Persian Princess Select bridegroom: About algorithms and major decisions

A math problem at college

My major in college is maths. Once, the professor gave the "Persian Princess selected bridegroom" title "1", as follows:

Persian princess to the marriageable age, to choose Bridegroom. There are 100 candidates, all of whom the princess has not seen. hundred people in random order, from the princess in front of each passing. Whenever a man passes in front of the princess, the princess either chose him as a bridegroom or not. If he were chosen, the rest of the men who had not yet appeared would be dismissed and the bridegroom of the election was over. If not, the current man will leave, that is, pass this person, the next person to debut. The princess is not allowed to go back and choose from the pass-off. The rule is that the Princess must elect one of the hundred to do bridegroom, that is to say, if the first 99 people the Princess can not see, she must choose the 100th man for bridegroom, no matter how ugly he is.

The task is to design a choice method for the princess, so that she has the highest probability of choosing the most handsome men of the Hundred bridegroom.

The point is, there is no choice to ensure that the princess must choose the most handsome handsome guy. For any selection method, there are always some appearances in order to let the princess and the handsome miss. Therefore, the question is not to win the election method (because it does not exist), but the highest probability of the election method.

Because it's not about math, I'm going to give you a straight answer: The best option is to pass off the first 100/e man (e = 2.718 ...). is the natural logarithm, i.e. 100/e is approximately equal to 37). But the most handsome of the 37 men were recorded. Then came the filed of the men, the first to appear handsome beyond all the first 37 people, that is, bridegroom. If everyone is gone and there is no such thing as Mr right, then you have to choose the 100th man.

This is the best choice, followed by a very interesting mathematical deduction. If you are interested, see note "2".

The way of thinking behind the positive solution

Mathematical inference and whatever, the answer behind this is a widely used method of thinking. The difficulty of the Princess's choice was that she did not know how handsome the hundred people were distributed and in what extent. So her best strategy was to pass off the first 37 men, but to see them as a representative sample, to understand the approximate distribution of the hundred people's looks. Then make a choice based on this cognition.

The real romance is certainly not a simple pageant. Ordinary people can not be like the Persian nobility who have who. But the way of thinking is common. If you are a girl, the first time in love, perhaps you think the boyfriend is not delicate, puzzled amorous feelings. But you can not judge is, whether the world boys mostly so, or you are particularly unlucky to meet such a need "3". Only after you have tried three or five, can you have a global judgment on the male species. So, when you break up with your first boyfriend, you don't have to compassion or disappoint the world's men. The right attitude is: Okay, I now have a data point, and now I'm looking for more data points"4".

How long does it take to study?

How many data points do you find? In other words, when will we learn to trust our judgments about the world? In this little story, we can see how nature solves this problem.

In the winter of 1944, World War II came to an end. The Germans cordoned off supplies from the Dutch German-occupied area. The winter of 1944-1945 is known as the "Hunger Winter" (winter of Hunger). 4.5 million of the Dutch were starving, and 18,000 were starving. In 1945, Germany was defeated and the blockade was lifted.

But the effects of this hungry winter have been preserved for decades. Women who were pregnant with children during the blockade, who were developing embryos in their stomachs, were unconscious and experienced the disaster. Decades later, when these children became 50-year-olds, scientists found that they were fatter and more likely to have cardiovascular disease "5" than their former, or later-born, Dutch children.

One explanation for this is that when you are still in your mother's belly, our body is learning what a world it is: a world of food and clothing, and no worries? Or a world with a last meal? These babies born in the Dutch famine that year, their bodies learn: "This is a world of food scarcity." Even after their adulthood, the Netherlands is already a surplus developed country, their bodies still do not forget the experience of early starvation, will try to store fat, ready for the next hungry winter coming. As a result, the population is more prone to obesity and is more prone to obesity-related cardiovascular diseases.

Interestingly, the study of the abundance of food was completed in the October, and could not be reversed for decades. This learning window is determined by our body, our genes. Children learn things quickly because their bodies and brains are specialized learning machines. The frontal lobe, which is responsible for abstract thinking in the human brain, is "6" at the age of 25, a study says. In other words, before the age of 25, our minds, especially those of advanced cognitive abilities, are constantly changing. And many of these changes come from our environment. The change is that we are learning about where we are, what kind of world we are in, and how we think and act in this world.

From the perspective of human brain development, after 25 years of age, at least physiologically, this learning has stopped. The deadline depends on the gene, and the gene comes from millions of years of evolution. For millions of years, the average human lifespan is hovering at the age of 20-30. This may be why our study, from the design of our bodies, is at the age of 25.

We can not affect our own physical and physical learning, but some of the learning, but it is something that we could influence, and should be affected. What kind of work do you choose? Which city do you live in? What kind of partner are you looking for? These do not seem to be in a hurry to catch a deadline (especially the 25-year-old deadline) to decide. You'll make a lot of comparisons before you decide to buy a car or a house. And the work, the partner, these more important decisions, of course, you have to compare more, understand what you are in a world, before making a decision.

Perhaps you are 30 years old, no marriage objects, do not like the work being done, but there are various pressures to look forward to your "don't toss, settle down." The pressure may come from a social tradition that has not been a personal choice, or from an evolutionary stress of a life expectancy of only 30 years. But all this has changed: society has more and more choices, we can also be expected to live to 80, 90 years of age.

Perhaps you should consider the Persian princess seriously: Should I continue to collect data points? Or is it time to make a life-long decision?

Back to the title of the Persian princess

The question of the Princess of Persia teaches us at least another point, that even if your method is optimal, you will never be the most handsome bridegroom ever. In the optimization of the selection method, the princess is only about 40% of the possibility to choose the most handsome man "7". That is, if this choice 10 times, each time the hundred men appear in random order, of which 6 times, the princess will choose not the most handsome bridegroom.

Life is risky, not measurable. It seems to be a blow, but it is also a relief. All personnel, Ann. It's not your fault if you do it in the right way, even if the results are not good.

The other thing I learned was that if I was chosen as a party (like the 100 men), timing was crucial. Here is a simple and multi-topic:

If you are one of these hundred men and you are able to decide where you are going to play, when will you choose to make the most of your chances of being selected?

The answer is 38th place. You will not choose before 38, because the probability of your being chosen is 0 (assuming our princess has studied advanced mathematics and knows the best choice). You will not choose after 38, because each person in front of you, it means a little more than the Princess chose his chance "8".

If you have a lover, of course you have to strive to pursue happiness, but you may also want to think about it, is this the best timing?

37% rule "measured"!

What is the effect of the 37% rule? We programmed the program on the computer to simulate the process of selecting n = 30 o'clock using the 37% rule (if MM never accepts the courtship, the last suitor is automatically selected). The smaller the number of boys the more times, the number of 30 boys is the best choice. Program run 10,000 times, unexpectedly have about 4,000 times to select the best boys, visible 37% law really effective AH.

Results obtained after computer simulation 10,000 times

The problem was first proposed by the mathematician Merrill M. Flood in 1949, which he named the "fiancee Problem". The subtlety of this question is that the natural base e, masterful in the Calculus world, unexpectedly appears in this seemingly unrelated issue. Do not know this issue, Geek between men and women will be more than a reason to break up: Sorry, you are the 37% of people??

Note:

"1" The original question is the Prince of Persia to choose imperial Concubine.

"2" This topic in the scientific squirrel meeting.

"3" I can now do a solution: The world's boys do most of the same, but after training, can be improved.

"4" This approach, which is the core of Bayesien statistics, modifies the perception of the world based on the constant enrichment of information.

"5" "Effects of prenatal exposure to the Dutch famine on adult disease in later Life:an overview", T.J. Roseboom et al., M Olecular and Celular Endocrinology, vol 185 DEC 2001, Pages 93-98

"6" "Why does they act?: A survival guide to the adolescent brain for you and your teen", David Walsh. New York:free Press, 2004.

"7" this 40% or so figure, can be deduced. But it can also be estimated by simulation. See Science Squirrel will text: http://songshuhui.net/archives/57722

"8" Thank you downstairs Zhongyu Wang's reminder. The 100th place is also a good position, that is, if the most handsome pot appears in the top 37, then the princess will definitely choose the 100th place. So, the 100th position is at least 37% of the probability of being selected. This may be better than the 38th location! Conversion to the pursuit of the opposite sex, do not know whether it means that you have to wait until the end of your loved one to reappear:)

How to find the best K value?

The large mathematician Euler has studied a mysterious mathematical constant e≈2.718, and this figure has a direct connection to the problem of "refusing people".

For a fixed k, if the most suitable person appears in the first position (K < i≤n), in order to let him lucky just by the MM selected, you have to meet the former i-1 personal best person in the former K person, this has k/(i-1) possible. Considering all the possible I, we get the total probability P (k) of the best boys to be selected after the first K boys are tested:

By using X to represent the value of the k/n, and assuming that N is sufficiently large, the above formula can be written as:

To-x ln x, which is a derivative of 0, can solve the optimal value of x, which is the reciprocal--1/e of the mysterious constant of Euler's study!

In other words, if you expect the courtship to have n individuals, you should first reject the former n/e person and wait for the next person who is better than these people. Suppose you meet a total of about 30 suitors, you should reject the former 30/e≈30/2.718≈11, and then start with the 12th suitor, and once you find someone better than the previous 11 suitors, accept him decisively. Since 1/e is approximately equal to 37%, this love Dafa is also called the 37% law.

However, the 37% rule has a small problem: if the best candidate is already in the 37%, miss the 37% of the people, she will never touch the better. But in the course of the game, she did not know that the best person has been rejected, so she will always wait. That is, MM will have a 37% probability of "failure to exit", or to be forced to choose the end of the last suitor ended.

Persian Princess Select bridegroom: About algorithms and major decisions