Tanya Khovanova's Math Blog

Recently I published my ideas for a Flexible Zipcar Algorithm that allows customers to rent Zipcars one-way. I had a dream the night after I published it: the Zipcar research lab invited me to give a presentation about my algorithm and they asked me a lot of questions. I would like to answer these questions now, but first let me give you a short description of the algorithm:

Some locations are flexible. The number of cars assigned at a flexible location is not fixed, but rather a range between the minimum and the maximum. A Zipster would be allowed to use a car one-way from one flexible location to another, as long the number of assigned cars doesn’t move outside the range.

Question. Zipsters expect to pick up the exact vehicle they reserved. With vehicles changing locations you will break this system. I am operating on the assumption that Zipsters will continue to pick up the exact vehicle they reserved.

Question. What about people who get attached to the cars at their location? You can divide your pool of cars at a flexible location into two groups: fixed Zipcars and floating Zipcars. Only floating cars will be allowed to move. You should advise Zipsters who get attached to their cars to choose fixed cars as the object of their affection. This solution is less flexible, but it resolves your problem.

Question. You designed your algorithm using the min and max number of cars at a flexible location. Where is the desirable number of cars coming from? We need the desirable number of cars to calculate the financial incentives. For example, you can take the absolute value of the difference between the current number of cars at a flexible location and the desirable number of cars. Then sum it up over all locations. We can call this total the distance between the current state and the optimal state. If a one-way trip increases this distance, that Zipster pays extra and vice versa. You can also add a surcharge for all one-way trips to cover the extra expense for parking spaces.

Question. What if cars get stuck? For example, what if we always have the maximum number of cars at Harvard Square and the minimum number of cars at Coolidge Corner? You can always offer free rides from Harvard Square to Coolidge Corner. If every Zipster knows that you have a discount program which is easy to find, your cars will be in the right place in no time. For example, you can color code overstaffed and understaffed locations on your map; or you can add RSS feeds for people who are interested when cheap rides from Harvard Square to Coolidge Corner are available.

Question. Are you saying that instead of hiring drivers to move our cars around we find a much cheaper Zipster who needs to go in the same direction? Yes. If you go even further and offer a token payment or some Zipcar credits, you can have Zipsters driving your cars to oil changes and car washes for you.

Question. Why do you always talk about a small number of flexible locations? I would limit the number of flexible locations for your initial stage, so that for a small investment you get something new, you can experiment, you can observe and study trends in how cars are migrating, you can tune your financial incentives system and you can design and test your web support for this algorithm.

Question. What if someone reserves a car for a one-way trip one month in advance? What happens with the car during this month? If your car – let’s name it “Comfy” — is reserved from the Harvard Square location for a month from now, you need to lock this car into the Harvard Square location for a month. That means, no one can take Comfy for a one-way trip during this month. This looks like a bad case of inflexibility. To resolve it you can have a time-limit for one-way reservations. For example, you might limit one-way reservations to not more than three days in advance.

Question. Suppose we have four cars at the Harvard Square location and the desirable number is three. Then four people reserve four different cars at this location for round-trips on Labor Day. Are these four cars stuck at Harvard Square for a month until Labor Day? I can offer one solution using the earlier idea of fixed and floating cars. You can allow very advanced reservations for fixed cars, but floating cars can only be reserved three days in advance. Another possible solution is that you do the same without dividing your cars into fixed and floating. For example, you allow people to have very advanced reservations for any car which is currently assigned to the Harvard Square location. As soon as three cars are reserved for more than three days in advance, the fourth car is no longer available for very advanced reservations. The fourth car temporarily starts behaving like a floating car, without that being its permanent designation.

Question. Suppose today is Monday and Jack reserves Comfy one-way from Harvard Square to Coolidge Corner for Thursday. To which location is Comfy assigned for three days between Monday and Thursday? On one hand it should be assigned to Harvard Square as it takes a parking space there and we can’t allow another car to take up that space. Also, Comfy is available for reservations before Thursday at Harvard Square. At the same time, Comfy can’t be counted as an extra car anymore as we know that it is guaranteed to be moved. That means that even though Harvard Square actually houses an extra car for these three days, it shouldn’t be flagged as a location with extra cars that need to be reassigned. On the other hand, Comfy will take a spot at Coolidge Corner soon, so we can’t allow other cars to move into that spot. Even though Coolidge Corner doesn’t yet have the complete set of cars, we can’t allow other cars to take up Comfy’s future spot.

Question. It sounds complicated. Can we make it simpler? You can simplify this design for the first implementation. For example, you can only allow immediate one-way reservations. That eliminates any problem with conflicting reservations or an extra car holding up a space for three days. Also, keep in mind that the user doesn’t need to see all this complicated stuff. They are only interested in knowing whether or not they can reserve a car.

Question. What if someone reserves the car one-way then cancels the reservation? This only becomes a problem if someone else reserves the same car for a later time at the destination. In this case, you might institute a very big fee for cancellations of one-way reservations, so that you can either hire a driver or offer a super deal to other Zipsters. Of course, if you allow only immediate reservations for one-way cars this problem will be minimized.

Question. Suppose Jack reserves Comfy for two hours; his starting point is Harvard Square and his end point is Coolidge Corner. Where is Comfy assigned during these two hours? This is a great question. For round-trip Zipcars, you do not care if a person starts a half-hour late or returns the car one hour early. With a one-way car, you need to know when Comfy’s parking space will be available to other cars. To allow Jack to pick up his car as late as he wants or return it as early as he wants, the simplest solution is to assign Comfy for two hours to both locations. That is, no other car is allowed to move into Comfy’s parking spot at Harvard Square until the end of Jack’s rental period and the spot for Comfy should be ready at Coolidge Corner from the start of Jack’s rental period.

Question. You were talking about the simplest solution. Do you mean you have other solutions? I have many ideas, but I prefer to get some feedback from the car-sharing research community before continuing.

Hey, Zipcar! Let’s do it!

According to a famous study: ” [E]ach older brother increases a man’s odds of developing a homosexual sexual orientation by 28%-48%.”

This means that sexual orientation is not uniquely defined by genes in the way that our blood type is defined. Indeed, if homosexuality was similar to blood type, the probability of giving birth to a homosexual would be the same for the same parents, independently of birth order. Furthermore, the fact that identical twin brothers quite often have different sexual orientation also supports my claim that it isn’t like blood type. On the other hand, there is some research that shows that there is a genetic component to sexual orientation. This might mean that there are genes that increase a predisposition in a man to become gay or it could be that there is a gay gene which determines their homosexuality for some part of the gay population — or both.

The research into a genetic component of gayness shows that there might be some genes in the X chromosome that influence male homosexuality. It also shows that the same genes might be responsible for increased fertility in females.

By the way, this fascinating research provides an explanation of why “gay genes” — if they exist — do not die out, as evolutionary laws might lead us to expect.

The theory that the probability that someone will be gay is dependent on the fraternal birth order impacts in several interesting ways on the whole field of research into the genetics of homosexuality.

Let us create a theoretical model where homosexuals would be born randomly with a fixed probability if they are the first sons of a woman, and increasing probability for subsequent sons. In this model there would be two interesting consequences:

• Mothers of homosexual men would be more fertile on average than mothers of men in general. Remember, this random consequence can be misinterpreted as a genetic correlation between homosexuality and fertility.
• If a homosexual person has a brother, then the probability that this brother is homosexual might be very different from the probability that a random person is homosexual. Again, this might create the suggestion that there is a genetic component, when it is not there.

My point is that all genetics research on homosexuality should take into account these two consequences and adjust for them. That means that it is not enough to show that brothers are more likely to be homosexual in order to prove that there is a genetic component. It needs to show that the correlation between relatives is much stronger than the correlation resulting from the birth order.

For people who are not mathematicians, I will build simple models that illustrate my points. Let us consider some extreme theoretical examples first.

First model. Suppose mothers only give birth to sons and only to one or two sons. Suppose the first son has a zero probability of being gay (which means that first sons are never gay) and the second son has a probability of one of being gay (which means that second sons are always gay). Then all mothers of gay men will have two sons, while mothers of random boys will have somewhere between one and two sons. Another result will be that all gay boys in this model will never have a gay brother.

Second model. Suppose mothers only give birth to one or three sons. Suppose the first son has a zero probability of being gay and the second and the third sons have a probability of one of being gay. Then all mothers of gay men will have three sons, while mothers of random boys will have somewhere between one and three sons. Another result will be that all gay boys in this model will always have a gay brother.

Third model. Let us take some more realistic numbers. Let us consider only the case of women who have one or two boys. Let a be a probability of a woman to have one boy, and correspondingly, 1-a to have two boys. Suppose N is the total number of women in consideration. Suppose x is the probability of the first boy to be homosexual and y is the probability of the second boy to be homosexual. We assume that these probabilities are independent of each other.

Let us first estimate the total number of boys born. It will be aN + 2(1-a)N. The number of homosexuals that are born is expected to be xN + y(1-a)N. The probability of a random boy to be a homosexual is (x + y(1-a))/(2-a).

Let us see what happens with fertility. For a random boy (including both homosexual and non-homosexual boys), what is the average number of sons his mother has? It would be one son for aN boys who are from one-boy families and 2 sons for 2(1-a)N boys who are from two-brother families. Hence a mother of a random boy has on average 2 – a/(2-a) children.

Let’s see what happens with homosexual boys. We have axN homosexual boys from one-brother families and (1-a)(x+y)N gay boys from two-brother families. Hence the average number of sons for their mothers is (axN + 2(1-a)(x+y)N)/(axN+(1-a)(x+y)N) = (ax + 2(1-a)(x+y))/(ax + (1-a)(x+y)) = 2 – ax/(x+y-ay). If we denote by r the ratio y/x, then the result will be 2 – a/(1 + (1-a)r). If r is more than 1, then mothers of gays are more fertile then average mothers. If we assume that y = 1.5x and a = .5, then the average number of sons for a mother of a random boy is 1.667 and for a homosexual boy, it is 1.714, a 3% increase. The impact will be stronger if we take into consideration three-boy families.

Now let us look at brothers in two-son families. There is a total of (1-a)N such families and they have a total of (x+y)(1-a)N gay boys. We have 2xy(1-a)N gay boys who live in families in which both brothers are gay. Hence, the probability of a gay boy who has a brother to have a gay brother is 2xy/(x+y). We saw earlier that the probability of a random person to be gay is: (x + y(1-a))/(2-a).

Let’s look at these numbers more closely. You can easily see that if x = y these two probabilities are the same — as they should be. If x = 0 and y > 0, then a gay person never has a gay brother, suggesting negative correlations with a genetic component. Suppose y = rx, where r is a constant. Then the first number is 2rx/(1+r) and the second number is: x(1+r(1-a))/(2-a). We see that the ratio of these two numbers doesn’t depend on x and is equal to 2r(2-a)/(1+r)(1+r-ar). Suppose r = 1.5, then the ratio is 2.4(2-a)/(5-3a). If a = .5, this ratio is 1.03. So, in this model, for a gay boy who has a brother, the probability that this brother will be gay is 3% higher than the probability that a random man will be gay.

I am so fascinated with the fact that a property that depends on the birth order can create an illusion of a genetic component. I am not discounting the possibility of a genetic component for male homosexuality, but I urge researchers to recalculate their proofs, adjusting for the impact of the fraternal birth order.

To be fair, the female fecundity correlation with male homosexuality was shown not only in mothers of gay men, but also in maternal aunts. Also, the increase in the probability of being gay for a brother of a gay man is very much higher than 3% in my model. That means these researches might survive my critique. Still, they ought to look at their numbers again.

Finally, this discussion is not really about homosexuality, but about any property that depends on birth order. In this case, such a property might imply a genetic component that doesn’t really exist.

I am now a proud owner of a Zipcard. My old car died a horrible, screaming death recently and I decided to try to go carless as an experiment, hoping to save some money or to lose weight.

Zipcar is the modern way to rent a car. You reserve a car by the hour or by the day through the web, arrive at the site, swipe your card and drive away.

My biggest problem with being a Zipster is that the closest Zipcar location to my home is about one mile away. On second thought, I am losing weight.

But for other Zipsters, the biggest problem is that you have to return the car at the exact location where you picked it up. Obviously, if you allow renters to return their car to a different location the Zipcar company might run out of paid parking spaces in a particular location or, even worse, the cars might migrate to certain places, leaving other locations without any car.

I would like to propose an algorithm that will add some flexibility to where you can return a car, without overwhelming the system.

Here’s how it would work. Suppose we have three cars currently assigned to the Mt Auburn/Homer Ave location. I suggest we name three as a desirable number, but actually allow from three to four cars to be assigned to this location at any particular time.

Now suppose I want to pick up a car at the Mt Auburn/Homer Ave location and to return it to the Somerville Ave/Beacon location. If the number of currently assigned cars to Mt Auburn/Homer Ave location is three (at the lower limit), the Zipcar reservation webpage tells me, “Sorry, you can’t use this location unless you return your car back here,” and shows me the closest location with extra cars. The same goes if the number of assigned cars at the Somerville Ave/Beacon location is at its upper limit. If my starting point has more cars assigned to it than its lowest limit and my destination point has fewer cars assigned to it than its upper limit, then I am allowed to take a car from my starting location and return it to my destination. Zipcar can throw in some financial incentives. If my choice disrupts the most desirable balance of car assignments, I have to pay a fee. If my choice restores the balance, I get a bonus discount.

It would be so cool if zipcars were flexible. I understand that the average cost to the company of parking each car might go up with my flexible algorithm as Zipcar will need more parking spaces than cars. But Zipcar can start implementing this flexibility with a small number of flexible locations. It would be a great feature.

Hey, Zipcar algorithm designers! Can I get a bonus if you implement my algorithm? I can also design a financial incentives formula for you.

I was waiting for a train in Newark. On the platform, there was an LCD screen that flashed advertising. The ad I was staring at was titled, “Reasons to take NJ Transit to Prudential Center, #15.” I was impressed that the NJ Transit sales people were working so hard to invent that many reasons.

I waited for my train for half an hour. It turned out that the NJ Transit advertising people were not working hard after all. The screen was flipping between four reasons, numbered 3, 6, 12 and 15. This is a case of false advertising. You look at reason number 15 and think that there must be a lot of reasons. They fool with your head. Cheaters.

I hope you noticed that I did the same thing with this posting — purely in order to illustrate my point.

The first time I heard about the “stopping problem,” many years ago, it was in this version: The king announces that it is time for his only daughter to marry. Shortly thereafter 100 suitors line up in a random order behind the castle walls. Each suitor is invited to the throne room in front of the eyes of the princess and the king. At this point, the princess has to either reject the suitor and send him away, or accept the suitor and marry him. If she doesn’t accept anyone from the first 99, she must marry the last one. The princess is very greedy and wants to marry the richest suitor. The moment she sees the suitor she can estimate his wealth by his clothes and his gifts. What strategy should she use to maximize the probability of marrying the richest person?

The correct strategy is to reject the first 37 suitors and then marry the one who is better than anyone else before him. Generally, if there are N suitors the number of people to skip is about N/e — slightly more than one third of the whole group.

However, the greedy princess never received a good mathematical education. It is clear that her goal should have been to maximize the expected wealth of the future husband. In this case the strategy would be different; in particular, it changes significantly at the end. Let’s imagine that when the line of leftover suitors thins, she realizes that she’s already rejected the best one. In this case, it would be in her interest to consider the second best and, closer to the end, even the third best.

In real life, marrying the second best creates a burden of regret and bitterness. Let us assume that we all want to marry the best we can. But of course in our cases, the best does not necessarily mean the wealthiest. Also, we do not know how many suitors are lining up for us. Back in Russia I heard that a woman, on average, receives ten proposals. So, we should skip the first three, then marry the best person after that.

I do not know how many proposals an average American woman gets, but nowadays she doesn’t have to wait for an official proposal before deciding whether a guy is right for her. The question becomes: which marriage candidate should a woman try to marry?

If we assume that marriage candidates are distributed evenly in time and girls are seriously hunting for husbands between ages 20 and 35, then the above math advice can be applied in the following way: From age 20 to about 25 or 26, just look around and see what life offers you. After that, marry the one who is better than all the previous ones.

This is a very mathematical piece of advice. The idea makes sense: first you sample your options then you target the best. The problem is that these assumptions do not cover our real-life situations. Let’s look at some realistic adjustments and how they affect the age at which you stop sampling and become more active.

• You might be OK without marrying at all. In this case you can afford to be much pickier and skip the first 60% of the candidates.
• Potential husbands are not spread evenly in time. In this case try to estimate the distribution and act accordingly. For example, if you expect more men around you while you are in college, your cut-off age goes down.
• You change with time. If you think that you will lose your freshness and charm with age, your cut-off age goes down. If you think that you’ll become more experienced and effective at seducing men later in life, your cut-off age goes up.
• Your values change with time. You might be interested in looks now and value a big heart later. Because of this, it is better to delay your choice until you know yourself well and your opinion of life has stabilized.
• The value of a man changes with time. A high school drop out can become a very successful businessman and the brightest student in college can become an alcoholic. You might benefit from waiting to see how he stabilizes, thus moving your cut-off age up.
• You can divorce. This allows you to make a mistake on the first try and moves your cut-off age for this first try down.

Based on the mathematics and discussion above, here is the advice I would give to my teenage daughter — if I had one:

Take your time looking around and sampling your boyfriends. Constantly analyze the pool of your boyfriends as a whole. If there are strange patterns — for example, all of them look exactly like your father or you always pay for their dinners — start psychotherapy and work it out. As soon as your boyfriends start to look different from each other — except for things that are important to you, like education — compare your dream husband to the pool. If you dream about a Nobel prize winner who is not older than 30 and on the list of the 100 sexiest men alive, you should adjust your expectations to your chances. You will choose a guy who is better than anyone before, but it is unrealistic to expect him to be much better. As soon as you get to a cut-off age, which you have estimated using the suggestions above, stop sampling and start deciding. As soon as you find someone who is better than anyone else before, go for it — marry him.

That was advice from my brain, now I will give you advice from my heart:

Use mathematics to guide you, but make the final decision with your heart.

My English teacher and editor Sue Katz wrote a funny blog entry about masturbation: “Sex and the Single Hand: Stroke Your Way to Health”

I followed the link of one of the studies she mentions to the BBC article “Masturbation ‘cuts cancer risk'”, where ” … They found those who had ejaculated the most between the ages of 20 and 50 were the least likely to develop the [prostate] cancer.”

When I hear such results, my first question is, “How was the study conducted?” It appears that “Australian researchers questioned over 1,000 men who had developed prostate cancer and 1,250 who had not about their [past] sexual habits.” The problem with asking people about their sexual habits 30 years ago is that there are a large number of dead people you can’t ask. What if the most active masturbators have died from fatigue?

Should you masturbate more to reduce your cancer risk as the BBC suggests?

Prostate cancer might not be related to masturbation at all, but rather to something else that correlates with masturbation.

• It could be that men who have a higher libido have less prostate cancer.
• Or that men who have more free time have less prostate cancer.
• Or that men who are not depressed have less prostate cancer.
• Or that men who have higher speed Internet connections have less prostate cancer.

In case you are wondering how one’s Internet connection is related to all this, let me remind you of a joke about a conversation between two geeks.

— “When you look at a girl, what do you notice first?”
— “Her hair, then her eyes, then her nose, then her lips — I have dial-up.”

One thing I know for sure: women who masturbate have even less prostate cancer than men who masturbate. Hooray for masturbation!

I stumbled upon an article in the Boston Globe (February 29, 2008) titled “Study: Spanking children affects their sex lives as adults.” Here are some quotes:

New research by a University of New Hampshire domestic abuse expert says spanking children affects their sex lives as adults. …[C]hildren who are spanked are more likely as adults to coerce partners to have sex, to have unprotected sex and to have masochistic sex.

The classical method to prove scientifically that spanking affects something is to find many parents of newborn identical twins and persuade them to treat their children the same way, with the exception that they spank one and do not spank the other. The researchers should then compare these twins in their adulthood. Such a project is impossible, as parents are likely to start feeling guilty towards the child they spank, for you can’t separate spanking from the whole package of how parents treat their children.

I decided to study the study. I found a more detailed description of the study in the Concord Monitor. As I suspected, the study was a survey. The survey found a correlation between spanking and “undesirable” sexual behavior. As every statistician knows, correlation doesn’t prove causality.

Here’s another quote from the study: “The best-kept secret in child psychology is that children who were never spanked are among the best behaved.” Did it occur to anyone that the best behaved children do not need spanking?

Could it be that parents who spank their children are tired, impatient and less loving? Could it be that not being loved as a child affects your sexual behaviour as an adult much more than spanking?

Could it be that parents who spank their children are more aggressive in general? Could it be that they pass their aggressive genes to kids and their kids’ aggressive behavior is related not only to upbringing, but to genetics?

Do not get me wrong. I do think that spanking is bad. I am saying that the study doesn’t prove that spanking is affecting anyone’s future sex life.

I am surprised that so many magazines republished the article without thinking. Now all the country is fooled into believing that the easy way to improve their kids’ future sex life is to stop spanking.

Go ahead! Stop spanking. Love your children too.

I tried to enroll on a website recently, but they didn’t allow me to continue without choosing five security questions out of about ten samples they supplied. I started in good faith to do what they asked.

Question: What is your father’s middle name?
Answer: They do not have middle names in Russia; they have something called “otchestvo” and I know seven different ways to spell my father’s.

Question: What is the name of the street on which you were born?
Answer: I am glad it was not Lenin Street, but it was equally bad. Besides, it was renamed and I am not sure which name to choose.

Question: What is the name of your high school?
Answer: Finally, an easy question. In Russia we didn’t have names, but rather numbers for schools. I happily entered 444, and oops — the applet wouldn’t accept numbers.

I couldn’t find five questions that I could answer uniquely and reliably. I felt that the designers of these questions were clueless and disrespectful to other cultures. Then I thought about whether I really wanted some creepy database to know the name of my best friend. No, I didn’t.

Now I have established a file where I put the answers to security questions and I can have all the fun I want with my new biography. I can name my first dog Tom Cruise and have my wedding date be 20 years before I was born. I can name my husband Freedom Of Speech and my city of birth IHateSecurityQuestions. Maybe next time I will switch: Freedom Of Speech will be my dog and Tom Cruise my husband.

If you are lazy like me, you can choose your questions so you have the same answer for everything. This way you do not need to type much into your file. For example, you can name your city, your cat and your best friend George Washington. Or, if you are really lazy, God.

I just read the following in Women’s Health Magazine (March, 2008; page 54): Visa conducted a study of 100,000 fast food restaurant transactions. They found that people who pay with credit cards spend 30% more on food than people who pay cash.

The article concludes with the suggestion to pay cash, so you spend less and lose weight.

My question is: Who is more incompetent, Visa or Women’s Health Magazine?

Perhaps people who do not have credit cards are poorer and more price-conscious; hence, they spend less on food. This might explain the correlation. Here’s another possible explanation: people who are ordering for large groups might prefer to pay with a credit card. Or, maybe stores do not like using credit cards for small transactions, so they encourage people to pay cash for modest orders.

The main rule of statistics is that correlation doesn’t mean causality.

There are several possible answers to my question about incompetence:

• The study wasn’t described correctly in the WH magazine. In this case we can’t say much about the competence of Visa, but WH looks bad.
• The study was described correctly, but the conclusion belongs to WH. In this case Visa is innocent and WH is incompetent.
• The study was described correctly and Visa suggested the conclusions. In this case both are incompetent  — Visa for its conclusions and WH for printing them.

It could well be that paying cash makes you stingier, or at least more price-conscious, but I can’t trust Women’s Health Magazine any more. One thing I know for sure is that math can help you lose weight. Math allows you to differentiate a good study from a dumb study.

Alice and Bob are good friends. Bob caught a cold and called Alice for help. He wanted Alice to go to a pharmacy and bring him some cold medicine. Alice did that and I would like to assign a number to this act of giving. How can we quantify this favor?

First, we need to choose a scale. Usually favors cost us in time, money and emotions. Alice spent half an hour driving around, plus $5 on the medicine (we’ll skip the cost of gas for simplicity). It also cost her emotionally, especially because the traffic was really bad.

Measuring everything on three scales is complicated. I would like to convert everything to one scale, because in the future I intend to compare this act of giving to other favors Alice does. For example, Alice knows for sure that this favor for Bob was a less costly favor than her phone call yesterday to her ex-mother-in-law, even though the phone call took only five minutes and didn’t cost any money.

We probably can convert everything to dollars, but I am trying to resist this money-driven society that measures everything in dollars. So, I prefer to use points. Each dollar translates to one point, but time and emotion are more subjective.

Alice makes two calculations in her head: what she really spent and what she is owed.

Here’s what she spent: Alice counts 5 points for the medicine. She also views her time as money. She charges $100 an hour for consulting and values all her time at this rate. Hence, she adds 50 points for time spent. Traffic was bad, but not so bad. She thinks that her traffic stress cost her 15 points. Since she also had to cancel her date with her boyfriend, she estimates her annoyance with this at 100 points. On the other hand, she got this warm feeling from helping Bob and she was happy to see him. So she thinks that she got back 30 points. Adding all this up, we get a total of 140 points. This is how much Alice thinks she spent for this particular favor.

Does it mean that Alice thinks that Bob owes her 140 points? Usually not. The calculation of how much Alice thinks Bob actually owes her is completely different. She thinks that he owes her 5 points for the cost of medicine. Also, she knows that Bob earns much less than she does and values time differently, so she think that he owes her 30 points for her time. Since Bob is not responsible for traffic, she doesn’t add traffic points. Also, she never told Bob that she had to sacrifice her date for him, so she doesn’t think it’s fair to want Bob to be thankful for the sacrifice he doesn’t know about. At the same time she hopes that one day Bob will sacrifice something for her. She can’t ignore this sacrifice completely, so she adds 10 points for that. Altogether she thinks that Bob owes her 45 points.

Do you think Bob feels as if he owes Alice 45 points? Like Alice, he also has two numbers in his mind. One number is the amount of points he received as a result of this favor and the other number is how many points he officially owes Alice.

He actually was planning to ask his neighbor to buy the medicine, but for some reason he called Alice first and she offered help. Alice was delayed at her work and arrived at Bob’s place much later than he expected. She also brought the worst flavor of the syrup. Bob doesn’t value time as much as Alice, so he thinks that Alice spent 10 points driving and 5 points on the medicine. Bob felt ill throughout Alice’s visit and did not enjoy seeing her. Combining that with her late arrival with the wrong syrup, he thinks that he was annoyed for about 15 points. So he thinks that he got zero points from this transaction.

At the same time he wants to be fair. Bob knows that Alice did her best to help him; besides he never specified the flavor he likes. As a result, he doesn’t count his annoyance in how much he owes Alice. So he thinks that he owes Alice 15 points. What Bob really did to thank Alice, I will discuss in a later blog entry.

In conclusion, let me remind you of my system. I measure all favors in points. And for each favor I assign four numbers:

• the giver’s official favor value (in our example 45 points)
• the giver’s hidden favor value (in our example 140 points)
• the receiver’s official favor value (in our example 15 points)
• the receiver’s hidden favor value (in our example 0 points)