Maximizing Expected Performance of a Workteam: Implications for Population Representation

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[1], and thus the value of A[1] here is 10, the value of A[2] 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[1] and B[1], and the remaining two members can come from either A or B as follows:

Team: A[1], A[2], A[3], B[1], expected performance 38

Team: A[1], A[2], B[1], B[2], expected performance 38

Team: A[1], B[1], B[2], B[3] 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[1], B[x], where x=1,2,3 or 4

Observation 5: For this example, the optimal four person team is

A[1] and 3 members of (B[1], B[2], B[3], B[4])

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.

 

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