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 xmax – 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 xmax, 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 xmax – somewhere between xmax and x + v, our expected increase is actually a positive number. The probability of such an event is (x + v – xmax) / (2v). Furthermore, since the distribution is uniform, the expected value of the next candidate will be exactly half-way between xmax and x + v, i.e. (x + v + xmax) / 2. The expected increase in value will then be (x + v + xmax) / 2 – xmax = (x + v – xmax) / 2.

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

[(x + v – xmax) / (2v) ] * [(x + v – xmax) / 2 ] = (x + v – xmax)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 xmax. 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 xmax to produce a new xmax.
2b. Calculate a new current expected payoff from the new current xmax.
2c. Increment counter.
3. In the end, the counter variable will contain the expected number of interviews until the manager’s optimal stopping pont.  ♣