not that marriage theorem, another one
February 8, 2011 11:46 AM Subscribe
A friend in college once mentioned a theorem for deciding the number of people to meet before deciding on the person to marry. Was it just a joke, or is there really such a theorem?
Every now and then I wonder about this, and something reminded me of it today. It is not the marriage theorem unless it is and I've misunderstood something.
It could have been just a joke but I think there really was something he was referring to. It was something about how many appointments should the matchmaker arrange before the person selects the next match.
Every now and then I wonder about this, and something reminded me of it today. It is not the marriage theorem unless it is and I've misunderstood something.
It could have been just a joke but I think there really was something he was referring to. It was something about how many appointments should the matchmaker arrange before the person selects the next match.
This seems to be a discussion of what you're talking about, with a link to a mathematical treatment of the question on Google Books
posted by logicpunk at 11:52 AM on February 8, 2011
posted by logicpunk at 11:52 AM on February 8, 2011
Yes, I think adipocere has what you were thinking of.
That said, there's a number of problems with applying the formalized secretary problems to real-world situations. Most notably (from the Wikipedia article): "The objective is to select the best applicant. The payoff is 1 for the best applicant and zero otherwise." In other words, in the formalized problem, the only positive outcome is to choose the best possible candidate/spouse; choosing the second-best candidate/spouse, or choosing the worst candidate/spouse, or not choosing a candidate/spouse at all, are all seen as negative outcomes, and more importantly for real-world practicability, they are all seen as equally bad outcomes. The problems inherent in applying this assumption to real-world situations should be obvious.
posted by DevilsAdvocate at 11:59 AM on February 8, 2011 [2 favorites]
That said, there's a number of problems with applying the formalized secretary problems to real-world situations. Most notably (from the Wikipedia article): "The objective is to select the best applicant. The payoff is 1 for the best applicant and zero otherwise." In other words, in the formalized problem, the only positive outcome is to choose the best possible candidate/spouse; choosing the second-best candidate/spouse, or choosing the worst candidate/spouse, or not choosing a candidate/spouse at all, are all seen as negative outcomes, and more importantly for real-world practicability, they are all seen as equally bad outcomes. The problems inherent in applying this assumption to real-world situations should be obvious.
posted by DevilsAdvocate at 11:59 AM on February 8, 2011 [2 favorites]
Listen to DevilsAdvocate's point. It's a good one. Also, hiring or rejecting an interviewee on the spot for a position so important that only the best is good enough is bad HR. Really cool math, though.
posted by monkeymadness at 12:06 PM on February 8, 2011
posted by monkeymadness at 12:06 PM on February 8, 2011
How to Find a Spouse: A Problem in Discrete Mathematics With an Assist From Calculus
posted by callmejay at 2:00 PM on February 8, 2011
posted by callmejay at 2:00 PM on February 8, 2011
Response by poster: You guys are awesome! I need to wait until I get home and have time to review each answer in order to decide on the best.
posted by bleary at 2:13 PM on February 8, 2011 [2 favorites]
posted by bleary at 2:13 PM on February 8, 2011 [2 favorites]
Is there any variation on this where N is unknown? It would better approximate actual marriage or hiring that way.
posted by tylerkaraszewski at 3:30 PM on February 8, 2011
posted by tylerkaraszewski at 3:30 PM on February 8, 2011
Can someone explain this in layman's terms? I can't understand the wiki.
posted by meadowlark lime at 3:37 PM on February 8, 2011 [1 favorite]
posted by meadowlark lime at 3:37 PM on February 8, 2011 [1 favorite]
In the article about the Secretary Problem it states that one of the conditions is that the total number of applicants is known ahead of time.
But when it comes to looking for a spouse, you don't know how many suitable partners you'll meet in your lifetime. So you can't calculate n/e since n is unknown.
posted by Chocolate Pickle at 4:34 PM on February 8, 2011
But when it comes to looking for a spouse, you don't know how many suitable partners you'll meet in your lifetime. So you can't calculate n/e since n is unknown.
posted by Chocolate Pickle at 4:34 PM on February 8, 2011
I need to wait until I get home and have time to review each answer in order to decide on the best.
No, you should just stop reading after the first 37% answers, and mark the next one as best answer.
posted by suedehead at 4:51 PM on February 8, 2011 [10 favorites]
No, you should just stop reading after the first 37% answers, and mark the next one as best answer.
posted by suedehead at 4:51 PM on February 8, 2011 [10 favorites]
Well, there's a bunch of numbers in that theorem.
Here's an alternate theorem:
Sympathetic pheromones + similar long-term goals = marriage.
posted by ovvl at 6:11 PM on February 8, 2011 [1 favorite]
Here's an alternate theorem:
Sympathetic pheromones + similar long-term goals = marriage.
posted by ovvl at 6:11 PM on February 8, 2011 [1 favorite]
I was going to mention the Jung Marriage Test but it seems there are many possible and (correct) answers out there...
posted by Unicorn on the cob at 8:49 PM on February 8, 2011
posted by Unicorn on the cob at 8:49 PM on February 8, 2011
Response by poster: tylerkaraszewski: "Is there any variation on this where N is unknown? It would better approximate actual marriage or hiring that way"
The wiki page mentions some variants but doesn't go in to a lot of detail so I followed some links.
Optimal Stopping and Applications, section 2.3 discusses variations including the case where N is unknown.
On the best choice problem when the number of observations is random
posted by bleary at 9:51 PM on February 8, 2011
The wiki page mentions some variants but doesn't go in to a lot of detail so I followed some links.
Optimal Stopping and Applications, section 2.3 discusses variations including the case where N is unknown.
Another variation lets the total number of objects that are going to be seen be un- known, violating assumption 2 of the CSP. It is assumed instead that the number of objects is random with some known distribution. These problems were introduced by Presman and Sonin (1972). (See Exercise 5.)I looked up the names from Exercise 5 and got
On the best choice problem when the number of observations is random
Abstractthough to be honest it's been a while since I've had to do any real math so I don't know how much focus I will be able to muster to absorb all of this.
We consider the problem of maximizing the probability of choosing the largest from a sequence of N observations when N is a bounded random variable. The present paper gives a necessary and sufficient condition, based on the distribution of N, for the optimal stopping rule to have a particularly simple form: what Rasmussen and Robbins (1975) call an s(r) rule. A second result indicates that optimal stopping rules for this problem can, with one restriction, take virtually any form.
posted by bleary at 9:51 PM on February 8, 2011
Response by poster: I hope smart and knowledgeable people are still reading this. So, what reminded me of the topic was a discussion about customer service some of us were having. I am not a customer service person, and I think I would suck as one because I'd want to give most everyone a refund or replace their broken stuff, &c.
clearly I cannot do that, but I am sure there is some degree of freedom I'd have for giving refunds. To operationalize the concept to help me think about it, I thought, perhaps I'd have x degrees of freedom (dollars, whatever), and would have n customers needing help in a day. I don't know what n is. if I give out too many refunds up front I won't be able to help people who call later. what do I do? it kind of reminds me of the marriage thing except that you aren't only doing a function to pick 1, you have some quantity and get to spread it out, but only of a subset of your n people.
There is probably a name for the concept and what is it?
posted by bleary at 8:13 AM on February 9, 2011
clearly I cannot do that, but I am sure there is some degree of freedom I'd have for giving refunds. To operationalize the concept to help me think about it, I thought, perhaps I'd have x degrees of freedom (dollars, whatever), and would have n customers needing help in a day. I don't know what n is. if I give out too many refunds up front I won't be able to help people who call later. what do I do? it kind of reminds me of the marriage thing except that you aren't only doing a function to pick 1, you have some quantity and get to spread it out, but only of a subset of your n people.
There is probably a name for the concept and what is it?
posted by bleary at 8:13 AM on February 9, 2011
I don't know if there's a name for that problem, but I have some thoughts on it:
I think it's impossible to come up with any reasonable strategy if you have absolutely no idea about what n will be. However, if you can make some assumptions about the likely distribution of n—even if it's something like "I won't get more than 100 customers in a day"—it might be possible to begin to approach a formal solution. You would also need information, or at least some assumptions, about what customers are likely to want—is every customer who comes in looking for a $20 refund, or does it vary by customer? If it does vary, how?
Also, you'd need to define exactly what it is you're trying to maximize, mathematically. For example, you might try to define "customer satisfaction" formally. There's more than one way you might do this.
Consider a customer's desired solution, i.e., what the customer would consider an optimal outcome. (Let's say you can ask the customer about this and the customer answers honestly.) The customer's optimal outcome would cost you $C. One definition of customer satisfaction might be that it's linear from 0 to 1 depending on the amount you actually give to/spend on the customer ($S); then the customer satisfaction would be $S/$C (with a maximum of 1, i.e., it doesn't help more if you give the customer more than they'd like). So if a customer wanted a $40 refund (or other solution which would cost $40), and you gave them $30 (or a remedy which cost $30), that customer's satisfaction would be 0.75.
An alternate definition of customer satisfaction might be that they are completely satisfied (satisfaction=1) if they receive their optimal solution, and entirely unsatisfied (satisfaction=0) if they receive anything less; i.e., a customer who wants $40 leaves just as angry if they get $39.99 as they would if they had got nothing.
In either of these cases, your goal might be to maximize the average customer satisfaction. (These are not the only two possibilities; there's many others. Maybe you're trying to maximize total, rather than average, customer satisfaction. Maybe you're trying to minimize total customer dissatisfaction. Maybe customer satisfaction is measured on the actual amount they receive, rather than being normalized to a 0-1 scale for every customer. Maybe some sense of "fairness" needs to be taken into account in what you're trying to optimze.) Your ideal strategy will be different depending on what, precisely, you're trying to optimize. Although for the two I've outlined here, it seems likely that part of the solution will likely be to always give the full refund requested to customers who are asking for small refunds ("small" based on both the amount you have to give out, and the number of customers who are likely to make a request in a day). If you have $100 to give out each day, and you aren't likely to see more than 10 customers a day, you should almost certainly give $0.10 to the person who asks for it—a fully satisfied customer for only 10 cents!
There's also the assumption in play that customer requests are independent of your strategy—if that assumption doesn't hold (say, people find out you give $0.10 to anyone who asks for it, and a bunch of people start asking for it even if they don't really have a complaint; or, if people find out you're more likely to give large refunds in the morning, and people who want large refunds then all make a point of coming in in the morning) the analysis becomes more complex still.
posted by DevilsAdvocate at 8:41 AM on February 10, 2011 [1 favorite]
I think it's impossible to come up with any reasonable strategy if you have absolutely no idea about what n will be. However, if you can make some assumptions about the likely distribution of n—even if it's something like "I won't get more than 100 customers in a day"—it might be possible to begin to approach a formal solution. You would also need information, or at least some assumptions, about what customers are likely to want—is every customer who comes in looking for a $20 refund, or does it vary by customer? If it does vary, how?
Also, you'd need to define exactly what it is you're trying to maximize, mathematically. For example, you might try to define "customer satisfaction" formally. There's more than one way you might do this.
Consider a customer's desired solution, i.e., what the customer would consider an optimal outcome. (Let's say you can ask the customer about this and the customer answers honestly.) The customer's optimal outcome would cost you $C. One definition of customer satisfaction might be that it's linear from 0 to 1 depending on the amount you actually give to/spend on the customer ($S); then the customer satisfaction would be $S/$C (with a maximum of 1, i.e., it doesn't help more if you give the customer more than they'd like). So if a customer wanted a $40 refund (or other solution which would cost $40), and you gave them $30 (or a remedy which cost $30), that customer's satisfaction would be 0.75.
An alternate definition of customer satisfaction might be that they are completely satisfied (satisfaction=1) if they receive their optimal solution, and entirely unsatisfied (satisfaction=0) if they receive anything less; i.e., a customer who wants $40 leaves just as angry if they get $39.99 as they would if they had got nothing.
In either of these cases, your goal might be to maximize the average customer satisfaction. (These are not the only two possibilities; there's many others. Maybe you're trying to maximize total, rather than average, customer satisfaction. Maybe you're trying to minimize total customer dissatisfaction. Maybe customer satisfaction is measured on the actual amount they receive, rather than being normalized to a 0-1 scale for every customer. Maybe some sense of "fairness" needs to be taken into account in what you're trying to optimze.) Your ideal strategy will be different depending on what, precisely, you're trying to optimize. Although for the two I've outlined here, it seems likely that part of the solution will likely be to always give the full refund requested to customers who are asking for small refunds ("small" based on both the amount you have to give out, and the number of customers who are likely to make a request in a day). If you have $100 to give out each day, and you aren't likely to see more than 10 customers a day, you should almost certainly give $0.10 to the person who asks for it—a fully satisfied customer for only 10 cents!
There's also the assumption in play that customer requests are independent of your strategy—if that assumption doesn't hold (say, people find out you give $0.10 to anyone who asks for it, and a bunch of people start asking for it even if they don't really have a complaint; or, if people find out you're more likely to give large refunds in the morning, and people who want large refunds then all make a point of coming in in the morning) the analysis becomes more complex still.
posted by DevilsAdvocate at 8:41 AM on February 10, 2011 [1 favorite]
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posted by adipocere at 11:49 AM on February 8, 2011 [6 favorites]