toilet seat. I've been browbeaten by various women for the past 60
yr about proper seat etiquette, starting with my mother. If I forget to
put the seat down even once, my wife reminds me for hours about this
life-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 gave
you exclusive ownership of the bathroom? If men are nice enough to put
the 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 and
shot. It's time to rebel!--Fed Up in Salem, Ore.
Dear Fed Up: What is it about toilet seats that excites people? We
received hundreds of letters on this subject and decided the "last
word" 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 a
question that arises when members of the opposite sex share the same
toilet. For some reason, this seemingly trivial question elicits passion
from all sorts of people. It has become a topic of national debates in
popular syndicated columns by Ann Landers and TV sitcoms such as
ABC's "Home Improvement" and NBC's "3rd Rock
from the Sun." It is clear that this age-old debate is divided by
the gender. Women complain that it should be the man's
responsibility to lower the toilet seat after use. "Leaving the
toilet seat up" is often described as a problem, and there is even
a toilet seat that goes down automatically after about 2 min, claiming
that it has the perfect solution to the problem. Men seem to question
why 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 the
bedroom. It may be the bathroom. The seat-up versus seat-down debate
rages on ..."
Despite high emotions in the debate, scientific inquiries into this
issue are sparse. In fact, it is not obvious why there should be a
presumption that men are expected to leave the toilet seat down after
use. Internet search generated the following noneconomic/scientific
reasons for the down rule. First, there is an argument that being
considerate to one's love partner's needs supports things
going 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 being
asked!" Second, it is not good Feng-Shi to leave the toilet seat
up. Third, a toilet is not the most attractive household appliance.
Closing the lid improves its appearance and prevents things from falling
into the bowl. The last argument, however, proposes not only the seat
down but also the lid down.
In this paper, I investigate whether there is any justification for
the down rule based on economic efficiency. I find that the down rule is
inefficient unless there is large asymmetry in the inconvenience costs
of shifting the position of the toilet seat across genders. I show that
the "selfish" or the "status quo" rule that leaves
the toilet seat in the position used dominates the down rule in a wide
range of parameter spaces including the case where the inconvenience
costs are the same. The intuition for this result is easy to understand.
Imagine a situation in which the aggregate frequency of toilet usage is
the same across genders, that is, the probability that any visitor will
be male is 1//2. With the down rule, each male visit is associated with
lifting the toilet seat up before use and lowering it down after use,
with the inconvenience costs being incurred twice. With the selfish
rule, in contrast, the inconvenience costs are incurred once and only
when the previous visitor is a member of different gender. The worst
case under the selfish rule would occur when the sex of the toilet
visitor strictly alternates in each usage. Even in this case, the total
inconvenience costs would be the same as those under the down rule if
the costs are symmetric. If there is any possibility that consecutive
users are from the same gender, the selfish rule strictly dominates the
down rule because it keeps the option value of not incurring any
inconvenience costs in such an event. This logic can be extended to the
case of asymmetric aggregate frequency of toilet usage across genders.
The remainder of the paper is organized in the following way. In
Section II, I compare three plausible rules for the toilet seat
position--up, down, and selfish--on an efficiency criterion. I show that
the selfish rule always dominates the other two if the inconvenience
costs of changing the toilet seat position are the same across genders.
In Section III, I characterize the optimal rule for the toilet seat
position. It turns out that the selfish rule is the most efficient rule
in a wide range of parameter spaces. I also derive the condition that
the down rule can be the most efficient one when the inconvenience costs
are asymmetric. Section IV extends the analysis to the case where the
inconvenience 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 the
opposite sex.
Assume that the proportion of male to all users of a certain toilet
is given by [alpha]. Let me assume the frequency of using a toilet by
male and female is the same without loss of generality. If one gender
uses the toilet more often, this asymmetry can be reflected in [alpha].
Thus, the parameter [alpha] represents the relative aggregate frequency
of male using the toilet. (2)
I analyze an infinite horizon discrete time framework where the
toilet is used once in each period. The discount factor is given by
[delta]. With the assumption about the relative frequency of the toilet
usage by each gender, the probability that the user is male in each
period is given by [alpha]. (3) The inconvenience cost of lowering the
toilet seat for women is given by [c.sub.f]. The corresponding cost of
lifting the toilet seat for men is given by [C.sub.m]. Even though I use
the term inconvenience costs, [c.sub.f] and [C.sub.m] can encompass
other types of costs such as "unwittingly placing one's bottom
directly on the porcelain" and risk of falling in by sitting down
without looking when the seat is up or "leaving sprinkles on the
seat" when it is down, respectively.
My goal in this section is to compare the expected aggregate
inconvenience costs of three rules--down, up, and selfish--concerning
the position of the toilet seat. In this comparative analysis, I
abstract from other considerations such as being considerate to members
of the opposite sex, aesthetic aspects, the wear costs of the seat
hinge, etc.
A. The Down (Female-Friendly) Rule
This is a rule that leaves the position of the seat down after one
is done with the bathroom task. In particular, this rule implies that
each visit by a male member will be associated with the inconvenience
costs 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 value
functions with the down rule when the particular user in the current
period is male and female, respectively. Then, these value functions
satisfy 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 is
done with the bathroom task. In this case, all the inconvenience costs
are incurred by females. The case is a mirror image of the down rule and
the 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 when
the particular user is male and female, respectively. Then, these value
functions 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 following
proposition.
PROPOSITION 1. The down rule is more efficient than the up rule if
and 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 value
functions when the particular user is male and female, respectively,
under the selfish rule. Then, these value functions satisfy the
following 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 following
result. See also Figure 1.
PROPOSITION 2. If the inconvenience costs are the same across
genders ([C.sub.m] = [C.sub.f]), the selfish rule dominates both the up
and down rules.
[FIGURE 1 OMITTED]
The intuition for Proposition 2 is easy to understand. With either
up or down rule, each member of one gender group has to incur the
inconvenience costs two times with each usage. This practice can be
obviously inefficient in the event that consecutive users are from the
same gender to which the inconvenience costs are attributed. This
inefficiency can be avoided by using the selfish rule because the
inconvenience costs are incurred only when the consecutive users are
from different genders. Even in such an event, the aggregate costs would
be the same as those under the up or down rule if the inconvenience
costs are the same across genders.
I cannot rule out the optimality of, say, the down rule if the
inconvenience costs are asymmetric across genders. My analysis, however,
suggests that to justify the down rule on efficiency grounds, the
inconvenience costs for female should be very high relative to those for
male. More precisely, the condition for the down rule to dominate the
selfish 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 same
frequency ([alpha] = 1/2), the inconvenience costs for female should be
three times higher than the corresponding costs for male to justify the
down rule.
Up to now, I have considered only three potential mechanisms. These
three rules, however, are not the only rules we can entertain. For
instance, I can imagine a rule such that the position of the seat should
be restored to the prior position before use. Alternatively, I can also
consider a mutually considerate rule in which male users leave the seat
down, whereas female users leave the seat up after use. In the next
section, however, I show that all these rules are dominated by one of
the three rules I have considered. Thus, restricting my attention to the
three rules does not entail any loss of generality in the analysis.
III. CHARACTERIZATION OF THE OPTIMAL RULE: A MECHANISM DESIGN
APPROACH
In the previous section, we compared three simple rules that can be
used for the toilet seat position. The task of this section is to derive
the most efficient rule among all possible mechanisms. I show that one
of the three rules discussed in the previous section is always optimal.
Thus, restricting my attention to the three rules does not entail any
loss of generality if the only concern is to minimize the aggregate
inconvenience 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 down
after use when the position of the seat before use is i and the visitor
is j, where i = u, d and j = m, f. The first subscripts u and d denote
up and down, respectively, and the second subscripts m and f denote male
and female, respectively. The objective is to search for the best
mechanism that minimizes the aggregate inconvenience costs.
In the Appendix, I prove that the position of the seat before
one's use should not count in the optimal rule.
LEMMA. The optimal rule should depend only on the gender of the
user, not the position of the seat before one arrives.
With the help of lemma, I can restrict my search for the optimal
mechanism 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 the
probabilities that the toilet seat should be in the down position after
usage 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 present
discounted value when a particular user in the current period is male
and 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 the
following:
Let [gamma] = [c.sub.f]/[c.sub.m] be the relative cost of changing
the toilet seat position for male and female. Then, the optimal rule is
characterized 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 put
the toilet seat down after use if [C.sub.m]
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You are got a coupon.":"Glückwünsche! Du hast einen Gutschein.","Congratulations! You are got all coupons.":"Glückwünsche! Du hast alle Gutscheine.","Continue":"fortsetzen","Continue Shopping":"mit dem Einkaufen fortfahren","Copy the code and use it directly in the shopping cart.":"Kopieren Sie den Code und verwenden Sie es direkt im Warenkorb.","Country":"Land","Coupon code":"Gutscheincode","Coupon List":"Couponliste","Date":"Datum","days after receiving":"Tage nach dem Empfang","Design customization":"Design-Anpassung","Do not use any discount":"Verwenden Sie keinen Rabatt","Earliest":"Früheste","Export successful":"Erfolgreich exportieren","Export failed":"Export fehlgeschlagen","email":"Email","email format does not match":"E-Mail-Format stimmt nicht überein","Estimated Delivery Time":"Voraussichtliche Lieferzeit","Effective Order Count":"Effektive Auftragszahl","Effective Sale Amount":"Effektiver Verkaufsbetrag","Expense":"Kosten","expired":"abgelaufen","export a report?":"einen Bericht exportieren?","Failed to upload files.":"Failed to upload files.","FAQ":"FAQ","Find Parts":"Teile finden.","for 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wählen Sie Ihr Land","Please upload the invoice file":"Bitte laden Sie die Rechnungsdatei hoch","Processing":"wird bearbeitet","Product Name":"Produktname","Please fill in the delivery address before selecting the payment method":"Bitte geben Sie die Lieferadresse aus, bevor Sie die Zahlungsmethode auswählen","promotion center":"Promotion Center","Promotion Link Click Amount":"Promotion -Link Klicken Sie auf Menge","Promoted link clicks":"Beförderte Linkklicks","Promotion Order Count":"Promotion Order Count","Promotion Reports":"Werbeberichte","read more":"Weiterlesen","Received commission":"Provision erhalten","Refund":"Erstattung","Refuse":"Sich weigern","Region":"Region","Register Success":"Erfolg registrieren","Remittance":"Überweisung","Reviews":"Bewertungen","reports":"Berichte","Sale ends in":"Verkauf endet in.","Save in wishlist":"Speichern in Wunschzettel.","Search":"Suche","swift code":"SWIFT-Code","Select how to share":"Wählen Sie, wie Sie teilen können","Select premium 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Bitte wenden Sie sich an den Kundendienst, um sie zu aktivieren","the content can not be blank":"Der Inhalt kann nicht leer sein","The coupon code has been copied and used in the shopping cart.":"Der Gutscheincode wurde im Warenkorb kopiert und verwendet.","The file name cannot exceed 100 characters":"Der Dateiname darf 100 Zeichen nicht überschreiten","The file size cannot exceed 2MB":"Die Dateigröße darf 2 MB nicht überschreiten","The number of withdrawals on the day has been capped":"Die Anzahl der Abhebungen an diesem Tag wurde begrenzt","The subscription is successful, thank you for your participation":"Das Abonnement ist erfolgreich, danke für Ihre Teilnahme","The user center is out of service. Please contact customer service":"Das Benutzerzentrum ist außer Betrieb. Bitte wenden Sie sich an den Kundendienst","There is no amount to withdraw":"Es gibt keinen Betrag, um sich zurückzuziehen","There is no data to export":"Es gibt keine Daten zum Exportieren","This is Text":"Dies ist Text.","This is title":"Dies ist Titel","This transaction has failed.":"Diese Transaktion ist fehlgeschlagen.","Time to shop":"Zeit zum Einkaufen","Tips":"Tipps","To be commented":"Kommentiert werden","Total":"Gesamt","Tutorial":"Lernprogramm","This Supplier/Shipping Company does not deliver to your selected Country/Region.":"Diese Lieferant/Reederei liefert nicht in Ihr ausgewähltes Land/Ihre Region.","Update password success":"Passwort-Erfolg aktualisieren.","Upload Image":"Bild hochladen","Upload up to 6 pictures":"Laden Sie bis zu 6 Bilder hoch","uploading":"Hochladen","used":"benutzt","user center":"Benutzerzentrum","Upload Invoice":"Rechnung hochladen","valid now":"JETZT Gültig","Validity period":"Gültigkeitszeitraum","View Cart & Checkout":"Warenkorb ansehen","views":"Ansichten","Valid for":"Gültig für","Welcome to the website":"Willkommen auf der Website","Western Union":"Western Union","When your buyers received and confirmed orders, you can get commission right now!":"Wenn Ihre Käufer Bestellungen erhalten und bestätigt haben, können Sie jetzt Provision erhalten!","Withdrawal":"Rückzug","Withdrawal success":"Rückzugserfolg","Withdrawal Method":"Rückzugsmethode","Write a Review":"Eine Rezension schreiben","Withdrawal Amount":"Auszahlungsbetrag","Yes":"Ja","Yesterday":"Gestern","You are clicking too fast":"Sie klicken zu schnell","You are got a coupon.":"Du hast einen Gutschein.","You can select a maximum of 90 days":"Sie können maximal 90 Tage auswählen","You can withdraw the commission to your Paypal account.":"Sie können die Provision auf Ihr PayPal -Konto abheben.","You haven’t chosen an address yet":"Sie haben noch keine Adresse ausgewählt","You haven’t selected a product yet":"Sie haben noch kein Produkt ausgewählt","Your rating":"Deine Bewertung","Your review":"Deine Bewertung","Your shipping address error":"Ihr Versandadressenfehler"}}