}) = μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{n}; ϕ_{n + 1}; . . .; ϕ_{N} | γ) 𝑃𝑟 (γ \in γ^{*})

(B.1)

Let us introduce the following set of $n$ functions:

f_{i} = μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{i}; 1; 1; . . . . 1 | γ)

i = 1,2, . . . ., n

(B.2)

representing the conditional job effectiveness when there are $i$ humans and the machines performing the remaining $(N - i)$ tasks have perfect proficiency. Then:

f_{i + 1} = μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{i}; ϕ_{i + 1}; 1; 1 . . .; 1 | γ) = μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{i}; (1 - (1 - ϕ_{i + 1})); 1; 1 . . .; 1 | γ)

(B.3)

and assuming that all humans are sufficiently proficient to ensure that $(1 - ϕ_{i + 1})$ is relatively small:

f_{i + 1} = μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{i}; 1; 1; . . . 1 | γ) - {(\frac{\partial μ}{\partial ϕ_{i + 1}})}_{ϕ_{i + 1} = 1} (1 - ϕ_{i + 1})

(B.4)

so that all the $f$ functions defined above satisfy the following iterative equations:

f_{i + 1} = f_{i} - {(\frac{\partial μ}{\partial ϕ_{i + 1}})}_{ϕ_{i + 1} = 1} (1 - ϕ_{i + 1})

(B.5)

Since, clearly:

f_{0} = 1

(B.6)

we get that:

f_{n} = 1 - {\sum_{j = 1}^{n} (\frac{\partial μ}{\partial ϕ_{j}})}_{ϕ_{j} = 1} (1 - ϕ_{j})

(B.7)

and therefore the conditional job effectiveness becomes:

μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{n}; 1; 1; . . . 1 | γ) = 1 - (1 - ϕ_{j}) {\sum_{j = 1}^{n} (\frac{\partial μ}{\partial ϕ_{j}})}_{ϕ_{j} = 1}

(B.8)

The corresponding unconditional job effectiveness can then be written as:

μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{n}; 1; 1; . . . 1) = [1 - (1 - ϕ_{j}) {\sum_{j = 1}^{n} (\frac{\partial μ}{\partial ϕ_{j}})}_{ϕ_{j} = 1}] π^{(N - n)}

(B.9)

if we are willing to assume that each machine operates independently of all other participants in the job and if we denote by $π$ the probability that the creator of each machine has included in his vision the prevailing circumstance $γ$ . If we finally assume that all humans have roughly the same proficiency, we can write with good approximation that:

μ (ϕ_{1}; ϕ_{2}; . . .; ϕ_{n}; 1; 1; . . . 1) = (1 - n (1 - ϕ) {(\frac{\partial μ}{\partial ϕ})}_{ϕ = 1}) π^{(N - n)}

(B.10)

This expression for the effectiveness with which the job gets done is bimodal in the number of humans assigned. When all participants are machines, job effectiveness is equal to $π^{N}$ . As humans are added to the job, the first factor in our expression for job effectiveness decreases linearly with the number of humans added because humans are assumed to be less proficient than machines. The second factor, however, increases as the negative power of $π$ because humans are more flexible than machines. The product will therefore have a maximum at a value of $n$ given by:

\bar{n} = \frac{(1 - ϕ) {(\frac{\partial μ}{\partial ϕ})}_{ϕ = 1} + 𝑙𝑛 π}{[(1 - ϕ) {(\frac{\partial μ}{\partial ϕ})}_{ϕ = 1}] 𝑙𝑛 π}

(B.11)

Figure B.1 shows how the job effectiveness depends upon the number of humans employed in its performance for the case in which:

(1 - ϕ) {(\frac{\partial μ}{\partial ϕ})}_{ϕ = 1} = 0.06

π = 0.85

(B.12)

Figure B.1: Balancing Humans and Machines

For machines with low flexibility, the probability $π$ that the creator of each machine has included in his vision the prevailing circumstance $γ$ is small, and the corresponding value for $\bar{n}$ is larger than the total number $N$ of participants. Under these circumstances, job performance is ever increasing with the number of humans assigned to the job and, as expected, the best performance is attained when all participants are human. For machines with high flexibility, on the other hand, $π$ is near one, the corresponding $\bar{n}$ is near zero, and performance ever decreases with the number of humans assigned. Therefore, best performance is attained when no humans are assigned to the job. In general, however, $\bar{n}$ is somewhere in between zero and sixteen as indicated in Figure B.1.