## Making the best choice #math

##### February 12, 2017

EVERY VALENTINE'S DAY, I open the Journal of Mathematics to read how to make best choices. TL/DR: Take the square root of the choices and discard those from the initial batch.

This process is "an extension of the secretary problem in which the decision maker (DM) sequentially observes up to n applicants whose values are random variables X_{1}*;*X_{2}*;*...*;*X_{n }drawn i.i.d. from a uniform distribution on ½0*;*1. The DM must select exactly one applicant, cannot recall released applicants, and receives a payoff of x_{t}, the realization of X_{t}, for selecting the tth applicant. For each encountered applicant, the DM only learns whether the applicant is the best so far. We prove that the optimal policy dictates skipping the first sqrt(n)-1 applicants, and then selecting the next encountered applicant whose value is a maximum." [1]

I'm introducing this mathematical analysis to a group of business students in the Limerick Institute of Technology and also sharing it with a group of interested listeners on Anchor.fm on the run up to Valentine's Day. The business students are studying Decision Support Systems. The listeners are rewinding their love lives to see if they might have made better decisions.

This peer-reviewed research is very relevant with Valentine’s Day approaching, should you wonder how many relationships must come and go in your search for an ideal mate. Probability calculus will help you decide.

As numbers columnist Jo Craven McGinty explains, "The solution uses probability calculus to figure out how to maximize the odds of choosing the best option from a series of prospects, which could include a string of dates, a pool of job applicants or, in one novel example by my favourite Numberphile, a field of portable toilets". [2]

The research studies how a decision maker (DM) might observe "a sequence of up to n applicants whose values are random variables X_{1}*;*X_{2}*;*...*;*X_{n }drawn i.i.d. from a uniform distribution on ½0*;*1. As in the standard secretary problem, the DM has two choices for each applicant: accept or reject. The DM’s payoff for selecting an applicant with X_{t }¼ x_{t }is itself x_{t}. Once an applicant is selected the problem terminates; if reached, the nth applicant must be accepted; and, once rejected, an applicant cannot be recalled. Importantly, however, at each stage t the DM only observes an indicator of X_{t}, where I_{t }¼ 1 if and only if x_{t }¼ maxfx_{1}*;*x_{2}*;*...*;*x_{t}g; otherwise, I_{t }¼ 0. In other words, the DM only learns whether each observed applicant is the best so far. Her objective is to maximize her expected payoff."

This research recalls the classical secretary problem [3] . The major distinction in Bearden's research is the payoff for his DMs is equal to the selected applicant’s underlying ‘‘true’’ value, "whereas in the classical secretary problem the DM earns a payoff of 1 if she selects the best overall applicant and earns nothing otherwise."

Bearden's research helps DMs discern an optimal selection if there is a specific time limit or a known number of options to be considered. Bearden's research considers a trader

(hirer) who wants to sell an asset when its price (applicant) is at its maximum during some period of time ½t_{min};t_{max}. Though the price ranks are salient in deciding when to sell, presumably she will derive utility that is strictly increasing in cardinal selling price. The nothing-but-the-best payoff scheme of the classical secretary problem fails to capture this.

The best strategy, according to the formula, is to reject the first 37% of the prospects, then select the next person who is better than everyone in the initial group.

“You’re using them to learn what qualities matter to you and what the range of quality is like in the population,” said Neil Bearden, a professor of decision sciences at Insead, a graduate business school, in Singapore. “They’re like a training set,” Bearden told the WSJ.

When I give students the documents appended to this blog post, I tell them that they should first decide on a time frame for a decision or to select a number representing the maximum acceptable number of dates. Then apply Bearden's formula. That means you calculate the square root of the number of dates. Skip anyone in that pool of dates. Then really start looking seriously. So if you are out on a summertime work placement and figure you can date 16 people during that time, you reject the first four people. The calculus says you will finish the summertime work placement with someone who is quite acceptable as a love interest.

And you need to know that calculus does not always produce the best answer. However, in the case of long-term commitments and getting married to start a family, the calculus helps slow down a love-sick process and stops decision makers from rushing into things.

How about you? Did you stop searching too soon?

The Supremes: Can't Hurry Love

[Bernie Goldbach teaches creative media for business in the Limerick Institute of Technology. The image with the love hearts is from the Wall Street Journal.]

- J. Neil Bearden, "A new secretary problem with rank-based selection and cardinal payoffs" in Journal of Mathematical Psychology 50, (2006), 58-9. [Download Bearden New Secretary Problem.docx (180.9K)]
- Jo Craven McGinty, "In Love, Probability Calculus Suggests Only Fools Rush In," in The Wall Street Journal, February 10, 2017.
- For reviews of secretary problems see Ferguson, 1989; Samuels, 1991. Download Samuels Secretary Problem.docx (668.2K)
- Stephen M. Samuels, "Secretary Problems as a Source of Benchmark Bounds," Purdue University, 1993.