Up Or Down? a Male Economist's Manifesto on the Toilet Seat Etiquette.
Dear Annie: I read with interest the letters about putting down thetoilet seat. I've been browbeaten by various women for the past 60yr about proper seat etiquette, starting with my mother. If I forget toput the seat down even once, my wife reminds me for hours about thislife-threatening situation. I know you said the last column was the final word on the subject,but I hope you'll reopen the issue. I want to ask women: Who gaveyou exclusive ownership of the bathroom? If men are nice enough to putthe lid down, why can't you ladies lift it up when you are done?When I suggested this to my wife, she wanted to have me taken out andshot. It's time to rebel!--Fed Up in Salem, Ore. Dear Fed Up: What is it about toilet seats that excites people? Wereceived hundreds of letters on this subject and decided the "lastword" would have to wait--Kathy Mitchell and Marcy Sugar. (1) I. INTRODUCTION Should the toilet seat be left up or down after use? This is aquestion that arises when members of the opposite sex share the sametoilet. For some reason, this seemingly trivial question elicits passionfrom all sorts of people. It has become a topic of national debates inpopular syndicated columns by Ann Landers and TV sitcoms such asABC's "Home Improvement" and NBC's "3rd Rockfrom the Sun." It is clear that this age-old debate is divided bythe gender. Women complain that it should be the man'sresponsibility to lower the toilet seat after use. "Leaving thetoilet seat up" is often described as a problem, and there is evena toilet seat that goes down automatically after about 2 min, claimingthat it has the perfect solution to the problem. Men seem to questionwhy women should be the free-riders all the time. To quote Larry James (2004), a personal relationship counselor, "The most hotly contested battlefield in the gender wars may not necessarily be in thebedroom. It may be the bathroom. The seat-up versus seat-down debaterages on ..." Despite high emotions in the debate, scientific inquiries into thisissue are sparse. In fact, it is not obvious why there should be apresumption that men are expected to leave the toilet seat down afteruse. Internet search generated the following noneconomic/scientificreasons for the down rule. First, there is an argument that beingconsiderate to one's love partner's needs supports thingsgoing well in and out of the bedroom. To quote a phrase in the Internet(available at "Foreplay begins with putting the toilet seat down without beingasked!" Second, it is not good Feng-Shi to leave the toilet seatup. Third, a toilet is not the most attractive household appliance.Closing the lid improves its appearance and prevents things from fallinginto the bowl. The last argument, however, proposes not only the seatdown but also the lid down. In this paper, I investigate whether there is any justification forthe down rule based on economic efficiency. I find that the down rule isinefficient unless there is large asymmetry in the inconvenience costsof shifting the position of the toilet seat across genders. I show thatthe "selfish" or the "status quo" rule that leavesthe toilet seat in the position used dominates the down rule in a widerange of parameter spaces including the case where the inconveniencecosts are the same. The intuition for this result is easy to understand.Imagine a situation in which the aggregate frequency of toilet usage isthe same across genders, that is, the probability that any visitor willbe male is 1//2. With the down rule, each male visit is associated withlifting the toilet seat up before use and lowering it down after use,with the inconvenience costs being incurred twice. With the selfishrule, in contrast, the inconvenience costs are incurred once and onlywhen the previous visitor is a member of different gender. The worstcase under the selfish rule would occur when the sex of the toiletvisitor strictly alternates in each usage. Even in this case, the totalinconvenience costs would be the same as those under the down rule ifthe costs are symmetric. If there is any possibility that consecutiveusers are from the same gender, the selfish rule strictly dominates thedown rule because it keeps the option value of not incurring anyinconvenience costs in such an event. This logic can be extended to thecase of asymmetric aggregate frequency of toilet usage across genders. The remainder of the paper is organized in the following way. InSection II, I compare three plausible rules for the toilet seatposition--up, down, and selfish--on an efficiency criterion. I show thatthe selfish rule always dominates the other two if the inconveniencecosts of changing the toilet seat position are the same across genders.In Section III, I characterize the optimal rule for the toilet seatposition. It turns out that the selfish rule is the most efficient rulein a wide range of parameter spaces. I also derive the condition thatthe down rule can be the most efficient one when the inconvenience costsare asymmetric. Section IV extends the analysis to the case where theinconvenience costs are heterogeneous even within the same gender.Section V contains concluding remarks. II. THE BASIC MODEL I consider the usage of a toilet that is shared by members of theopposite sex. Assume that the proportion of male to all users of a certain toiletis given by [alpha]. Let me assume the frequency of using a toilet bymale and female is the same without loss of generality. If one genderuses the toilet more often, this asymmetry can be reflected in [alpha].Thus, the parameter [alpha] represents the relative aggregate frequencyof male using the toilet. (2) I analyze an infinite horizon discrete time framework where thetoilet is used once in each period. The discount factor is given by[delta]. With the assumption about the relative frequency of the toiletusage by each gender, the probability that the user is male in eachperiod is given by [alpha]. (3) The inconvenience cost of lowering thetoilet seat for women is given by [c.sub.f]. The corresponding cost oflifting the toilet seat for men is given by [C.sub.m]. Even though I usethe term inconvenience costs, [c.sub.f] and [C.sub.m] can encompassother types of costs such as "unwittingly placing one's bottomdirectly on the porcelain" and risk of falling in by sitting downwithout looking when the seat is up or "leaving sprinkles on theseat" when it is down, respectively. My goal in this section is to compare the expected aggregateinconvenience costs of three rules--down, up, and selfish--concerningthe position of the toilet seat. In this comparative analysis, Iabstract from other considerations such as being considerate to membersof the opposite sex, aesthetic aspects, the wear costs of the seathinge, etc. A. The Down (Female-Friendly) Rule This is a rule that leaves the position of the seat down after oneis done with the bathroom task. In particular, this rule implies thateach visit by a male member will be associated with the inconveniencecosts of 2 [c.sub.m], whereas female members will incur no costs. Let [V.sup.DOWN.sub.m] and [V.sup.DOWN.sub.f] denote the valuefunctions with the down rule when the particular user in the currentperiod is male and female, respectively. Then, these value functionssatisfy the following recursive relationships. (1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2) [V.sup.DOWN.sub.f] = [delta][[alpha] [V.sup.DOWN.sub.m] (1 -[alpha]) [V.sup.DOWN.sub.m]] By solving these two equations, we can get (3) [V.sup.DOWN.sub.m] = - (1 - [delta][alpha]/1 -[delta])(2[c.sub.m]) (4) [V.sup.DOWN.sub.f] = - (1 - [delta][alpha]/1 -[delta])(2[c.sub.m]) Because the probability of a particular arrival being male is[alpha], the value function associated with the down rule is: (5) [V.sup.DOWN.sub.f] [alpha] [V.sup.DOWN.sub.m] (1 - [alpha])[V.sup.DOWN.sub.f] = - [alpha]/1 - [delta](2 [c.sub.m]) B. The Up (Male-Friendly) Rule This is a rule that leaves the position of the seat up after one isdone with the bathroom task. In this case, all the inconvenience costsare incurred by females. The case is a mirror image of the down rule andthe value function of this rule can be derived in an analogous way. Let[V.sup.UP.sub.m] and [V.sup.UP.sub.f] denote the value functions whenthe particular user is male and female, respectively. Then, these valuefunctions satisfy the following relationships. (6) [V.sup.UP.sub.m] = [delta][alpha][V.sup.UP.sub.m] (1 -[alpha])[V.sup.UP.sub.f]] (7) [V.sup.UP.sub.f] = 2 [c.sub.f] [delta][[alpha][V.sup.DOWN.sub.m] (1 - [alpha])[V.sup.UP.sub.f]] By solving these two equations, I can derive (8) [V.sup.UP.sub.m] = - [delta](1 - [alpha])/1 - [delta](2[c.sub.f]) (9) [V.sup.UP.sub.f] = - (1 - [delta](1 - [alpha])/1 - [delta])(2[c.sub.f]) Because the probability that a particular arrival is male is[alpha], the value function associated with the down rule is: (10) [V.sup.UP] = [alpha] [V.sup.UP.sub.m] (1 - [alpha])[V.sup.UP.sub.f] = -(1 - [alpha])/1 - [delta] (2[c.sub.f]) A comparison of Equations (5)and (10) yields the followingproposition. PROPOSITION 1. The down rule is more efficient than the up rule ifand only if [c.sub.f]/[c.sub.m] > [alpha]/1 - [alpha]. C. The Selfish (Status Otto) Rule This is a rule that leaves the position of the seat as it was used. Let [V.sup.SQ.sub.m] and [V.sup.SQ.sub.f] denote the valuefunctions when the particular user is male and female, respectively,under the selfish rule. Then, these value functions satisfy thefollowing relationships. (11) [V.sup.SQ.sub.m] = - (1 - [alpha])[c.sub.m] [delta][alpha][V.sup.SQ.sub.m] (1 - [alpha]) [V.sup.SQ.sub.f] (12) [V.sup.SQ.sub.f] = [alpha][c.sub.f] [delta][alpha][V.sup.SQ.sub.m] (1 - [alpha]) [V.sup.SQ.sub.f] By solving these two equations, I get (13) [V.sup.SQ.sub.m] = - [(1 - [alpha])(1 - [delta])(1 -[alpha]))/1 - [delta] [c.sub.m] [delta][alpha](1 - [alpha])/1 - [delta] [c.sub.f]] (14) [V.sup.SQ.sub.f] = - [[delta][alpha](1 - [alpha])/1 - [delta][c.sub.m] [alpha](1 - [delta][alpha])/1 - [delta] [c.sub.f]] Because the probability that a particular arrival is male is[alpha], the value function associated with the down rule is: (15) [V.sup.SQ] = [alpha][V.sup.SQ.sub.m] (1 - [alpha])[V.sup.SQ.sub.f] = - [alpha](1 - [alpha])/1 - [delta]([c.sub.m] [c.sub.f]) Comparisons of Equations (5), (10), and (15) give me the followingresult. See also Figure 1. PROPOSITION 2. If the inconvenience costs are the same acrossgenders ([C.sub.m] = [C.sub.f]), the selfish rule dominates both the upand down rules. [FIGURE 1 OMITTED] The intuition for Proposition 2 is easy to understand. With eitherup or down rule, each member of one gender group has to incur theinconvenience costs two times with each usage. This practice can beobviously inefficient in the event that consecutive users are from thesame gender to which the inconvenience costs are attributed. Thisinefficiency can be avoided by using the selfish rule because theinconvenience costs are incurred only when the consecutive users arefrom different genders. Even in such an event, the aggregate costs wouldbe the same as those under the up or down rule if the inconveniencecosts are the same across genders. I cannot rule out the optimality of, say, the down rule if theinconvenience costs are asymmetric across genders. My analysis, however,suggests that to justify the down rule on efficiency grounds, theinconvenience costs for female should be very high relative to those formale. More precisely, the condition for the down rule to dominate theselfish rule is [gamma] = [c.sub.f]/[c.sub.m] > 1[alpha]/1-[alpha].For instance, if male and female users visit the toilet with the samefrequency ([alpha] = 1/2), the inconvenience costs for female should bethree times higher than the corresponding costs for male to justify thedown rule. Up to now, I have considered only three potential mechanisms. Thesethree rules, however, are not the only rules we can entertain. Forinstance, I can imagine a rule such that the position of the seat shouldbe restored to the prior position before use. Alternatively, I can alsoconsider a mutually considerate rule in which male users leave the seatdown, whereas female users leave the seat up after use. In the nextsection, however, I show that all these rules are dominated by one ofthe three rules I have considered. Thus, restricting my attention to thethree rules does not entail any loss of generality in the analysis. III. CHARACTERIZATION OF THE OPTIMAL RULE: A MECHANISM DESIGNAPPROACH In the previous section, we compared three simple rules that can beused for the toilet seat position. The task of this section is to derivethe most efficient rule among all possible mechanisms. I show that oneof the three rules discussed in the previous section is always optimal.Thus, restricting my attention to the three rules does not entail anyloss of generality if the only concern is to minimize the aggregateinconvenience costs of toilet users. The general rule can be considered a collection of four numbers([[sigma].sub.um], [[sigma].sub.dm], [[sigma].sub.uf], [[sigma].sub.df)where [[sigma].sub.ij] denotes the probability that the seat be downafter use when the position of the seat before use is i and the visitoris j, where i = u, d and j = m, f. The first subscripts u and d denoteup and down, respectively, and the second subscripts m and f denote maleand female, respectively. The objective is to search for the bestmechanism that minimizes the aggregate inconvenience costs. In the Appendix, I prove that the position of the seat beforeone's use should not count in the optimal rule. LEMMA. The optimal rule should depend only on the gender of theuser, not the position of the seat before one arrives. With the help of lemma, I can restrict my search for the optimalmechanism to a class of rules that can be written as ([[sigma].sub.m],[[sigma].sub.f]), where [[sigma].sub.m] and [[sigma].sub.f]are theprobabilities that the toilet seat should be in the down position afterusage by a male and a female, respectively. Let [[sigma].sub.m]([[sigma].sub.m], [[sigma].sub.f] and[V.sub.f]([[sigma].sub.m], [[sigma].sub.f]) be the corresponding presentdiscounted value when a particular user in the current period is maleand female, respectively. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Then, the corresponding value function for the rule([[sigma].sub.m], [[sigma].sub.f]) can be written as V([[sigma].sub.m], [[sigma].sub.f]) =[alpha][V.sub.m]([[sigma].sub.m], [[sigma].sub.f]) (1 -[alpha])[V.sub.f] ([[sigma].sub.m], [[sigma].sub.f]) = [-[alpha]M (1 - [alpha]F)]/(1 - [delta]), where M = [[alpha][[sigma].sub.m] (1 - [alpha])[[sigma].sub.f]][c.sub.m] [[sigma].sub.m][C.sub.m] and F = [[alpha](1- [[sigma].sub.m]) (1 - [alpha](1 - [[sigma].sub.f])][c.sub.f] (1 -[[sigma].sub.f])[C.sub.f]. The search for the optimal mechanism is equivalent to solving [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] PROPOSITION 3. The optimal toilet etiquette is given by thefollowing: Let [gamma] = [c.sub.f]/[c.sub.m] be the relative cost of changingthe toilet seat position for male and female. Then, the optimal rule ischaracterized by two critical values of [gamma] ([[gamma].bar] and[bar.[gamma]]) such that: (1) The toilet should be down if [gamma] > [bar.[gamma]] =1[alpha]/1-[alpha] (2) The toilet should be left as it was used if [alpha]/2-[alpha] = [gamma] [alpha]/1-[alpha],women should leave the seat as it was used and a male visitor should putthe toilet seat down after use if [C.sub.m]