The-Ave.US

THE THREE BODY PROBLEM IN ECONOMICS*

Richard O. Zerbe**
Daniel J. Evans Distinguished Professor Emeritus, The Evans School

A classical yet unsolved problem in economic efficiency, the three body problem, involves the seemingly simple question of whether or not the toilet seat should be left up or down in the bathroom.[1] Three categories of genders are considered: men, women, and others. In accord with usual mathematical analysis, these bodies of different genders are referred to as particles. A solution to this problem has important implications for economic efficiency as an incorrect solution has the power to cause dissolution of marriages and relationships, with huge physical and emotional losses and indeed the loss of life as we have known it. Moreover no family is immune to it. Some people prefer using Japanese toilets though to help with this problem, particularly as the Japanese toilet seat helps with this simple question.[2]

I became interested in this problem when I retired from the University and my School moved my office to a closet (I mean small office) across from the men’s and women bathroom. (I am lucky and grateful to have this closet, make no mistake.) This new location has in fact proved fortuitous as it has allowed me to gather substantial data through direct observation (about the types of restroom users, their behavior, and even the usage of various items for bathroom stall items like glass, stainless steel or phenolic toilet partitions), through hearing relevant activity and through questionnaire. A small part of the more extensive questionnaire is shown in Appendix B.

One Body

In the simplest form of the problem, only one body is involved. If the body is a female particle, the answer is straight forward, leave the seat down. In approximately 98.77% of the cases, women particles use the toilet with the seat down.[3] For an example see below:[4]

The General Model
Let ? be the proportion of users that prefer the up position. The proportion supporting the down position will be (1- ?). The value of the up or down position is then given by the value function:

V = ? V _? – (1- ?)V_{(1- ?)}. (1)

For the one body problem if ? = zero then this is negative, just – (1- ?)V_{(1- ?)}, so the up position is rejected and the down position is preferred. The down position, as noted, is favored by most women and men who need to perform numbers 2 or both 1 and 2. The correct solution for the women only bathroom is thus: seat down. For the men only bathroom ? will be close to 1 in which case ? V _? > (1- ?)V_{(1- ?)} so the correct solution will be: seat up. Let the cost of changing the seat position be c, and assume that this is the same for women and for men. The value of the Down Rule is the –(2cm ?) and the value of the Up Rule is –2c(1- ?). Which rule is better then depends solely on ?.

The social importance of these rules can be shown by the following benefit-cost analysis. I limit my calculations to the women only bathroom. From the questionnaire I find the average time for lowering or raising a raised seat is about 0.60 seconds.[5] This total time cost is then about 1.2 seconds. For simplicity assume conservatively all of the (1- ?) users are women. Women earn about $710 per week in the US. This is about $36,000 per year. This is a gross estimate but less then men’s earnings due to the fact that many more women work part time and do more child care. Basically the discrepancy between men’s and women’s earnings are due to this fact. [6] Of course this raises the issue of why more women particles work part time. A friend who will remain anonymous, but I refer to under the pseudonym Jane Zerbe, notes

“I’d say the time women have to endure taking their young kids to bathrooms, and all of the gross things that happen or are prevented in these bathrooms should easily close the wage gap regardless of part time status. There are some horrifying stories of what can go wrong in a dirty public restroom with a potty trained 2 year old – seats up or down is irrelevant.”

This of course raises the issue of why women spend more time than men with their children. It could be women’s preference or due to men’s tyranny, or just something women feel they must do given the failure of men. In any event I suggest either a new regulation requiring that men spend equal (or more) time with the children or a regulation that forbids children. But I digress.

So, to continue, there are about 162,130,770 women particles in the United States. Now the number of times a women particle wees per day will depend on her age, and size and how hydrated she stays according to Dr Yik Lim[7]. Based on the important work of Dr. Yik Lim, consultant urogynaecologist at Melbourne’s Mercy Hospital for Women, I estimate that the mean woman wees about 10 times per day.[8] From this information and the questionnaire (see Appendix B) the cost of requiring a raised seat on a toilet used only by women would be as follows: Find the time lost from toilet lowering and raising, multiply by the wage for woman and adjust for the number of women. This is shown below:

Table 1

The Cost of Raising the Toilet Seat Where Only Used by Women.

Category		Women
time for raising and lowering toilet seat		1.2 seconds
number of wee times per day for women		10
seconds per day lost		12
total wage cost per day		0.005
total wage costs per year per women		$10
Number of Persons		$ 162,130,779.00
COSTS PER YEAR FOR US		$ 1,621,307,790.00
PV of cost at 7%		$ 23,161,539,857.14
PV of cost at 2%		$ 81,065,389,500.00

The cost per year in the value of lost time is then about $1.6 billion.[9] We can now calculate the present value of this cost once the simple issue of deciding on the correct discount rate is decided. According to Burgess and Zerbe (1013)[10] and to Harberger and Jenkins (2015) this should be about 7-8% real. Arguments for lower rates by Arrow, Weitzman and many others, especially environmental economists whose biases are large and well known, are, not to put a too fine point on it, full of it. At this rate the present value of the cost of requiring raised seats in toilets calculated as a perpetuity will be about 23 billion give or take. [11] Note that this could fund an extensive space program, which would open up further toilet possibilities.

From these calculations we can make a clear policy statement about a rule that would require raised toilet seats used exclusively for women: It is a very bad idea! We leave it as an exercise to the reader to consider other reasons for this conclusion.

Two Bodies

The two body problem, namely men and women with joint use, the value of the up position remains V = ? V _? – (1- ?)V_{(1- ?)}.

The (1- ?) group will be larger given that women wee more than men pee, and that the cost of raising and lowering the seat is the same for all users the Down Rule is superior to the Up Rule. This would be the case a fortiori if the inconvenience cost of raising or lowering the seat is greater for women than men. The safest assumption however is that this cost is the same for both groups. Nevertheless the down rule dominates the up rule for mixed use.

The Selfish Rule

There is now a third rule to consider, developed by Choi, the Selfish Rule. The
Selfish Rule is to leave the seat as used. This rule is superior to either the down or up rules. With the up or the down rule, the disfavored group of either gender will have to change the seat before and after use, whereas with the selfish rule there will be cases in which the disfavored gender use will follow one another eliminating the need to change seat position. More formally, let the inconvenience cost of changing the seat position be c. To derive the Selfish Rule we determine the selfish rules for both the ? and (1- ?) groups respectively and add them. Following Choi, the values to the two groups are

V _? ^sr = – (1- ?)c +[ ?V^sr_? +(1- ?)V^sr_{(1- ?)}] (3)

V^sr_{1- ?)} = – ?c + [ ?V^sr_? +(1- ?)V^sr_{(1- ?)}]

The value of the selfish position is found by adding these values and solving for total V^sr. This gives,[12]

V ^sr = – ?(1- ?)c (3)

This will be larger than either the Up or Down rules as ? (1- ? ) will be less than ? or (1- ?) alone, which separately are the multiplies in the Up and Down rules. Suppose, for example, that the ? and (1- ?) are equal at 0.5. The multiplier for the inconvenience costs is just 0.25 less than for the Up or Down rules. Thus the Selfish Rule dominates.[13] However, with the Down Rule and the Selfish Rule, women will benefit more as they use the facilities more. That women are favored raises issues of equity which are unlikely to be addressed, particularly by women.

Three Bodies

Women and men are not the only particles to be considered. There are at least three law suits about correct bathroom use by transgender people. [14] For example, the school board in Glouster County Va has demanded that Gavin Grimm a transgender boy be barred from using the boy’s restroom at school. The suit, filed by the American Civil Liberties Union on his behalf, concerns whether or not the disbarment is unlawful discrimination. We must then consider gays, lesbians, bi-sexuals and trans-genders, hermaphrodites and children with their unique needs. There are transgender women particles with male genitalia some of whom wish to use the women’s’ facilities and some the men’s, along with transgender men particles with female genitalia some of whom wish to use the women’s facilities and some the men’s. We will call this class, other particles.[15] Complicating any solution, according to Dr Yik Lim there’s an ideal pee window. Just as metabolism adapts to how much you eat, your bladder can be ‘taught’ how much urine it should hold before feeling full and needing to go, says Dr Lim. So if you’re going ‘just in case’, you’re training your bladder to become smaller, just as you’d train your body to survive on less food. As a result, your bladder will be able to hold less fluid, meaning lots of extra dashes to the loo. Lim says, “What I tell my patients is if you don’t stretch the bladder up over time it may not hold as much [urine] as you like.” “But going too late is not a good idea either”, says Dr Lim. “This is because leaving urine in your bladder for ages makes you more likely to develop a urinary tract infection (UTI).”

Thus the timing of going as well as the number of times of going needs to be modeled in the three body problem. We must ask, which group is more likely to have an urgent need? If men the up rule gains value and vice versa. However, the solution to the timing problem, if people would only follow it, is straightforward as Dr Lim notes: “You should only go when you need to.” “Who’d have thought, right?” In addition we must consider, as Dr Yik Lin notes, the amount of urine your bladder can hold. This turns out to depend on how old you are, your sex and your size: the older smaller the person, the smaller their bladder capacity. He further notes,

“For instance, I would expect a young 30-year-old woman to be able to hold 500 to 600 ml with relative ease. But if they get up to age 80, it’s probably around the 300 ml mark”. However, your bladder capacity also depends on how tall you are: “So a five-foot-tall person would have a smaller bladder than a six-foot-tall person.”[16]

The full consideration then of the problem requires a general equilibrium solution that takes into account the age and size distribution of the particles, and the likelihood of urgency by particle group.

The importance and difficulty of this problem is shown by the existence of more than 800 papers, between 1750 and the beginning of the twentieth century, invoking a roll call of many distinguished mathematicians and astronomers. The three body problem, was described by Whittaker as “the most celebrated of all dynamical problems”.[17] The three body problem has a further attribute noted by Grelutzelius in Analytical Mechanics: “its intimate link with the fundamental question of the stability of the solar system.”[18] (I assume this is the same as the martial system.)[19].

The difficulty of this problem when fully considered is shown by the findings of mathematicians Heinrich Bruns and Henri Poincaré who believed they showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals.[20] We too have failed to provide a solution but are working hard on it. For now the lesson is clear: be selfish when it comes to the position of the toilet seat if there is any chance you will be followed by a similar particle.

References

1. Barrow-Green, J. (2010). The dramatic episode of Sundman, Historia Mathematica 37, pp. 164–203.

1. Bruns, Heinrich, “Über die Perioden der elliptischen Integrale erster und zweiter Ordnung (Dorpat 1875)
2. Burgess, David and Richard Zerbe (2013 “The Most Appropriate Discount Rate” Journal of Benefit Cost Analysis volume 4,(3) pp. 391-400

4. John Cartwright: March 2013, http://news.sciencemag.org/physics/2013/03/physicists-discover-whopping-13-new-solutions-three-body-problem
5. Choi, Jay, “Up or Down?A Male Economist’s Manifesto on the Toilet Seat Etiquette_Michigan State University, November 2002
6. Cole, Daniel, (2012) ,Law, Politics, and Cost-Benefit Analysis, 64 Ala. L. Rev. 55 (2012)

1. For further details on Lim’s wok contact Melbourne Urogynaecology: see e.g. http://c.ymcdn.com/sites/www.iuga.org/resource/resmgr/Brochures/eng_bladderdiary.pdf

8. Poincare’, Henri, 1913.“The New Mechanics,” The Monist, Vol. XXIII. 9 English translation)
9. Wolfram,Stephen, “A New Kind of Science Notes for Chapter 7″: Mechanisms in Programs and Nature Section: Chaos Theory and Randomness from Initial Conditions, Page 972

Appendix A

Here is a picture of Hill’s region which Poincare’ used in his discussion of the three body problem. Note the toilet like shape. The picture is due to Rick Moeckel University of Minnesota [email protected] in a presentation appropriately called, “Chaos in the Three Body Problem.”

Appendix B

Persons answering this questionnaire will not be identified except by a full-proof encrypted series of digits. For example Jane Zerbe is identified by the digits 10,26. It is clear to us that the connection to the name is beyond the limits of human or machine intelligence so all respondents are assured anonymity.[21]

	Yes	No	Explain
1. Are you male or female or other? *
2. You just used the sanitary facilities?
3. Did you pee or defecate or both?
4. How many times have you peed or weed today?
5. How many times do you think you will pee before the day is over?
6. Same for poop?
7. Did you find the toilet seat up?
8. Did you find the toilet seat down?
9. If up did you lower it?
10. If down did you raise it?
11. How long did these actions take?
12. Do you prefer to find the toilet seat up?
13. Why?
14. Why not?

*Some problems arose from this question as many responses inappropriately just said yes.

[1] The New York Times editorial of July 27,,2015 considers “America’s Restroom Battles” p.A16

*I wish to thank anonymous referees Jane Zerbe and Gary Mason and Judy and Bill who responded the same day as I submitted, thereby setting a record for response several orders of magnitudes better than is the case with most articles. It is possible, though highly unlikely, that there are mistakes here. If so they are entirely due to my research assistants Amy Glaskova and Andrew Overmyer or my colleague Sandra Archibald. If any are found please send to my wife who is well able to handle any criticism. Or send them to me for burial at the UW, i.e. the University of Washington not the now defunct University of Wisconsin.

** Richard Zerbe was the Daniel J. Distinguished Professor in the Evans School at the University of Washington. He is now not yet dead but close since he has announced as a Republican candidate for the presidential nomination. His by-word is “Have you looked at the other ‘guys'”:.

[2] In 1912, the Finnish mathematician Karl Sundman proved there exists a series solution in powers of time^1/3 for the 3-body problem.^[10] This series is convergent for all real t, except initial data that correspond to zero angular momentum. However, these initial data are not generic since they have Lebesgue measure zero. It turns out, moreover, that this solution is useless as convergence to any result happens very very slowly. See Barrow-Green, J. (2010). The dramatic episode of Sundman, Historia Mathematica 37, pp. 164–203. This is the same Barrow who suggests there are 10⁵⁰⁰ universes in his new (and quite nice) book, The Book of Universes, 2011, The Bodley Head, (not head as in loo): London.

[3] It is a fascinating question what the other 1.33% of women do with the seat up, but no evidence exists to provide an answer. Suggestions are welcome.

[4] My friend Ms Cole who is an expert on bathrooms around the world notes that there are in some China bathrooms that request, “please to not stand on the toilet to pee”. Moreover she notes, [S]ome toilets do away with the problem. No matter which particle should use it. (1st class car, railway from Yangon to Mandalay). No toilet seat at all. No up and no down.

[5] See questionnaire in appendix. I am sorry to report that the questionnaire has raised spurious and incomprehensible issues with the Human Subjects Committee of the University so this study can probably never be replicated.

[6] See Zerbe, Chapter 12, “The Rise of Women” in The Path of Human Progress, forthcoming in a major book, 2016 or later.

[7] An important sematic problem arises here whether or not to say a women pees or wees. I think wees is the more gentle word and in deference to women’s more delicate nature will hence forth use the word wee when women pee.

[8] A less mean women would wee less.

[9] All P values are significant at the 5% level or higher. By higher I mean as high as 50%.

[10] The early date here indicates how long this issue has been a problem. A 2013 reprint is available.

[11] If we use, say, the 2% favored by the many biased or unscientific environmental economists or lawyers (See work by Daniel Cole), the present value would be larger by factor of 4, or 11.12 X 10¹¹. Given the size of this figure one might speculate if a worthwhile investment at least by environmental economists would be to develop a solution in which we no longer have to pee or wee at all and only poop in the woods.

[12] This result is due to Choi.

[13] This is because the selfish rule includes the term ?(1- ?) while the up and down rules include only ? which will be larger.. This is practically the same as Choi’s rule, except that in his formulation ? is the proportion of men while in mine it is the proportion of those favoring the up position, which, as we have seen, will include some women and a few men who poop a lot. In addition, Choi adds a discount factor which I omit as it adds nothing.

[14] See The New York Times July 27,,2015, p. A16.

[15] Ms Cole informs me that the Whitney museum in NY City has three bathroom, men’s, women’s and others.

[16] Lim also has important work on the best color for pee and wee. See also Dr Lim’s contribution to Randomised trial of a Single incision – versus an Outside-in Transobturator Mid-urethral Sling in Women with Stress Urinary Incontinence -12 months results”., in the Journal of Obstetrics and Gyneclogy, 2015, impact factor 3.97.

[17] It fulfilled for the great mathematician Hilbert the necessary criteria for a good mathematical problem; it can be simply stated: three particles (bodies) moving in space under their attraction to the toilet where their location and the location of the toilets are endogenous. The problem now is, given initial conditions, determine the location of facilities and person particles by type, age and size that will optimize economic efficiency. Like many mathematical problems, the simplicity of its statement belies the complexity of its solution.

[18] My apologies to Grelutzelius for any mistakes in proper attribution. I also note that although I admired his or her work, I did not understand a damn bit of it.

[19] I report an actual case of a women who worked with me who conveniently stated that she would never, never marry a man who did not always, always put the toilet seat down. I found this such an unattractive sentiment that it probably saved me from a sexual harassment suit because I had accidentally used the work “lust”- (a typo)-instead of the word” just” in an email to her. Due to the vigilance of the sexual correctness authorities I was able to settle this matter for a mere $11,000. Indeed one of the administrators at the University used the word “lust” instead of “must” to me in an email . I reframed from suing but let this be a warning to others.

[20] Another solution is provided by an anonymous reader, whom I will call Gary Mason, who notes that we should just use the forest as our toilet as do thousands. As he notes, mis-citing John Muir, “When hiking in the forest, the whole world is your toilet.” After all he asks, “Do bears defecate in the woods?” Why not us?

[21] We induced participation by offering to explain the Selfish Rule to any completing the questionnaire.

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