# Sequential mutually exclusive signing-bonus offers (A variant on the Secretary Problem)

How much should you offer a potential hire in a signing bonus?

Imagine you are interviewing a list of candidates for a particular job. Each candidate has a "lifetime value", and probability of capture. You are able to invite candidates in order from most promising to least. The only incentive you can offer is a one time check at signing - salary and other benefits are immutable. As soon as both you and the candidate agree on the terms, your search stops and you only pay the incentive offered to that employee. You are able to invite candidates in order from most promising to least. Under these conditions, how would you calculate the maximum signing bonus incentive for each employee?

Examples

• If there was only 1 employee in the list of candidates, the maximum signing bonus would be the employee's value, after all if the employee says "no" you are giving up on 100% of the value that employee would have delivered.

• If there were only two employees in the list, and the second employee was worth half of the first employee, the maximum signing bonus for the first employee would be something less, as you know there is another candidate waiting in the lobby and you know the probability of the second candidate accepting the job. I'm not sure how to calculate this value for a given candidate in a given finite list

• If there was a near infinite stream of candidates with equal probabilities and equal values, the signing bonus would be zero as you could simply continue to interview until you found someone who said yes.

For the sake of simplicity, we can imagine the probability of the employee accepting/rejecting the offer to be the same for all candidates (although a robust model should probably take this into account).

Although I've described the problem in a "hiring" context to make it a little less compliced to explain, I'm applying this type of analysis to a sales problem (offering one-time incentives).

I'm not sure the "name" of problems like this or if there is some body of research that might help! How would you go about solving this problem?

(I've cross-posted this in Finance's exchange as well -- I wasn't sure which community was a better fit. Let me know if I should not have done this and I'll delete one of them)

• I'm confused. Why should I offer a bonus at all if it won't affect the probability of capture? – Lior Kogan May 4 '15 at 19:43
• The bonus will affect the probability of capture for that employee, but the effect is unknown -- the simplifying assumption that the probability of accepting the offer is the same for all candidates is without the bonus. The goal is to identify the upper bound for the bonus (the bonus that would have the greatest effect on the probability of capture) – Robert May 4 '15 at 19:46
• What is your goal? Are you trying to maximize your revenues, or just trying to maximize the probability of having some revenue? – Lior Kogan May 4 '15 at 20:01
• The problem is about finding the upper bound of incentive to offer... it's the answer to the question "the most you should offer this employee is \$x" – Robert May 4 '15 at 20:03