Mathematical Appendices

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**Mathematical Appendix For Topic 10**

One of the measures used in Slide 3 of Topic 10 is the mean payoff a manager can expect to get from the next interview. Its value can be calculated from two pieces of information: the value of the best interviewed candidate so far, and the probability distribution of the remaining candidates' values. Let us calculate this measure assuming the uniform distribution of Topic 10's simulations.

**Given:** Each candidate is equally likely to exhibit
any value between x – v and x + v, where we shall call x the mean of our
distribution, and v – its variance. Furthermore, the value of best candidate so
far is x_{max} – a number that must also fall between x – v and x + v.

**Find: **The expected increase in the value of best
candidate during the next interview. This value, minus the interview’s cost,
becomes the manager’s expected payoff should she choose to meet with another
applicant.

**Solution:** If the value of next applicant happens to
be below x_{max}, the increase in the value of best candidate so far is
zero – a case we may, therefore, ignore. If, on the other hand, the next
applicant’s value is above x_{max} – somewhere between x_{max}
and x + v, our expected increase is actually a positive number. The probability
of such an event is (x + v – x_{max}) / (2v). Furthermore, since the
distribution is uniform, the expected value of the next candidate will be
exactly half-way between x_{max} and x + v, i.e. (x + v + x_{max})
/ 2. The expected increase in value will then be (x + v + x_{max}) / 2 –
x_{max} = (x + v – x_{max}) / 2.

Putting the two pieces together, the expected increase in
best candidate value is equal to the probability that next candidate is better
than x_{max}, times the expected value increase in that case:

[(x + v – x_{max})
/ (2v) ] * [(x + v – x_{max}) / 2 ] = **(x + v – x _{max})^{2}
/ (4v)** ♣

Using the expression above for expected value increase, we can run a simple algorithm to calculate the expected number of interviews until optimal stopping point:

1. Use the formula above to compute the expected payoff
from next interview, given current x_{max}. Set a variable counter equal
to zero.

2. While current expected payoff is greater than the cost
of next interview,

2a. Add this expected payoff to current x

2b. Calculate a new current expected payoff from the new current x

2c. Increment counter.

3. In the end, the counter variable will contain the expected number of interviews until the manager’s optimal stopping pont. ♣

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