As long as males and females have been living with indoor plumbing they have argued over the placement of the toilet seat. Most often, males leave the toilet seat up and females leave the toilet seat down. Males may not necessarily have a problem with the toilet seat down, but then females will suffer from wet bottom syndrome from time to time. If males leave the seat up, females may “fall” into the toilet particularly at night. A solution to this problem is to always leave the toilet seat in a particular position: the toilet seat remains down and males must lift the toilet seat to urinate, but then return it to its down position; alternatively, the toilet seat remains up and females must always place the seat down to use the toilet and return it to its upright position when done.

The trouble with this solution is: in which position should the toilet seat be placed? This decision has, no doubt, been the source of many arguments in male-female households. Previous scientific research has been undertaken on this household problem. Choi (2002) uses an optimization method to identify the efficient placement of the toilet seat. He finds that unless the costs of changing the toilet seat position are asymmetric across the parties involved, the optimal toilet seat placement follows the selfish rule: do not change the toilet seat position when you are finished using the toilet. Harter (2005) and Siddiqi (2006) both use game theoretic models to show that optimal toilet seat placement is up. However, Harter (2005) does note that in order to avoid marital conflict the toilet seat may best be in the down position.

In this paper, an efficiency-based argument is used to show which position the toilet seat should be in, depending on the composition of the household. This is done through a mathematical modeling approach that extends previous research by considering households with more than one male and more than one female. Because it takes effort to raise and lower the toilet seat, the toilet seat should be left in the position that minimizes the number of times it must be moved. It is shown that the optimal toilet seat placement depends on the ratio of males to females.

In order to determine the optimal toilet seat placement, a mathematical modeling approach is taken. In order to perform such modeling, a number of axioms must be made.

Axiom 1: Females always use the toilet with the seat in the down position.

Axiom 2: Males urinate with the toilet seat up in the up position and defecate with the toilet seat in the down position.

Axiom 3: Males and females defecate once per day and urinate 7 times per day.

Axiom 3 is clearly critical for the results, but in a sensitivity analysis the results presented below were shown to be robust. Considering these axioms, the toilet seat ratio is calculated as follows:

This ratio is bounded by zero and unity. If the TSR is greater than 0.50, the optimal toilet seat placement is up; if the TSR is less than 0.50, the optimal toilet seat placement is down; and if the TSR is equal to 0.50 the optimal placement is either up or down. The TSR is calculated for all combinations of 1 – 5 males and 0 – 6 females.

The results of the mathematical modeling are shown in Figure 1 and Table 1. Figure 1 also shows the 0.50 mark (grey line) and all TSR values greater than 0.50 in Table 1 are in bold.

Figure 1. Toilet Seat Ratio

Table 1. Toilet Seat Ratios, ad nauseam

The first point to notice in this analysis is that the claims of previous research have not been replicated here: when there is one female and one male in the household the optimal toilet seat placement is down. However, as evident in Table 1, all hope for having the toilet seat in the up position is not lost for males.

Overall, the general results clearly show that if the number of females is greater than or equal to the number of males the optimal placement of the toilet seat is down. Additionally, when males outnumber females, the optimal toilet seat placement is not always up: when there are four males in a household, the optimal toilet seat placement is only definitively up when there are two or fewer females; and when there are five males in a household, the optimal toilet seat placement is only definitively up when there are three or fewer females.

Through the use of mathematical modeling, the analysis in this paper has shown that the optimal placement of the toilet seat can be calculated based on the number of males relative to the number of females. The general result (that is not sensitive to reasonable changes in Axiom 3) is that when the number of females in a household is greater than or equal to the number of males the optimal placement of the toilet seat is down. Therefore, there is no longer any need for males and females to argue over the placement of their toilet seat as long as they are concerned with the efficient expenditure of household energy.

Choi, J.P. (2002). Up or down? A male economist’s manifesto on the toilet seat etiquette. Department of Economics, Michigan State University Working Paper.

Harter, R. (2005). A game theoretic approach to the toilet seat problem. The Science Creative Quarterly.

Siddiqi, H. (2006). The social norm of leaving the toilet seat down: a game theoretic analysis. MPRA Paper No. 856.

I obtain much pleasure from writing.

Also from reading books.

Literature is the greatest art form in the world.

And so I imagine a writer living in a small efficiency apartment in the southern part of the United States.

A third-floor apartment, 33 steps up black cast iron stairs in a back alley.

The apartment has a main bed/living-room and a chair, a small kitchen, a tiny refrigerator, a stove, a small bathroom.

No television, no cell phone, no computer.

But the writer has a typewriter and an old radio setting on a small table with a lamp next to it.

He has a few tattered books, some clothes, an abstract painting on the wall, not much else.

The man has loved literature since he was a small boy.

Escaping into stories and novels and picture books.

But he is also a man of action, not a wimp.

He lifts weights regularly and occasionally gets into brutal fist fights in crappy bars.

The man has been writing ever since the age of 11; and found his “literary voice” at a young age.

Every day the man sits at his typewriter and writes for hours – brilliant short stories and minimal poems come reeling off his typewriter.

There is a primitive beauty to the prose in his stories, a roughness to the visage of his piercing minimal poems, all of it fresh and compelling and packed with genuine emotion, which showcases his unique perspective of the world.
He sends his work out to magazines and journals once a week by regular mail.

And soon he finds that the literary game is NOT RIGGED.

Editors and other writers actually RECOGNIZE and appreciate his literary genius.

They publish his poems and stories in their magazines and quickly he gains the respect and attraction of literary agents and five major NY publishers.

Three small books come out in rapid succession: a novella, a book of short stories, a book of minimal poems. They gain mostly positive reviews in the best periodicals in spite of the revolutionary way the young writer has re-evaluated and almost “attacked” modern literature.

The writer receives large advances and generous royalty checks and moves out of his tiny efficiency apartment.

He gets married and buys a large house and an expensive sports car and continues writing every single day.

And the writer does not become an alcoholic, he does not develop a horrible drug habit, he does not commit suicide.

He only works harder at his writing and fully nurtures his devastating talent.

One summer he decides to buy an old laptop computer equipped with only a primitive word processor and a spell check – no other programs.

He writes a 40,000-word brutalist novel in two months that becomes a true classic of southern gothic literature.

And the writer goes on to become one of the greatest authors America has ever produced and wins numerous major awards and becomes a multi-millionaire.

The End.

Theorem: It is impossible for engineers, scientists and the professoriate to earn as much money as business executives.

Proof:

Postulate 1: Knowledge is Power.
Postulate 2: Time is Money.

As every engineer knows,

Since philosophers and economists have amply demonstrated that Knowledge = Power, and Time = Money, we have via substitution,

Solving for Money, one obtains:

and taking limits,

Result: The less you know, the more you make.

Quod Erat Demonstrandum

Whenever I think of Weird Tales magazine, I recall my father telling me old Conan the Barbarian stories that he read back in the early 60s, which I assume were first published in Weird Tales. I loved the Conan stories my father would recount but I was disappointed that he didn’t have any of the original books; and it struck me as bizarre that I couldn’t simply go out and purchase Robert E. Howard’s books and read them myself. Only later were Howard’s original Conan tales collected together and republished. But after finally reading them, I realized my father had remembered the stories differently than the originals, yet I liked his versions a little better.

Weird Tales also reminds me of the enigmatic and reclusive writer, H.P. Lovecraft, who adored macabre fiction throughout the ages and collected every copy of Weird Tales since the magazine’s birth. At one point, Lovecraft was actually offered the lead editor position at Weird Tales by publisher J.C. Henneberger, the man who set up the magazine, but Lovecraft turned it down, not wanting to relocate to Chicago. Farnsworth Wright, a journalist, accepted the position in Lovecraft’s place and became the most famous editor associated with Weird Tales. But Wright showed signs of being a literary incompetent and a subnormal when he rejected excellent stories by Lovecraft and Howard that later went on to become classics in the field of fantastical literature, such as “The Shadow over Innsmouth,” by Lovecraft and “The Frost Giant’s Daughter” by Howard. In fact, toward the end of H.P. Lovecraft’s career, Wright rejected his work so frequently that Lovecraft actually STOPPED WRITING FICTION ENTIRELY, convinced that he had no talent and was a literary failure. So tragic, what a waste. Wouldn’t it be incredible to actually hold and read one of the original issues of Weird Tales from the 20s or 30s that featured one of Lovecraft’s tales? I’ll have to look on Ebay soon for some past issues.

During the course of my writing career I have sent about ten or twelve fiction submissions to Weird Tales magazine, but none were ever accepted. Recently however I had a new idea. I would send Weird Tales one of my infamous ‘concrete primes,’ hoping they would print it along with my byline. A concrete prime is a mathematical entity I invented that involves prime numbers – integers with no divisors other than themselves and one – along with a visual component to the layout of the number; i.e. the digits are arranged in such a way that a word or phrase is “pictured” in the decimal expansion (see an example in my letter below).

Now why would I want to send Weird Tales a concrete prime? Isn’t that too weird even for Weird Tales? Nope. They print stories with a science fiction tinge all the time, I’m sure. And math is a large part of science. Also I wanted to send them a prime because I’m a prime hunter. I like to find large primes with interesting or strange properties. The primes I “build” to spell out certain words are the ones I call examples of “concrete math” because they remind me of concrete poetry (see the letter below for a definition). And because my concrete primes are sufficiently weird and also have a macabre element, I thought Weird Tales might like the chance to print one, or at least send me an interesting response. They probably won’t publish my prime, but it was still fun to try. Here is the letter I sent to the Weird Tales offices via snail mail:

Dear Editor of Weird Tales Magazine:

Please find below a prime number that spells out WEIRD TALES in its digits. I would be honored if you would print this prime in any section of your magazine with a byline stating, “Found by Jason Earls, author of Red Zen, Cocoon of Terror, and How to Become a Guitar Player from Hell.” Thanks for your consideration.

More info on the concrete prime above: Its full decimal expansion contains 2366 digits and it was found with the freely available program, WinPFGW. Classes of numbers that possess a certain visual component when the digits are arranged in a specific way is a concept closely related (in my opinion) to “concrete poetry” (poems in which the typographical arrangement of symbols or words plays a direct role in conveying the “meaning” of a poem); hence, I have dubbed numbers like the WEIRD TALES prime above, “concrete mathematics.” But I have also thought of simply calling them, “WEIRDematics.”

Regards,
Jason Earls

Surely you can see how ‘WEIRD TALES’ is spelled out in the bolded digits above? Cool, isn’t it. I hope Weird Tales prints it on the cover of their next issue with some wicked visual effects surrounding it. To tell you the truth, finding the Weird Tales prime above was not easy. At one point, my computer actually overheated and began to smoke. Then, like some numerical phantom of perversion, like a monstrous apparition from Hell almost unnamable in its atrocity, the Weird Tales prime number popped up on my computer screen and I stared at it with my body paralyzed with terror. The digits seemed to resemble clotted gray froth boiling in a wizard’s cauldron, or scales on the back of a glistening Extraterrestrial-God from the 24th dimension. Wait, I’m only kidding. I tried to get a little Lovecraftian influence in the text there. I am waiting now for Weird Tales response. Perhaps their editors are partial to mathematics. I know Lovecraft was interested in science and probably mathematics during his lifetime. Surely some of the editors will be able to appreciate math since we’re living in such a computer oriented age. And this prime is definitely weird. But on the other hand, perhaps primes are too nerdy even for Weird Tales. We’ll just have to wait and see. So far I have examined a couple of recent issues of Weird Tales and my prime has not appeared there yet. But maybe by the time you read this article they will have printed it on one of the covers. I hope so. I think I’ll do a Cthulhu prime next. Or one that spells out Conan in the digits. Yeah. Anyway, I’ll keep you posted on what happens.

KEYWORDS: Median, Mean, Census

1. BACKGROUND.

One June 28, 2007, Fryar, et. al. released a report that included the summary statement that “The median number of lifetime female sexual partners for men was seven and the median number of lifetime male sexual partners for women was four.” On August 12, 2007, Kolata reported this result in the New York Times, along with commentary that similar inequalities have been found in other surveys. Kolata’s article also included a claim by mathematician David Gale that the inequality is impossible, since there is a sense in which women have to have the same amount of sex as men. The current paper is a continuation where we prove that the inequality should actually be reversed.

We intend to compare statistical means, rather than medians, since our method of proof does not apply to medians. To be precise, we wish to compare the mean number of distinct partners of the opposite sex experienced by males (M) and females (W) over their lifetimes. The same data (CDC, 2007) that support the introductory quote about medians also support the statement that M= 23.4 and W=7.0 (the data are very skewed, hence the large differences between means and medians). Our contention is that M has to be smaller than W.

We emphasize that our conclusions depend on only one physical fact: there are more men born than women. Given that fact, no assumptions about the sexual habits of either men or women are needed, nor can any be conclusions about those habits be made from the truth of our title.

2. THE PROM THEOREM.

Like Gale, we make an analogy to a high school prom. A dance is a good starting point, since we know when it starts, we know when it ends, and it is very unlikely that anybody will die in the meantime. So suppose there is a prom, and that a chaperone wants to find the average number of girls that a boy dances with (call it M) and likewise the average number of boys that a girl dances with (call it W). Suppose two boys and one girl attend the prom. If both boys dance with the girl, then M=1 and W=2. If only one boy dances with the girl, then M=0.5 and W=1. If nobody dances, then M=0 and W=0. There are no other cases, so in no case is M greater than W. Why is this?

In general, let B be the number of boys at the prom and G the number of girls. Also let U be the number of unique pairings of boys dancing with girls (U is 2, 1, and 0 in the three examples above), and note that the words “boys” and “girls” could just as well be interchanged in the definition of U; if a boy dances with a certain girl for the first time, then that girl is also dancing with the boy for the first time. This basic symmetry is essentially Gale’s observation, and it would follow that M=W if there were the same number of boys as girls at the dance. The general formulas for M and W turn out to be M=U/B and W=U/G, as will be shown in the next section. Thus, regardless of who dances with whom, as long as 0/0 can be interpreted favorably, the ratio W/M is always B/G. To find W/M, we don’t need to measure U or pay attention to the dancing in any way, but only measure the number of boys and girls that attend the prom.

We will now show that the formulas for M and W are correct. Let M_i be the number of unique girls that the i^th boy dances with, and let W_j be the number of unique boys that the j^th girl dances with. Then, by definition, the average number of unique girls danced with by a boy is (M₁ + M₂ +…+ M_B) / B, and the average number of unique boys danced with by a girl is (W₁ + W₂ +…+ W_G) / G. But the two sums are both equal to U, the total number of unique couples dancing. The proof of this is essentially the observation that summation is associative — it is simply a matter of partitioning U in two different ways. Every time a girl and a boy dance for the first time, the M-sum is incremented by one because one of those boys has just danced with a girl, and likewise for the W-sum. Both sums are equal to U, the total number of unique couples that dance.

3. APPLICATION TO SEX.

Our object in this section is to apply the prom theorem to having sex. We assume that “having sex with” is well defined and symmetric, just like dancing at the prom. One of the reasons why survey results conflict with theory may be that this assumption is faulty — Bill may feel that he has had sex with Monica, for example, while Monica doesn’t feel the same way about it. We nonetheless assume symmetry. We also assume that there are only two possibilities for a human, to be either male or female when born. Regardless of sexual preferences or subsequent sex changes, we take the gender of any individual to be as when born. Any sex within a gender is simply ignored — we count only heterosexual couplings. Let M_i and W_j be the number of lifetime partners of the opposite sex for the i^th man and the j^th woman.

The prom analogy is best if we include all humans as subjects, since this eliminates the possibility of someone having sex with someone else who is not a subject. Therefore, in spite of the impracticality, let’s imagine including all humans who have ever lived or ever will live as subjects. This catholic viewpoint will lead us to including many humans who have had no sexual partners at all when they die, as well as a few who have had very many. If man number 3 dies at age 2, for example, then M₃=0. If we include all humans in our complete survey, regardless of the age at death, then, when the books are finally closed on the human race, we will have a perfect analogy to the prom. The ultimate ratio W/M will be the same as the ratio of the number of men who have ever lived to the number of women who have ever lived. If current trends continue, this ratio will be approximately 1.05, since that is the ratio of male to female births. In any case, the truth of falsity of our title depends only on this ratio.

The situation is a bit more complicated if we want to average over some finite time period. Suppose we take our subjects to be all people who die in that period. There may have been sex between subjects and other humans who are not subjects on account of outliving the period. In the prom analogy where U=1, suppose that the boy who dances turns out not to be a subject, so that the population consists of one boy and one girl. In that case we would find M=0 (the only boy in the population doesn’t dance) and W=1 (the girl still danced with somebody), which is confusing. If instead the girl is not a subject, then W is not even defined. In spite of these difficulties over short periods, there is no essential complication to our argument as long as the period is long compared to a human lifetime — almost everyone who has sex with a subject will also have died, and therefore also be a subject. Over a long period of time where more men than women enter the population, we will still find W>M.

Women live longer than men, to the extent that women outnumber men in the population in spite of the larger male birth rate. Might there be some circumstance where “mature” men have more sex than “mature” women, even though more men are born than women? Certainly there is. We might, for example, define “mature” to mean “over 30”, and imagine a world where women only have sex when under 30, and always die of it immediately. A survey of mature women will find only women who have never had sex and never will, whereas the same may not be true of mature men because some of them have had sex, albeit not with mature women. As long as we count sex between mature and immature individuals, this kind of example will always be possible because the immature individuals are not themselves subjects, and the prom theorem need not apply. The theorem still applies, however, if we count only sex between mature individuals. When someone dies, we ignore the event if his or her age is less than the maturity age t, or otherwise include him or her as a subject. The score for that subject may not be known for another t years, since it may not yet be clear whether some of the subject’s partners are themselves subjects, but is nonetheless well defined. If we define W_t and M_t to be the average scores of mature women and men, respectively, we will find W_t > M_t if and only if the rate at which mature men enter the population is greater than the rate at which mature women enter the population. This rate of entry is simply the population in year group t. According to the US census (Census, 2007), the smallest year group for which women outnumber men is 36, so the “maturity” age at which women and men have the same amount of mature sex over the rest of their lives is about 35. Conventional definitions of sexual maturity are well below age 35, and so, barring examples of the type that began this paragraph, the conclusion that women have more sex than men persists.

4. WHY THE CONFLICT?

We have argued above that the ratio W/M should be about 1.05, the same as the ratio of male births to female births. Surveys, on the other hand, usually verify the popular notion that men have more sex than women. Here are some possible explanations for the conflict:

1. The survey responses might not be accurate – perhaps men brag about sex, whereas women don’t.

2. As mentioned earlier, “having sex” might mean different things to men and women. Men may count some incidents that women don’t.

3. The surveyed subjects might not be a random sample. For example, they might not include prostitutes, and they certainly don’t include young children.

4. The surveys are not exit interviews. Instead of sex over a complete lifetime, subjects report only sex so far.

5. If the statistics for women were more skewed than for men, as they might be if a significant fraction of sex involved high-scoring female prostitutes, then the median ranking might reverse the mean ranking, thus preserving intuition. However, the CDC data (CDC, 2007) do not encourage this explanation. The largest score (9000) out of about 5,000 subjects is held by a 38 year old male, and the next largest score (1500) is also held by a male. On account of these large male scores, the mean scores mentioned in the introduction show an even greater disparity than the medians.

REFERENCES
CDC (2007), link, Center for Disease Control, “demographics” and “questionnaire” datasets for 1999-2000 and 2001-2002, accessed September, 2007.

Census (2007), “detail” file here, U. S. Census Bureau, accessed September, 2007.

Fryar, C.; Hirsch, R..; Porter, K.; M.S.; Kottiri, B.; Brody, D.; Louis, T. (June 28, 2007, revised September 11, 2007). “Drug Use and Sexual Behaviors Reported by Adults:United States, 1999–2002”, Center for Disease Control, Advance Data #384.

Kolata, G. (August 12, 2007), “The Myth, the Math, the Sex”, New York Times.

Arbesman Limit (keywords: science, eponym, immortality)

… the maximum number of concepts or ideas that can be named after a single person

link

Drugmonkey Scale (keywords: drugs, reaction to blog post, neuropsychology)

link

Fox Paradox (keywords: genomics, ethics)

Just because we’ve sequenced your genome, we don’t necessarily know your name.

Some notable exceptions:
Craig, James, Susie, Cinnamon, Twilight, Glennie, and RJF#256
link

Gorton’s Measure (keywords: marine life, edibility)

link

Gorton’s Constant (gamma) (keywords: marine life, edibility, lemon, butter)

link

Higgins-Levinthal Dictum (keywords: blog post, obnoxious content, comments)

link

Justapie’s Constant (keywords: computers, script, sign mistakes)

J = maximum number of lines of computation that can be done without sign mistakes

link

Ng’s Score (keywords: cup holders, transportation, social value)

link

Orzel Teammate Desirability Factor (TDF) (keywords: basketball, player assessment)

link

Redfield Factor (keyword: DNA, mass)

The number of kilobase pairs in a gram of DNA: 10¹⁸

link

Rowan Sarchasmic Index (keywords: sarcasm, irony, the British)

And accompanying ironic susceptibility value:

link

Sciencewoman’s Law (keywords: post frequency, work, kids, life)

link

Semeniuk-Bjorge-Colby Score (keywords: sex, hairyness, pity)

link

Stemwedel Index of Luddite Nature (keywords: luddite, technology)

link

In this paper, we internalize the cost of yelling and model the conflict as a non-cooperative game between two species, males and females.We find that the social norm of leaving the toilet seat down is inefficient. However, to our dismay, we also find that the social norm of always leaving the toilet seat down after use is not only a Nash equilibrium in pure strategies but is also trembling-hand perfect. So, we can complain all we like, but this norm is not likely to go away.

All hope is not lost though. An important issue regarding social norms is whether they are created to increase welfare. Are they society’s response to market failures? One such norm is tipping for service quality. Azar (2003) has shown that the norm of tipping increases social welfare. In this paper, we show conclusively that the social norm of leaving the toilet seat down after use decreases welfare and by doing that we hope to convince the reader that social norms are not always welfare enhancing. Hence, there is a case for scientifically examining social norms and educating the masses about the fallacy of following social norms blindly.

The Structure of the Game
The basic ingredients of the game are the same as in Harter (2005). Where we differ from Harter (2005) is the explicit modeling of the costs of yelling.

Consequently, as will become clear shortly, our game is a non-cooperative game of conflict, whereas Harter (2005) models it as a cooperative game. There are two people, one is a representative of the male species, call him John and the other is a representative of the female species, call her Marsha. Initially, they live alone and separately. Each has access to a separate restroom with one toilet. They use this toilet for two operations; #1 and #2. Marsha performs both with the seat in the down position whereas John performs #1 with the seat in the up position and #2 with the seat in the down position. That means, he must change the seat position appropriately before performing the corresponding operation [1]. Assume that the inconvenience cost of changing the seat position is C. Further assume that the need for #1 arises with a probability p. Let’s look at the average costs to John and Marsha:

The average cost to John as a bachelor
Doing #1 this time when he did #2 last time + Doing #2 this time when he did #1 last time:

The average cost to Marsha as a bachelorette
Obviously 0 since she performs everything with the seat in the down position.

Now, consider the situation in which John and Marsha decide to cohabit and both use the same toilet. This situation is popularly known as marriage [2]. That changes things for the worse for both parties as far as the toilet operations are concerned. John argues ‘Why does it matter if the seat is up or down? Let’s leave the seat in the position used.” Let’s call that strategy J. Marsha fights back, “It must be down or else.” Or she says, “If you love me then you would leave the seat down.” Let’s call this strategy M. Assume that both John and Marsha use the toilet with the same frequency.

The average cost to husband John from strategy J
Doing #1 and the last user was Marsha + Doing #1, the last user was John and he did #2 + Doing #2, the last user was John and he did #1:

The average cost to wife Marsha from strategy J
The last user was John and he did #1:

The marginal costs under strategy J

The average cost to husband John from strategy M
Doing #1:

The average cost to wife Marsha from strategy M
Obviously 0.

The marginal costs under strategy M

We differ from Harter (2005) in the following:
From this point, Harter (2005) considers the situation as a cooperative game. We differ and consider the situation as a 2-player non-cooperative game with 2 strategies for each player. Suppose Marsha (following M) goes to the toilet and finds the seat in the up position, consequently she yells at John inflicting a cost of D on him. Assume that D >>C . Table 1 shows the payoff matrix.

As can be seen from table 1, there are two Nash equilibria in pure strategies: (J,J) and (M,M).

Proposition 1: The equilibrium (J,J) is not trembling-hand perfect whereas the equilibrium (M,M) is trembling-hand perfect.

Proof: Let e be a very small probability that Marsha trembles to M. Since D >>C, it follows that the Nash equilibrium (J,J) does not survive the trembling hand. By a similar argument, the opposite conclusion is reached for the Nash equilibrium (M,M).

Proposition 2: The equilibrium (M,M) is inefficient compared to the equilibrium (J,J)

Proof: The total cost of strategy combination (J,J) is p² x C. The total cost of strategy combination (M,M) is higher: 2 x p² x C.

Discussion and conclusions
For “mankind”, the analysis in this paper has the following appeal: Once again, it has been found that the social norm of leaving the toilet seat down is inefficient; hence, “mankind” may feel vindicated.

For “womankind”, the analysis in this paper is appealing for the following reason: It has been shown that the social norm of leaving the seat down is a trembling-hand perfect equilibrium. Hence, this norm is not likely to go away, at least in the near future.

References

Azar, Ofer H. (2005), “Who Do We Tip and Why? An Empirical Investigation,” Applied Economics, 37(16), 1871-1879

Choi, P. (2002), “Up or down? A male economist’s manifesto on the toilet seat etiquette.” Michigan State Working Paper

Harter, R., (2005), “A game theoretic approach to the toilet seat problem.” Science Creative Quarterly

The law, in its simplest form, states:

“If anything can possibly go wrong, it will, and at the worst possible time”

After beginning work on this project, previous work was uncovered. The work is minimal, but there has been at least one other notable attempt1 at generating an equation for Murphy’s Law. This equation however does not fit recorded data very well, and is very complex. More careful analysis of the Law and recorded data, as well as personal experience, have led me to the following new equation (the derivation of which I have not included here), called Murphy’s Equation:

First, to start with a simple example, lets calculate the probability of the clutch on a 1989 Toyota Tercel ceasing to function 100km from home at night in the middle of a rainstorm. The importance of the clutch working in this situation is obviously high, but no one is dying, so lets estimate an 8. The system is fairly simple compared to other systems in the car, so C=5. It would be nice if the clutch worked soon, so the urgency U would also be around 8. Finally, the clutch only needs to work for one drive home, so the frequency is low, say F=1. Now, putting these parameters in Murphy’s Equation shows that P_M=1. Comparing to experimental data, this number matches exactly, as the clutch did indeed cease to function at this time. Repeating this calculation with I=7 also gives P_M=1, showing the robustness of this equation.

Moving to a more complicated example, lets examine the case where the clutch has failed in the above example, but calculate the probability that the flashlight needed to inspect the clutch doesn’t work, the front half of the hazard lights on the car don’t work and that the trunk where the few tools are kept has 3” of water in it (remember its raining). In this case, Importance is about the same, an 8 or so (its hard to justify higher than this without a life or death situation). Here, C=4, as the flashlight, hazard lights and trunk are all fairly simple systems. Because it is raining and the driver is now stranded, the urgency can be bumped up to a 9, while F=3 (three events need to occur). Putting these values into the equation gives a P_M of 1, which is indeed what did happen, showing that again this equation matches experimental data very well.

In summary, the equation presented here gives a new mathematical foundation to one of today’s most recognized physical laws, Murphy’s Law. Experimental data matches very nicely with the theory presented in this equation, and the author expects further testing will yield equally positive results, and hopefully the adoption of this equation as a metric for predicting how often things really do go wrong for no good reason.

References:

1) “The formula that proves that ‘Sod’s Law’ [Murphy’s Law] really does strike at the worst possible time” British Gas News, October 7, 2004.

Oh, telomerase,
You crafty soul,
Whose life-giving phrase
Like us, gets old
Lets polymerase,
With longlasting hold
Yank out from our fate
Our mortal coil

Syllable Count

5 Oh, telomerase,
4 You crafty soul,
5 Whose life-giving phrase
4 Like us, gets old
5 Lets polymerase,
5 With longlasting hold
5 Yank out from our fate
4 Our mortal coil
—-
37

First letter of each line, it’s order in the alphabet and mathematical commentary on the role of the telomerase enzyme itself

O Y W L L W Y O =
15(10)25(-2)23(-11)12(0)12(11)23(2)25(-10)15
10(-12)-2(-9)-11(11)0(11)11(-9)2(-12)-10
-12(3)-9(20)11(0)11(-20)-9(-3)-12
3(17)20(-20)0(-20)-20(17)-3
17(-37)-20(0)-20(37)17
-37(37)0(37)37
37(0)37

The card game of war is typically considered a children’s game, as it requires no skill to play and minimal understanding of playing card relationships in a standard deck. With skill eliminated as a factor in determining the outcome, it is often assumed that war is simply a random game of chance. Yet the game cannot be purely determined by chance, as the initial conditions of the game must have some bearing on the final state. The existence of random factors in the game do not allow for the claim that war is a deterministic game, yet it is still possible to quantify properties of the initial state that are indicative of a victory probability.

The Rules of War

The traditional game of war uses a standard 52-card deck of playing cards divided evenly and randomly between two players. Neither player is allowed to view their hand or rearrange it. The players simultaneously reveal the top card of their deck on the table, and the player with the higher value takes all cards on the table. Cards won from a particular trick are gathered and added to the bottom of the victor’s deck. Suits are disregarded in war, making it an ideal game for exposing young players to concepts of inequality. If both players reveal cards of equal value, a war commences: each player plays two face-down cards on the table and reveals a third card that decides the victor. It is possible to play two, three, or more iterations of a war before the revealed cards are not equal, but in most cases one iteration will suffice to decide the outcome. The victor gains all cards played in the war, including those played face-down. This is the only way in which a player may capture an opponent’s aces. A player loses the game when all of their cards have been captured or exhausted.

Variations in the game of war include varying the number of face-down cards in a war to one, three, a random value, or a plethora of other dependencies. The other major variation consists of placing all cards won from tricks in a separate pile until the player’s deck is extinguished. These cards are then shuffled and used to continue the game. None of these variations is significant enough to differ from the standard rules above when considering a long term analysis. Other variations of war exist in plenty, but most of these have additional rules that warrant their classification as a different game.

Collecting Game Statistics

A computer simulation of war provides a robust and rapid method for examining deterministic tendencies of the game. This simulation used the Mersenne twister random number generator, which has a period much greater than is required for an analysis of this magnitude. Ten million games of war generated a set of three characteristics. The first, and most obvious, statistic is victory. One might guess that in lieu of any deterministic factors, the outcome of a game of war would be no different than tossing a coin. Two other values were also calculated based on a player’s starting hand: deck weight and initial advantage.

Deck weight is a measure of the relative strength of a player’s starting 26 cards. Given that there are 13 different numerical card values (two through ace), define the weight of the card with the numerical value “8″ to be 0. Defined as such, the cards [9, 10, J, Q, K, A] would have weights of [1, 2, 3, 4, 5, 6] respectively, and likewise the cards [2, 3, 4, 5, 6, 7] would have respective weights of [-6, -5, -4, -3, -2, -1]. The deck weight is then defined as the sum of the weights of a player’s initial 26 cards. The weight of an entire 52 card deck is 0, as would be the case for each player in a perfectly divided game of war (for example, one player is dealt all the reds while the other all the blacks). In a game not so evenly divided, the weight of one player’s deck is equal to the opposite of the other player’s deck (since the weight of the entire 52 cards must vanish). The maximum possible deck weight for a player is 84, which would ensure certain victory. In general, a large deck weight may be indicative of a player’s advantage over their opponent, at least concerning card values.

While the deck weight may suggest a certain type of advantage, the ordering of the cards in war is equally important. The initial advantage is defined for a player as the number of tricks won minus the number of tricks lost, over the first iteration through their deck. If no wars occur in the first 26 cards of play, it would be expected on average for a player to have an initial advantage of 0. The maximum initial advantage of 26 would give a player sure and swift victory. For any particular game, a player with a large initial advantage will be in possession of a majority of the cards once all initial 26 have been played. After this point, though, the random element of placing cards won at the bottom of a player’s deck renders the initial advantage measurement useless any further prediction. At best, it is an indicator of the depedence of final victory on the initial ordering of the cards.

Simulation Results

The average cases for both statistics are as expected: the mean deck weight and mean initial advantage are both 0, with both statistics normally distributed about a mean of zero at a 99% confidence level. Deck weight is distributed with a standard deviation of 13.62 and initial advantage with a standard deviation of 4.69. It is interesting to note that even with 10 million games, the maximum and minimum values of the weight and initial advantage did not occur once. Of course, these states are also highly improbable. Neither histogram corresponds exactly to the normal distribution, but the agreement is well within acceptable statistical significance.

Figures 1 and 2 are plots of the probability of victory versus deck weight and initial advantage, respectively. The relationship between deck weight and victory shown in figure 1 is clearly linear, at least for a deck weight between -40 and 40 (or 3σ). At magnitudes greater than 40, the number of sample games decreases sharply allowing the bounds on the victory probability to be less constrained. Fitting a line to this data with linear least-squares yields the equation

where W is the deck weight and p(W) is the probability of victory. This line fits the data with a correlation of R²=0.994. With a deck weight W=0, the probability of victory is nearly 50% as expected. Although on average a player will win and lose an equal number of games, predictability is possible once the deck weight is determined. The probability of victory improves to 60%, 70%, and 80% respectively for deck weights one, two, and three, standard deviations from the mean.

The relationship between initial advantage and victory is illustrated in figure 2. This trend is even more markedly linear than figure 1, with no points significantly deviant from a linear regression. This can be attributed to the smaller range of values (-26 to 26) of initial advantage compared with deck weight. The fit linear relationship between these two variables is

where A is the initial advantage and p(A) is the probability of victory. This line fits the data with R²=0.999, even better than the relationship in figure 1. For this statistic, the victory probabilities at one, two, and three standard deviations from the mean are 54%, 58%, and 61%. Though the relationship is apparent between A and p(A), the degree to which initial advantage is useful in prediction is low. An initial advantage of 26 would win the game automatically, but a slightly lesser value of 24 does not even win 70% of the time. Prediction based on initial advantage seems to be of higher risk and lesser certainty.

Deck weight is a useful indicator of victory probability, and is easily obtained by keeping a cumulative total of the weight as the first 26 cards are played. However, the question of applicability now comes into focus: the weight of the deck is only known after play has begun. Perhaps the cunning player could casually make a friendly wager once they discover their own advantage. Or, those slight of hand could stack the deck in their favor enough to tip the balance—but not enough that suspicions are aroused. But of course, fair game play is always the best course of action. Perhaps the ability to impress an opponent by making an early prediction will suffice, as we sing with Edwin Starr: War. Huh. Yeah. What is it good for? Absolutely nothing!