Suppose we are tasked with assembling a workteam, and we can select for the workteam from two populations, say A and B. Suppose the expected performance of a person on a workteam can be reduced to a single value v between 0 and 10 (10 is best, 0 is worst) and suppose that value can be reliably and accurately determined through the selection process. And suppose the expected performance of a workteam equals the sum of the expected performance of individual team members. Thus, for example, if I have a two person team and person 1 has value v=7, and person 2 has value v=8, then the total value of this team is 7+8=15.
Case I: Two populations identical in size and with identical characteristics
Suppose in assembling the workteam we can select from population A with 6 members, or population B with 6 members.
Suppose population A has members with values (10, 9, 9, 8, 8, 7). We identify the first member of population A as A, and thus the value of A here is 10, the value of A is 9, and so on.
Suppose population B also has members with values (10, 9, 9, 8, 8, 7) – so population B is identical to population A.
What observations can we make?
Observation 1: When the supply of the highest value individuals within any given population is too little to staff the workteam(s), then assembling a team from a single population will be sub-optimal. In this example, to assemble a 2 person team, hiring from only A or only B will be suboptimal. Why? The two individuals with the highest performance values do not both belong to either A or B. If I choose the two highest valued members of A, the expected performance of the workteam will be 19, and same for B. However by drawing on A and B the expected performance of the workteam will be 20.
Observation 2: If I want to assemble a workteam of 4 people, the highest performing teams must contain A and B, and the remaining two members can come from either A or B as follows:
Team: A, A, A, B, expected performance 38
Team: A, A, B, B, expected performance 38
Team: A, B, B, B expected performance 38
Note that in 2 out of the 3 optimal solutions, the highest performing workteams will not be composed of equal members from A and B.
Case II: Two identical populations exist, however twice as many candidates from population B are available for selection.
In this case, the effective population size for say A is half that of B. Let:
A = (10, 9, 9, 8, 8, 7)
B= (10, 10, 9, 9, 9, 9, 8, 8, 7, 7)
Notice that B has twice as many members as A, and for every member of A with value v there are 2 members in B with the same value. Thus if I want the highest performing team of say size 3, I will pick 1 person from A (the one with value 10), and two persons from B (also with values 10). If I want the highest performing team of size 9, I will exhaust the available highly valued candidates from A after picking 3 people from A, so the other 6 will come from B. This leads to:
Observation 3: In general, the optimal team will be composed of 1/3 members of A, and 2/3 members of B simply because there are twice as many available candidates in B, even though A and B can be assumed to otherwise be identical populations. If I were to force an optimal workteam to be composed of 1/2 A and 1/2 B, I would have a sub-optimal solution because there would not be enough of the best candidates available in population A.
Case III: Two populations identical in size and with identical average characteristics but with different distribution of performance values
Suppose now population A has members with values (10, 8, 8, 8, 7, 7).
Suppose population B has members with values (9, 9, 9, 9, 6, 6).
A has characteristics: average value 8, max 10, min 7
B has characteristics: average value 8, max 9, min 6
Observation 4: For this example, the optimal two person team is
A, B[x], where x=1,2,3 or 4
Observation 5: For this example, the optimal four person team is
A and 3 members of (B, B, B, B)
Notice that even though the single most valued individual is in population A, and even though population A is identical in size and average value to population B, in this example the optimal workteam will be composed of 25% population A and 75% population B.