Wednesday, 24 July 2019

Toilet Seat Economics: Lifting The Lid On The Optimal Marital Policy

Every married couple knows that the most important question of all isn't 'Why is there something rather than nothing?", or "What is reality?" or "Will we ever fully understand consciousness?" - it is this epic question:

"Should men put the toilet seat down after they'd had a wee?"

I jotted down some thoughts, and my friend Tim Reeves commented. For anyone interested, here is our discussion:

Toilet Seat Economics
JK: You know the scene; Jack and Jill get married, and they have that discussion about what the toilet seat policy should be in the marital home. Jill usually says "Please leave it down after you use it", and Jack can respond in different ways. He can refuse in a carefree way (that's not going to help the relationship), he can acquiesce in a resentful way (that's not the best response either), he can refuse in a logical way (more on that in a moment) or he can acquiesce in a cooperative way because he loves her. One of the latter two ways is best - and I'm going to explore which one it is.

This kind of scenario is often thought of in terms of game theory strategy - but husbands who think this way are making an amateur mistake. In game theory, a state called the Nash equilibrium occurs when no agent in a situation can improve their strategy by unilaterally changing their actions. The optimal outcome of a situation is where there is no incentive to deviate from the initial strategy once he or she knows their opponent's decision. In the well known prisoner's dilemma, for example, the Nash equilibrium is not for both of you to remain silent. The Nash equilibrium in the prisoner's dilemma is for both you and Jill to betray each other, because even though mutually staying silent leads to a better outcome, if one chooses betrayal and the other stays silent, one of you has a worse outcome. In other words, the rational decision is to betray, because betraying can't make your situation worse, but it might make it better.

This also shows that rational decisions don't always lead to good outcomes. Imagine you're Jack. If Jill betrays you and you remain silent, you get three years in prison by not betraying Jill, and two years in prison by betraying her. Therefore it's better for you if you betray Jill. If Jill stays silent and you betray Jill you walk free, and if you also remain silent you receive one year in prison. Therefore it's still better if you betray Jill, and the Nash equilibrium occurs if you both betray each other. However, a husband and wife with a healthy relationship are a cooperative unit, not in competition, so Nash equilibria does not apply here, because it only applies when two agents are in competition.

TR: Agreed. What is “rational” here really depends on whether or not we are dealing with a single organism or two competing organisms. For organically joined players the cooperative strategy is the rational one, but for two separate competing organisms betrayal is the rational choice.

JK: And thankfully, married life need not be like a prisoner's dilemma - it should be an opportunity to bless each other, although sometimes the mathematics is fun too. To see why the toilet seat issue is an even bigger chance to bless each other than is often realised, let's assume for a second that the husband and wife are treating the toilet seat issue in a competitive prisoner's dilemma-style scenario. In competition, where the strategy is to touch the toilet seat as infrequently as possible, you'll find that acquiescing to the lady's request to always leave the toilet seat down is inefficient. Leaving the toilet seat the same as after you've used it is the most efficient system.

Here's why. If you leave it always down then any time the man goes consecutively he is inconvenienced by having to move it up again, even though he used it last. Similarly if you leave it always up, any time the woman goes consecutively she is inconvenienced by having to move it down again. Leaving it always down is inefficient to males; leaving it always up is inefficient to females; and leaving as you used it the most efficient system.

But we are only looking at this situation from the perspective of minimising the joint total cost, where cost is touching the toilet seat unnecessarily. The mathematics is fairly easy to enumerate on this: assuming for simplicity that both Jack and Jill use the toilet exactly the same number of times, then if both Jack and Jill leave the seat up every time they use it, both will touch it the most, and Jill will touch it about six times more than Jack, making it the least efficient of all. If the decision was decided by a coin toss, then both Jack and Jill should touch it roughly equal number of times, but they will jointly touch it the second most occurrences.

If Jack leaves it the opposite of how he found it, he will touch it about 75% of the time, and Jill 25% of the time; If Jill leaves the opposite of how she found it, she will touch it about 75% of the time, and Jack 25% of the time - and in both cases they will jointly touch it less than the two cases above. The last option though - leaving it as you last used it - is the most efficient way to minimise the joint total cost. Because Jill only needs it down, and Jack needs it both up and down, he will touch it a little more often than Jill will, but they will both jointly touch it the least often.

However, all this only factors in the scenario that Jack and Jill go equally and alternatively. If we allow for the fact that for all sorts of reasons, like different liquid consumption, and different times they are in the house, there will be many times when Jack and Jill will go consecutively, and that's even stronger argument for leaving it as you last used it. Take an extreme case: suppose Jack drinks lots of water and goes to urinate 5 times for every 1 time that Jill goes, then a policy in which he put the seat down every time for Jill means he is putting it down and then back up again 4 times, with him being the only one who has used it during that time.

Conclusion: leaving it as you last used it is the best policy for Jack and Jill.

Now, here's the rub: it's not quite that simple, because love is about more than logical reasoning - it is about putting each other first. A couple whose strategy was simply to minimise the joint total costs would quite naturally adopt the policy of leaving the toilet seat as you last used it.

TR: Conventional water closets get visited by two kinds of people. Namely “Down Persons” (=D) and “Up Persons”. You notice that I haven’t distinguished between male and female. When a male goes to the toilet he can be in either an “up” state or a “down” state; so as far as toilet usage is concerned he’s two different people in the up state and down states. Because we are thinking of the overall organic economy of the household we need not make the distinction between male and female just yet; in fact it’s a distraction from the core mathematics. So if we assume that Up and Down persons visit the cubicle in some pattern of visits we can represent that pattern as something like this:


The troublesome configurations are the pairs UD and DU as they entail the expenditure of energy and time by the household in order to change the state of the toilet seat. Interesting to note that it doesn’t matter who puts the seat up or down as the expenditure in energy & time is the same.  A D-person could put the seat up in advance just prior to the visit of a U-person. Or alternatively a U person visiting the cubicle could put the seat up after a D person has used it. (It’s contrariwise for the configuration UD, of course) The point is that in terms of time and energy it doesn’t make any difference whether U or D makes the necessary changes to the state of the toilet seat.

But there is a big “but” with what I have just said: It assumes that the pattern of visits is predictable to the extent that a householder knows whether the next visitor is an Up or a Down. This assumption, of course breaks down if the pattern of visits is random, in which case a cubicle user will not know whether the next user is U or D.

So what is the most household efficient strategy if the pattern is random? In this case the householder doesn’t know whether the next visitor is a U or a D. Hence if a householder changes the state of the seat after using it in preparation for the next visitor this may well be wasted time and energy, because the next visitor who comes in at random may have to wastefully change its state back to what it was. Because a user cannot anticipate whether the next user is a D or a U that user is not the best person to change the state of the seat because that state may have to be undone. The best person is of course the next person who visits the cubicle because they obviously know their requirements best and therefore no tricky precognition is needed by the previous visitor!

So to summarise, I agree that your strategy, which involves the last visitor leaving the toilet seat in the state he/she used it, is the most efficient scenario timewise and energetically.

JK: Moving on from mere economic efficiency to blissful marriage efficiency, a good strategy would be to supersede the good strategy with a better strategy whereby the husband insists on putting the toilet seat down for his wife. One objection might be that if the basis of a successful marriage is for each to put the other one first, then letting Jack put the toilet seat down every time is to make him complicit in an erroneous cost-benefit scenario, and that is not really an example of Jill putting his needs first.

TR: I’d go along with the general sentiment here that if “putting the other first” leads to wasteful inefficient scenarios it’s not actually that “loving”. After all, the overall household economy affects both players and if wasteful practices are adopted all parties can be adversely affected in the long term. So we are looking for efficient cooperative strategies. However there are other factors at work. Toilet seat changes involve very little energy and time and so the extra energy and time used by, say, U putting the seat down for D may perhaps be worthwhile for relationship purposes and help cement ties which could be of overwhelming concern in an organic whole that depends on  a cooperative household economy.

JK: The way to optimise this situation in accordance with mutual regard for the other, then, is to have a discussion with your beloved and agree on what is most important to them about this scenario, and who cares the most (because it will vary couple to couple). Because economics is primarily a study of human behaviour, and human behaviour is conditioned by our feelings, then your optimal system in the marital home must factor in levels of importance for each beloved. For example, in a typical marriage I'd predict that a lady cares a lot more about not having to touch the toilet seat than a man does about having to lift it up, so the most maritally efficient, and therefore also the most economically efficient in this case, is for Jack to show love and put it down after use, for Jill to love him gratefully in return for doing so, and for her to forgive him if he forgets from time to time, because she knows his intentions are the best for her. 

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