The mathematics of compromise

There’s a scenario I’ve been thinking about for years, but I don’t think I’ve ever posted it. (If I have, and someone can point me to it, you’ll have my thanks and apologies). I’m curious what y’all think.

Angie and Brian are a couple getting ready to buy a house together. They work identical jobs, where they can set their own hours and receive equal pay for the hours they work. You may think of permutations, but if you do, please keep the income part equal.

The problem, when they come to house-buying, is that Angie wants a bigger house than Brian. Brian would be happy with a house they could afford by working a combined total of 80 hours a week. Angie would be happy with a house they could afford by working a combined total of 100 hours a week. There are a whole bunch of houses in between, affordable by working different amounts of time.

Note that Brian would enjoy the larger house–it’s just not particularly important to him. Note equally that Angie would enjoy working fewer hours–it’s just not particularly important to her.

The question is, how many hours should they each work, and what kind of house should they buy?

The obviously correct answer is, “whatever they mutually agree on.” That’s obviously correct, and boring and cowardly. I want you to take a position! What would you find satisfactory, if you were Angie? If you were Brian?

Four possible choices:

  1. They each work the amount of hours they’re comfortable with. Angie works 50, Bob works 40, they get a house worth 90.
  2. Angie works 60 and gets her dream house; Brian works 40 and gets his dream hours.
  3. Angie and Bob both work 50 hours and get the dream house.
  4. They each work 45 hours and get the 90-hour house.

I personally find #4 the most satisfying–each person stretches a bit past their zone of comfort (Brian works more, Angie settles for less), and they share both the sacrifice and the benefit equally–but I’m curious what y’all think.

Well there is a math field that deals with this.

It does help to actually be able to quantify the values in order to come up with a solution that isn’t based upon fuzzy feelings. I’d go with 4 or 1 personally. And if I chose 1 I’d try to find something for Bob to do in that situation to make it seem more equitable. Maybe take care of the lawn and do some extra dishes or something.

The hard part is defining value especially when the value has diminishing returns.

I think compromise, in general, ends up getting the worst of both worlds rather than the best of both worlds. So I’m generally not in favor of it.

Angie and Brian should sit down and have a heart to heart talk: Which is more important - Angie’s enjoyment of a bigger house, or Brian’s desire not to work so hard?

Once they hash that out, then the outcome should either be 80 hours and a small house, or 100 hours and a big house. But any sort of 90-hour compromise is likely to just piss both off.

And if it’s the 100-hour solution, then I think they should both work 50 apiece.

Personally, if I were Brian, I wouldn’t mind working the extra hours. And if I were Angie I’d just chill with the smaller house. I guess I’m easygoing that way.

To answer the question, you’d have to assign values to how much they value of the extra space of the bigger house, and how much cost (not a monetary cost but a measure of hardship) they associate with working the extra hours.

The question is more easily examined if it’s viewed as a choice between two houses, the 80 hour house and the 100 hour house.

If the cost of the extra hours is zero, then Angie and Brian would work 50 hours each and go for the bigger house.

If Angie values the bigger house at 3 times her earnings, while Brian values the bigger house at 1 time his earnings, then Angie should work 15 hours extra/week while Brian should work 5 hours extra/week.

If Angie and Brian represent segments of a population, rather than individuals willing to compromise with each other, you’d probably find that the concept of diminishing returns applies. Angie is willing to work an extra 10 hours and Brian is willing to work an extra 5 hours for the bigger house. As individuals, they might each choose to work an extra 2 1/2 hours, or 10/3 vs 5/3 hours, as a compromise. However, a population making that decision would probably choose to select the smaller house, rather than working above their hours threshold. (Note that my answer is based on rational economics. For behavioral economics, you need to know if the work is done before the bigger house is purchased or after.)

If there are a range of houses between the 80-hour price and the 100-hour price, the optimal house price would be still be subject to each individuals value of the house versus cost of the labour and how the value/cost curve is determined. If you can quantify those curves, then the multi-objective optimization math cited by Octopus will provide you with an answer.

I tend to agree with this. If the 100-hour house is Angie’s dream, options 1 and 4 don’t fulfil that dream, and in option 4 Brian is working extra hours for little mutual benefit. In option 1, they get to spend a significantly smaller portion of their waking hours together and Angie still doesn’t have her dream house.

Option 2 does give Angie her dream house, but I think the huge difference in working hours is likely to lead to mutual resentment sooner or later. Plus, your OP states that Angie is happy with the house they could afford by working a combined 100 hour week - this does not necessarily imply she would personally be happy working a 60 hour week.

Option 3 gives Angie her dream house, but has Brian working significantly more hours than he would like.

I’m definitely Brian in this scenario, I would sit down with Angie and try to reason that working more hours for a nicer house is illogical, as it reduces the amount of time you have to enjoy living in it. Why isn’t there an option 5 - they both work 40 hours and get the 80 hour house? I suppose because that is then 100% Brian’s preference and Angie gets ‘nothing’. But that would be my advice to them, I think.

Another option would be: they both work 45 hours and get the 80 hour house. Then in a few years they might have saved enough for a big enough downpayment on the 100 hour house that they can buy it and still work 45 (or even 40) hours each. If recent history has taught us anything, it’s that it’s wise not to overextend oneself in the housing market if you can easily avoid doing so.

Bottom line - if you really want me to stick to one of the original 4 options, I agree with you that option 4 is probably best.

I mean, I suppose option 4 makes sense, if we presume the utility to B of working 40 hours is equal to the utility to A of having a bigger house. And vice versa on the lower end of the flip side of the spectrum.

Because that’s really what you want. The trouble is, you haven’t really explained a guiding principle to us. Do you want to maximize the satisfaction (utility) of A and B together, or do you want an equitable relationship whereby each gets the same overall utility from the compromise and its outcome?

Because if it’s just maximization, then technically (assuming the 40 hour point is exactly as satisfactory to B as the 100-hour home is to A, and vice versa, with a linear relationship throughout) then maximization would be achieved through a wide range of options.

But… assuming A and B are normal human beings, and that ingrained/foundational (to the home or job) unfairness would have a deleterious effect on their relationship, I think it’s reasonable to place a premium, or at least “first among equals” status and go with option 4 as well.

B’s Utility at 40 = -40 from working 40 hours, 80 for living in a home size 80 (40 total)
B’s Utility at 50 = -60 from working 50 hours, 90 for living in a home size 100 (30 total)
B’s Utility at 45 = -50 from working 45 hours, 85 for living in home size 90 (35 total)

A’s Utility at 40 = -30 from working 40 hours, 60 for living in a home size 80 (30 total)
A’s Utility at 50 = -40 from working 50 hours, 80 for living in a home size 100 (40 total)
A’s Utility at 45 = -35 from working 45 hours, 70 for living in a home size 90 (35 total)

Notably, the way I have set the numbers up, the overall utility for A and B combined is exactly the same (70) whatever the outcome. So the ultimate solution, 45 hours a week with a home size 90, presumes the parties will place a premium on fairness. But there are other ways to mix the numbers, such as if A and B each don’t like working 40 hours a week exactly the same, it’s just that each additional hour of work beyond 40 matters less to A than to B. That would tend to skew things towards a solution where it really would make sense for A to work more hours than B to get the bigger house, for both of them. And then what do you do? Will their relationship survive? Will B find some sort of in-kind payback he can perform for A, like always making dinner? Will A consider that an equitable tradeoff?

I think it’s a mistake to think this sort of question can be answered by math. What is it that keeps the 80 hour a week house from being Angie’s dream house - is it that it has four bedrooms instead of five and Angie’s dream house has five, but she’s fine with four? Or is it that the dream house has a larger kitchen and Angie will regret a cramped kitchen every minute she lives in the 80 hour a week house? Would Angie enjoy the large kitchen more than Brian dislikes working more hours ? Does Brian dislike working the extra 10 hours more than he would dislike Angie complaining every holiday about how they have to run dining tables into the living room, which doesn’t leave enough space for people to easily walk around the table ?

I’m going to come back to this later after more coffee but I think an important point is that Angie and Brian should split ownership according to the proportion they are paying as well. While you may say, “Duh. Of course.” It should be agreed to before hand and contracted.

IANAL but that may also play into joint tenancy with survivorship. If my mom Angie has 60% and ahole Brian (you’re not my real dad! you’re not even married to her!) has 40% and mom dies. He should now get 80% and I should get 20%?

If ANY of that is an issue then it should be equal proportions. Let me get more coffee.

I think that depends on a lot - if Angie and Brian are married, in my state they can be “tenants by the entirety” which is only open to married couples and has legal advantages ( for example, a creditor cannot force a sale unless both husband and wife owe the debt) which might be worth it to Angie and Brian even if they contributed different amounts of money.

Too late to edit.
Why does each want to buy a house? Investment? Not worry about landlords not renewing leases? Have a dog so no one will rent to them? Angie and Brian may see different benefits in home ownership so if Angie sees it as an investment and Brian is simply tired of landlords, then that would favor a split that maximizes Angie’s ownership amount and minimizes Brian’s. If they both just want a backyard for Ratty the Rat Terrier to run around in, then given what I said above, an even split may be the best.

Which after reading Investopedia would lead to an equal proportion solution because

  • Each spouse has a legal right to an equal portion of the property provided they were married at the time title was received in both their names.

That was worded poorly - further down it says

The condition of mutual ownership of the entire property means the spouses must be in agreement when making decisions about the property. For example, one spouse doesn’t have the legal right to sell off or develop part of the property without the other’s consent


There is no subdivision that separates the property into equal parts between the spouses. So, even if one spouse writes a will that grants an interest stake in the property to an heir, the power and rights of tenancy by the entirety creates a right of survivorship and invalidates and supersedes that aspect of the will.

The only way it would get divided is in the event of a divorce - if your mother Angie contributed 60% and was married to Brian who contributed 40% and they own the house as tenants by the entireties, Angie can’t leave you any stake in the house. Brian automatically gets it, no matter what she puts in her will.

I hope I’m not fighting the hypothetical or threadshitting by avoiding talking about mathematics or compromise. But it seems to me that, if they can afford the house Angie wants by working 100 hours but not 90 hours, that suggests that Angie’s house is at the upper limit of what they could afford even by working those 100 hours. And that means they don’t have much margin in their budget: it doesn’t leave much money to spend on other things, or for a cushion in case their expenses go up or their income goes down significantly. And that’s a situation that I, personally, would not like to be in.

Maybe - or it could mean that they could actually barely afford the house working 90 hours a week and the 100 hours is to provide a buffer.

I think that these side conversations are part of the equation. Brian wants to put more in his Roth IRA. Angie is wanting to travel. Kids in the future? All of this enters into the compromise.

This is my problem with the question. The OP expresses their opinon thusly:

“Brian works more, Angie settles for less” sounds like a fair compromise, but I’d argue that being at work for longer hours every week, in order to support a house you don’t actually want, or need, is a far greater cost than “settling” for a slightly smaller house. The sacrifices in this “compromise” are not equal.

Even if Angie were to argue that walking into a smaller house every day has an emotional impact on her equivalent to Brian being stuck at work extra hours every week, that would just convince me that Brian should dump her.

It’s interesting that folks are so into the issue of mortgages and property ownership. I see all of those as fundamentally beside the point: I assume that both Angie and Brian have already thought about these issues and factored them into their desires about house size and work hours. And it’s interesting to me that folks are taking Angie’s side or Brian’s side: I tried to make the hypothetical one in which both folks have legitimate, defensible views.

What got me thinking about this, years ago, is when I moved in with my wife. I’m naturally a person that tolerates a fair level of clutter but kind of hates cleaning. My wife is naturally a person who tolerates a fair level of cleaning but kind of hates clutter. I’m not living in squalor that breeds disease or anything, and she’s not a neat freak who covers the furniture in plastic or anything; from my perspective, we’re both in a normal, healthy range, and it’s just a difference of preferences.

So our choices were something like:

  1. I clean the amount that I’m comfortable with, and she cleans until she’s comfortable. In this scenario, I get the extra-clean house without doing any extra work; she gets the house she wants but does more work than she’d do if she lived alone.
  2. We each clean the amount we’d clean if we were living alone. I get my preferred hours and a small bonus from an extra-clean house; she works her preferred hours and gets a house less clean than she’d like.
  3. We each clean until she’s got the level of clean house that she wants. In this scenario, I do a lot of extra cleaning without much perceived benefit to me; she cleans her preferred amount and gets her preferred level of cleanliness.
  4. I clean a little more than I’d like, and she cleans a little less than she’d like. In this scenario, we each lose something of our preference. I get the small benefit of an extra-clean house (from my perspective), and she gets the small benefit of extra relaxation (from her perspective).

We’ve settled on 4, and I’ve become more tolerant of doing chores that I don’t always see as necessary, like sweeping floors several times a day, and she’s become more tolerant of clutter, like loose papers on the coffee table. It’s not ideal for either of us, but it’s where we’ve come out, and we’re both reasonably okay with it.

The OP scenario is an attempt to analogize the situation. I’m curious if others have encountered similar situations, and how you’ve resolved them.

Heh, your scenario with cleaning is virtually identical to how I would describe my wife and I. At the moment I would say we are firmly on option 4. I think the fact we have 2 young kids (8 and 3) has helped her accept that reality, in that (as far as we are concerned) it is literally impossible to attain a high level of cleanliness/tidiness without sacrificing something we both see as more important (spending time enjoying life). In all honesty I probably should/could do more, but I do sometimes make an effort to complete a chore because I feel it will make her happier, as opposed to because I think it needs doing right away (e.g. doing the washing up before going to bed rather than leaving it until the morning). There is also the fact that I work 40 hours a week and she (typically) works 15 hours a week, and therefore even with childcare considered, has more time to do chores than I do - which she understands and accepts.

My advice (for myself and you) in this situation is to collaborate on dividing the chores between you. We are lucky insofar as our standards in terms of satisfactory completion of a chore are similar, in that if I clean the bathroom (say), she doesn’t feel the need to redo it to a different standard. As long as everything on the chore list gets done, it doesn’t matter who does it from that perspective. So if there are 10 chores and I do 3 and she does 7, that can work. It’s an amount of labour and standard of tidiness that we can both cope with. If chores 11 and 12 (say, clearing cobwebs from the corners of the room, cleaning the oven) only get done every few months, she is OK with that.

To be honest, I don’t think your OP is a particularly close analogy to this because of the large financial implications inherent in property purchase, that don’t really apply to division of chores. The positions of Angie and Brian are both defensible, but the financial side is a confounding factor. Very interesting thread, nevertheless.

Similar but not quite the same and not resolved. I don’t mind loose papers on the coffee table and I don’t expect floors to be swept several times a day. I do, however, want a dining room table that functions as dining room table without having to spend ten minutes clearing it off a and a living room that doesn’t have perpetual baskets of laundry in it. I’d also like to have some flat surfaces not cluttered with things that belong elsewhere ( such as a cabinet in my dining room that has various tools on top of it) so I could display photos and decorative items. I would like the kitchen floor to be mopped more than several times a year. I have learned that I can only get what I am willing to do myself - and not even some of that, because if I put the tools somewhere else it will be the wrong place and if I get rid of a can of paint that we used ten years ago, there will be howls. So I live with a house that I don’t let people into without pre-planning which then takes hours of straightening up and cleaning to the point where it would actually have been less effort not to let it get so cluttered to begin with.

So I don’t even get option 1. He’s gonna get a surprise soon, because I am hiring a junk removal place to take out a home gym that I’ve been wanting to get rid of for ten years ( the kid who actually used it moved out) and I’m going to have them get rid of a bunch of other junk , too.

Yeah, cleaning the house is a different dynamic, because that does have direct, day-to-day impact on each person, that is of comparable seriousness. So long as one person isn’t really into one extreme or the other, you can find some balance of tolerance for each person.