This is a guest post by Bethany Soule with assistance from Daniel Reeves.

Prescript: We realize how crazy this all sounds! Nonetheless, we’re perfectly serious, and do actually pay each other to put our kids to bed and whatnot. We think we have quite clever mechanisms for deciding who does what and keeping things exquisitely fair, efficient, and resentment-free. And separate finances — i.e., financial autonomy — is a prerequisite. We tried it with our own fiat currency — yootles — and that didn’t work so now we just use dollars. [1] Some of this bears uncomfortable resemblance to the kinds of things two raving Libertarian nutjobs [2] would say. (It’s not like that, baby, we swear it.) Nonetheless, this being the single most demanded article by you, Dear Readers, we are finally caving to the pressure.

Merging all finances upon getting married is still the de facto standard in the modern world. Each contributes according to their ability and takes according to their need. Which is fine. It’s not fundamentally unreasonable for a relationship. Marriage is about becoming a team, becoming life-partners, and, yes, sharing your resources. But we also feel particularly strongly about maintaining autonomy even in a life-long committed relationship. Psychologists will tell you that perceived level of autonomy is a prime predictor of happiness with your life and situation. [3] Partnership and resource sharing may seem at odds with autonomy but I’m here to argue that they need not be!

Danny and I have been married for six years now, and we have two kids and a burgeoning startup that we co-founded. We work literally side by side every day. We fancy ourselves a pretty good team. Being financially autonomous [4] helps us do all that. I should emphasize that none of this has much to do with the fact that we’re business partners in addition to being life partners. We were financially autonomous for years before founding Beeminder — in fact, we always have been, since we never merged finances in the first place. I mention it only as evidence of our commitment and ass-kicking teamwitchery. [5]

Philosophical assumptions

Before we get too far into extolling the benefits of separate finances, let us explain how we operate on some basic shared philosophical assumptions.

Egalitarianism

Everyone’s happiness is fundamentally equally important.

Autonomy

People are free to make their own choices, their own mistakes, and have their own utility functions.

Fairness

Everyone in a team, contributing equally, should share equally in all the rewards of the teamwork.

I get a little giddy when I read that list. Seriously, we named our daughter Faire because we like fairness so much. It’s my litany against being mean. When I start to get het up over some perceived injustice, it almost always goes away when I stop to examine what underlying assumption I’m violating. “I can’t believe you…” oh wait. They’re an autonomous agent. Their happiness is just as important as mine. Stop being such a judgmental Bee!

Furthermore it may be the case that separate finances are a natural consequence of these three principles. If we pooled finances, I would benefit from Danny’s hard work, even when I don’t do anything. That certainly violates fairness. If I use joint resources to buy a banjo, Danny’s resources got allocated in a way that he didn’t choose, violating autonomy, and leaving room (and hours of my practicing) for resentment to build.

Benefits of separating finances

First, we find there is much less stuff to fight about. We’ve eliminated a wide array of guilt trips and I-always-and-you-never!-s from even being possible. There’s no arguing over how you spent our money. And if you are complaining about how your spouse is spending their own money, assuming it is not dangerous or self-destructive, excuse me but you’re being kind of a jerk. You’re probably either questioning the correctness of their utility function, or you’re just devaluing their happiness.

Aside from making it much harder to fight about money, thinking of your partner as an autonomous agent is a great innoculation against any sort of argument that starts with “but you should have…” You stop making assumptions about their utility, their thoughts, their emotions, and you do a lot more active communicating and wind up frustrated and disappointed much less.

Second, with separate money we’ve got a whole new class of conflict resolution. We still debate, negotiate, argue, and occasionally even plead (OK rarely; if you’ve hit pleading you should probably move on to a different resolution mechanism). But any decision has a fallback resolution method as definitive as voting would be if there were an odd number of us. That’s right, we use auction-based decision-making in our family. Where’s my nerdcrown?

Third, since my money is my own money, if I pay Danny it is actually meaningful. Transferable utility is pretty exciting! It means that any outcome can be made fair after the fact. You know, by paying each other. Without financial autonomy, conflict resolution often winds up at compromise (nobody is happy), or sacrifice (one person gets all the happy). By paying each other we’ve opened up a whole middle ground of less bad outcomes! For example, let one person have their way, but then transfer half their surplus utility to the person who lost out.

“All preferences are quantifiable in terms of money. If you think like an economist (and do we ever), that is obvious to you.”

Of course, these second two benefits rely on the assumption that all preferences are quantifiable in terms of money. If you think like an economist (and do we ever), that is obvious to you. But if you have never practiced quantifying your preferences before, it might seem impossible to put a price on, e.g., making sure your kid gets registered for school. It is true that coming up with valuations for nebulous things is hard and can sometimes seem arbitrary. However you can improve with practice, and we even have an exercise [6] utilizing binary search to help you home in on a reasonable value. Sometimes trying to quantify your preferences is little better than pure randomization, but it is still slightly better than actually flipping a coin. Plus, afterwards you make a payment to help the loser feel better. The more important the decision, the better idea you have how much you care, and the less like a coin flip it is. The closer your utilities are or the more arbitrary your assignment of utility, the more like a coin flip it is, but the less egregious that is.

Mechanisms for decision making and resolution of conflicts

The most general form of decision auction that we use works like this:

There are n participants, each with some share — i.e., some fraction — of a decision. Everyone submits a sealed bid, the second highest of which is taken to be the Fair Market Price (FMP). The high bidder wins, and buys out everyone else’s shares, i.e., pays them the appropriate fraction of the FMP.

We’re usually in the special case of two players and 50/50 shares and we call that case “even yootling” or just “yootling”, in reference to our ill-fated currency, the yootle. It works like this: Each person simultaneously states how much they’d pay. [7] The person with the higher number wins and pays the loser the loser’s bid.

We may do this multiple times per day, whenever there’s a good that we have shared ownership of and one of us wants to offload their shares onto the other person. The goods can be anything, e.g. the last brownie, but they’re more often “bads” like who will get up in the middle of the night with a vomiting child, or who will book plane tickets for a trip.

As for simultaneously stating bids, we do that by each sticking our hands behind our back and holding up a configuration of fingers to represent / commit us to our number and then we reveal on a count of three. We’re very loose with how we represent numbers with fingers. We might hold up two fingers to represent $20. Or a 1 on one hand and a 3 on the other to represent $13. It wouldn’t really be hard to cheat and make up a new advantageous interpretation of one’s finger configuration upon learning the other’s bid, but we trust each other not to be cheats and jerks. That’s true love, baby. [8]

We find this an elegant means of assigning loathed tasks. The person who minded least winds up doing the chore, but gets compensated for it at a price that by their own estimation was fair.

Bilateral trade and PandA

We almost always use 50/50 yootling to make decisions and allocate shared resources but once in a while we’ll use other kinds of auctions. One variant is the special case of the general decision auction where the shares are 100/0. In other words, plain old bilateral trade: you have something, I want it. If we can agree on a price, the good changes hands, if we can’t, no trade occurs. It’s an insanely useful transaction, and it’s not possible if you share your finances.

To see how it’s a special case of the general decision auction, take the case of me wanting to buy Danny’s old monitor. I say how much I’d pay, Danny says how much he’d sell for, and if we overlap then I buy it at the price he named (the lower price, which, recall, is taken to be the Fair Market Value). If we don’t overlap — if my willingness to buy was lower — then Danny buys my 0 shares for 0% of the Fair Market Price. I.e., nothing happens and he keeps his monitor.

Once in a while the trick with simultaneously revealing our fingers is too cumbersome. In that case we’ll use a mechanism we call PandA — for “Propose and Accept”. In the case of 50/50 shares, it works like this: One person names a price, and the other must choose to either pay the price and get the good (or get out of the chore), or accept the price and relinquish their share of the good (or do the chore). We don’t use this mechanism often, and we’ll leave it as an exercise for the reader whether you should prefer to be the proposer or the accepter. Seriously, this is homework. Get out your mechanical pencils and figure that shit out and tell us in the comments. Points will be awarded.

Besides the easier coordination, an advantage of PandA is that only one person has to pull a price out of thin air. Our squishy human brains are much better at saying yes or no to a given price than deciding how much something is worth, with nothing to anchor on. It can be agonizing to decide how much you care about running an errand, or doing some dishes or whatever. Of course we’ve gotten pretty good at that. [6]

Joint purchase auction

The decision auction and variants are about allocating shared or partially shared resources to one person or the other, or picking one person to do something. Once in a while you have the opposite problem: deciding on a joint purchase.

Suppose Danny thinks we need a new sofa (this is very hypothetical). I think the one we have is just fine thank you. After some discussion I concede that it would be nice to have a sofa that was less doggy. Danny, being terribly excited about getting a new sofa does a bunch of research and finds his ideal sofa. I think it is a bit overpriced considering it is going to be a piece of gymnastics equipment for the kids for the next 6 years. Conflict ensues! I could bluff that I’m not interested in a new sofa at all and that he can buy it himself if he wants it that badly. But he probably doesn’t want it that bad, and I do want it a little. If only we could buy the sofa conditional on our combined utility for it exceeding the cost, and pay in proportion to our utilities to boot. Well, thanks to separate finances and the magic of mechanism design, we can! We submit sealed bids for the sofa and buy it if the sum of our bids is enough. (And, importantly, commit to not buying it for at least a year otherwise.) Any surplus is redistributed in proportion to our bids. For example, if Danny bid $80 and I bid $40 to buy a hundred dollar sofa, then we’d buy it, with Danny chipping in twice as much as me, namely $67 to my $33.

Generosity without sacrificing social efficiency

“The payments are simply what keep us honest in assessing that.”

If you’re thinking “how mercenary all this is!” then, well, I’m unclear how you made it this far into this post. But it’s not nearly as cold as it may sound. We do nice things for each other all the time, and frequently use yootling to make sure it’s socially efficient to do so. Suppose I invite Danny to a sing-along showing of Once More With Feeling (this may or may not be hypothetical) and Danny doesn’t exactly want to go but can see that I have value for his company. He might (quite non-hypothetically) say “I’ll half-accompany you!” by which he means that he’ll yootle me for whether he goes or not. In other words, he magnanimously decides to treat his joining me as a 50/50 joint decision. If I have greater value for him coming than he has for not coming, then I’ll pay him to come. But if it’s the other way around, he will pay me to let him off the hook. We don’t actually care much about the payments, though those are necessary for the auction to work. We care about making sure that he comes to the Buffy sing-along if and only if my value for his company exceeds his value for staying home. The payments are simply what keep us honest in assessing that. The increased fairness — the winner sharing their utility with the loser — is icing.

Nitty gritty

That was all extremely high level. You probably want the real dirt. Like do you pay each other for sex? Or are there any legal teeth to our arrangement? No, it’s purely a gentlemen’s agreement. And in fact for whole classes of expenses, including everyday things like groceries, we do very little accounting. We used to split groceries and dinner bills 60/40, reasoning that Danny ate (or should be eating) rather more than me, being the larger one. Actually, rather than splitting the bill 60/40, when grocery shopping or eating out we’d randomize such that Danny would pay the whole bill with 60% probability and I’d pay with 40%. We’ve gotten even lazier and now we just wing it and figure it balances out over time. Or we pay each other stochastically — with 90% probability we pay nothing (and don’t even have to compute the actual transfer amount), and with 10% probability we pay ten times the amount. We wrote an Android app — Expectorant — to make that sort of thing easier. We impressed Andy Brett recently on a trip to Tahoe when I repaid Danny $100 stochastically (i.e., taking a 10% risk of owing him $1000).

For larger recurring expenses we put a repeating transfer on a ledger and forget about it. Rent, for example, we split. You could fix different rent percentages based on salary, and I believe this is what we did in NYC when I was in grad school and Danny was gainfully employed. Alternately, a joint-purchase auction about our housing options should make the socially efficient outcome happen, not least of all by encouraging us to look at our finances and make an informed decision about our utility functions with respect to housing.

Children are a joint expense, obviously. [9] If one partner is a stay-at-home-parent you pay the full-time parent something, possibly half the salary of the parent who works, if you don’t have a fairer principle to apply to pay for the parenting. A more fair principle might be to split the difference between a fair market wage for full time childcare with what they could be making if they were out working instead of staying at home to raise your (their) kids.

The most important thing of course is to come up with a principle that feels fair to both parties. Danny and I have a fun way to debate something like this. We switch sides and each argue for the other’s best interest. The spirit of our agreement is quite loving. We’re a team. We’re actually quite nice to each other. We just recognize that no matter how well we know each other, we don’t know the other’s utility function better than they do, and we shouldn’t presume to. When you make assumptions about someone else’s preferences, you fall into the trap of assuming they’re an extension of yourself, or a piece of the scenery. So let them speak for themselves. Feed your preferences into a decision mechanism, turn the crank, and trust that the optimal outcome comes out. Live the robot-mediated life!

Related Reading

• The Gender of Money
• Housewife Charged In Sex-for-Security Scam
• Five-part series on marital finances in Slate
• Yootopia!

Footnotes

Two years ago, when I still had a day job and Beeminder was just a side project, I wrote about my epic struggle with email:

My email is dysfunctional. I keep things in my inbox because I can’t afford for them to go out of sight, out of mind — but then that’s exactly what happens. They get buried deeper and deeper in my inbox by all the other messages I delusionally think I’m going to deal with.

That was the preface of my article, Inbox Zeroer, in which I proposed that we all declare email bankruptcy and stop adding things to each other’s task lists willy nilly. No, but seriously, I laid out a wish list for some tools to help me reach Inbox Zero, namely, email snooze, re-ping, and auto-expire. Today I’m happy to announce that I’ve implemented those features for Gmail, and you don’t even have to be a nerd to use them. It’s a one-click install, except for saying yes to all the scary sounding things it wants to do: [1]

The best part is that it’s completely unobtrusive. It doesn’t add or change anything in the Gmail interface, working purely off of standard Gmail labels.

How Gmail Snooze Works

Snooze

A thread with a single number as a label, say “7”, will reappear in the inbox in 7 days. To snooze for 7 days (assuming you have keyboard shortcuts turned on in your Gmail settings) just type: v7<enter>. That moVes the thread from the inbox to the label “7”.

Every night at midnight, everything will shift: threads labeled i will be moved to i-1, with new labels being created as necessary. And anything with label “1” will reappear in the inbox, with no more numeric label. [2]

Re-Ping

Re-Ping just refers to snoozing your own outgoing mail so it comes back to you and reminds you to follow up (or re-ping the person) if you haven’t gotten a response.

I originally conceived of this as an additional part of the interface of composing a message, since that’s when you want to decide when a follow-up would be needed. Instead, I implement this by having all of my own sent-mail get filtered so it lands right back in my inbox when I send it. That sounds annoying but it’s just two more keystrokes to get a message back out (“xy” — select and archive) if you won’t need to re-ping. If you will, snooze it like any other message. The idea is that all active conversations should initially be in your inbox until you explicitly allow them to go out of sight, out of mind.

Here’s how to create a filter to make your own sent mail go to your inbox:

1. Create a filter (available from the “show search options” dropdown on the search box)
2. Set the From to your own email address, or your username — something that will match all the email addresses you send from
3. On the next screen, check nothing but “Never send it to Spam”

I find this far superior to what I consider the next best alternative, FollowUp.cc, because there’s no external service to rely on, and no clutter, either in Gmail’s interface or in your email threads.

Auto-Expire

Auto-Expire is like the inverse of Snooze. A thread in your inbox with a label like “x90” will automatically archive itself in 90 days. To apply such a label (again, assuming keyboard shortcuts are enabled in your Gmail settings), type lx90<enter>. That applies the Label “x90”.

Similarly to Snooze, the number after the x will decrement nightly, until the number disappears and the label is just “x” (indicating auto-expired), when it will disappear from your inbox.

I don’t yet have much experience with Auto-Expire but the theory is sound: When a message first has your attention is the best time to make decisions about it. If it’s an email about an upcoming event, set it to leave your inbox sometime after the event will happen. It will then disappear from your inbox after it’s no longer relevant, with zero further thought from you. If you establish a habit of setting an auto-expiration for all incoming mail it would reduce the painful slogs through your inbox, clearing out no-longer-relevant email.

You can even use snooze and auto-expire together, for example, by setting a message to come back to your inbox a week, and then automatically leave again in a month: lx30<enter>v7<enter>.

“The fundamental problem that all this tries to solve doesn’t have anything to do with email processing tools, it’s my damn brain.”

I’ve been using some version of Email Snooze and Re-Ping for years now, in various hacky forms, and I think they’re really powerful tools. But guess what I found out? The fundamental problem that all this tries to solve doesn’t have anything to do with email processing tools, it’s my damn brain. Having an amazing IDE with built-in refactoring tools and a Turing-complete macro language won’t just make you produce code. (To rephrase that for the non-nerds, an amazing word processor won’t make you produce prose.) And as great as email snooze and company are, they don’t solve your fundamental procrastination problem.

Which sounds like a case for — Beeminder! Specifically, Beeminder’s Gmail Zero Project.

How Gmail Zero Works

Gmail Zero is almost as easy to get started with as Gmail Snooze. There’s the usual scary sounding permissions [3] as you connect your Gmail and Beeminder accounts, but then it’s just a matter of picking a target inbox size. Beeminder makes a graphical yellow brick road for you to follow to gradually get there. It periodically checks your inbox throughout the day and records the number of Read messages, plotting the counts along that road. It only counts the minimum of the day, so that you just have to hit the low point and stay there long enough for Beeminder to pick up on it. This lets you (and — once you add a pledge — forces you to) make slow but inexorable progress toward inbox zero.

Gmail Zero only works for people who (try to) diligently archive messages. It doesn’t count Unread messages against you since you don’t have control over those. For example, the 1000 messages you get on vacation? Not a problem — it won’t count them until you get home and start reading through them.

Ignoring unread messages is a double-edged sword, of course. One person using Gmail Zero reports that he finds himself leaving things as unread in order to avoid Beeminder’s sting. We suggest a simple trick to combat that tendency: mark a batch of unread messages as read at the end of each day, when you’re already safely on your yellow brick road, putting yourself on the hook to deal with them tomorrow. Of course Gmail Snooze itself lets you postpone Beeminder’s day of reckoning. But I find that if I’ve snoozed something often enough I’ll eventually admit that I’m never going to deal with it and I’ll just archive.

For me, Gmail Snooze and Gmail Zero are a beautiful combination. I have the option to procrastinate indefinitely on something, by repeatedly snoozing it. But Gmail Zero forces me to keep cycling through those things, never letting a deep pile accumulate where important items can get buried and forgotten about till they come back and bite me. I leave the biting (or stinging) to Beeminder. Here’s how it’s been working for me:

Try it yourself:

Related Reading

• Email gamification ideas
• Proposals for fixing email
• Don’t suck at email
• Other tools: FollowUp.cc, Streak, Boomerang, Rapportive

Footnotes

[1] It’s actually entirely safe except in the unlikely event that you happen to already use Gmail labels like “7” or “x7” — labels that are bare numbers or numbers with an “x” prefixed — in which case don’t use this or it will change those labels on you!

[3] This is worse than for Gmail Snooze, which is written with Google Apps Script and hosted by Google. For external apps, Google doesn’t give fine-grained enough permissions for Beeminder to ask just to see stats about your email. It’s all or nuthin’ with them. But Beeminder’s not actually reading your email, I promise! See the original blog post on Gmail Zero for more on exactly what it’s doing.

Illustration by Bethany Soule

This is a guest post by Jake Hofman, part of the original MESS in Messy Matters and now a researcher at Microsoft Research – New York City. We’re not sure where Jake’s own PRs would appear on the charts below but he crushed both Sharad and Dan this summer in the NYC Triathlon.

A recent New York Times feature compared the performance of Olympic medalists across time, highlighting, for example, how far back the 1896 gold medalist would have been if he were to have raced against Usain Bolt in this year’s 100 meter olympic event. Having seen the Times’ impressive visualization I wanted to play with the data myself, and so I scraped Olympic records for each sport over the past century from Sports Reference, a terrific site run by just a handful of folks who collect all kinds of fun information. [1]

While the Times looked at the absolute distance gap between current and past competitors within events, I was curious about the relative performance of medalists across different sports and events. For example, how much progress has been made in track events over Olympic history, and how does this compare to progress in other sports? The figure below addresses these questions through the percent difference in times for past medalists — indicated by gold, bronze, and silver points — relative to their 2012 gold medal counterparts, with a sample of track events in the top row and swimming events in the bottom row. Across sports, distances, and event types, we see a remarkably similar pattern of rapid inital progress followed by a short period of stagnation during World War II, after which successively smaller improvements in times bring us to present day.

This pattern of relative progress holds for other Olympic events as well, although the rate of improvement varies by sport, as shown by the difference in scales for the track and swimming events — the relative improvement in men’s swimming is about twice the observed rate for track events. For instance, comparing Olympic competitors in 1960 to today’s athletes, we see that sprinters were about 7% slower while swimmers were more than 15% slower. As Nate Silver has pointed out, this difference is likely due to a number of factors, including changes in swimming regulations and technology (e.g., pool designs, skin suits, etc.) as well as varying costs of participation for running compared to swimming. Note that the butterfly and individual medley events show similar patterns of progress to freestyle swimming events despite being established over 50 years later, further supporting this argument.

Taking a slightly different perspective on progress, the figure below shows medalist speeds for men’s and women’s track events over the last 50 years, from which we see that today’s sprinters are about 1 to 2 miles per hour faster than competitors in 1948. [2] Note that current day male sprinters in the 100 meter and 200 meter races run at nearly equal speeds of over 23 miles per hour, and longer distances are run at successively slower speeds. Somewhat surprisingly, 10K long distance runners maintain close to equal speeds to 5K competitors, despite running twice the distance. Also of note are FloJo’s long-standing 1988 world records in the 100 and 200 meters women’s races.

The other clear feature from these plots is the relative speed between male and female sprinters for the same event. As mentioned in a recent piece by The Atlantic, while one might imagine that this gender gap would decrease as we hit potential limits of human performance, men appear to consistently perform about 10 to 15 percent faster than women, as shown in the figure below. (A similar trend is found in swimming and other olympic sports.)

While the gap between top men and women is significant, don’t make the mistake of thinking it holds for all individuals. Indeed, to contrast Dan’s somewhat misleading introduction, I’ll note that Bethany destroyed each and every male member of Team Beeminder in the swim portion of the triathlon!

Footnotes

[1] Code to scrape the data and generate the plots is available on GitHub.

[2] While it’s initially tempting to place best-fit lines over these points, one quickly runs into the controversy around human speed limits, although this paper looks like an interesting approach.

Illustration by Kelly Savage

In the 15 years since Hoffman and Novak highlighted the digital divide between those with and without internet access, use of the web has risen dramatically in the United States, increasing from roughly 30% to 80% of adults regularly going online.^[1] Even with this tremendous growth in access, however, substantial inequalities persist across demographic groups. For example, in contrast to the 80% of adults in the general population who use the internet, only about 70% of Blacks and 40% of people over 65 do so. Assessments and discussions of the digital divide typically focus on these disparities in basic access. In a recent paper with Jake Hofman and Irmak Sirer, we investigate a related but distinct question: Among those individuals who are already online, how do usage patterns vary across the population?

Our analysis of web browsing behavior is based on complete activity logs for approximately 250,000 people in the Nielsen MegaPanel, who in aggregate generated over three billion pageviews over the course of a year. In addition to these web browsing histories, individual and household-level data were collected from each panelist, including age, sex, race, educational attainment, and household income. Leveraging these data, we were thus able to systematically investigate online browsing behavior at scale across a representative sample of the online population.

“Adults with a post-graduate degree spend more than three times as much time on health sites than those with only some high school education, and Asians read more than 50% more online news than Blacks and Hispanics.”

Concern over the digital divide typically stems from the belief that internet access improves quality of life along several core dimensions, including health and education, and that lack of access thus leaves certain subpopulations at a disadvantage. We examine usage in three areas that directly relate to these particularly consequential outcomes — news, health (e.g., WebMD), and reference (e.g., Wikipedia) — and find large differences across demographic groups. Adults with a post-graduate degree, for example, spend more than three times as much time on health sites than those with only some high school education, and Asians read more than 50% more online news than Blacks and Hispanics, exacerbating disparities in basic access between these groups. On the other hand, differences between Whites and underrepresented minorities largely vanish among those who are already online.

These results illustrate that the digital divide is more than a simple question of access, and highlight some of the issues that remain even among those with access to the web. In many ways, however, our findings raise as many questions as they answer. Perhaps most importantly, what are the effects of these observed differences in web browsing behavior? While it is certainly reasonable that easier access to information on, for example, nutrition and contraception would lead to better health outcomes, rigorously establishing this relationship is not easy. In this regard, accurately measuring online behavior is an important, but still just a first step in making informed policy decisions.

Footnotes

[1] These and related data are available from the Pew Research Center.

Illustration by Kelly Savage. Big thanks to Mainak Mazumdar and the Nielsen Company for providing the web panel data. Large chunks of this post were shamelessly plagiarized from my paper with Jake Hofman and Irmak Sirer, “Who Does What on the Web: A Large-scale Study of Browsing Behavior”.

We have worked out what we believe is the fastest fair way to split a restaurant bill! You know what the state of the art is like so this is quite a breakthrough. To be clear about the “fastest fair way” it helps to compare to the extremes. First, the slowest fair way to split a bill is to compute exactly what everyone ordered. Groan. Dividing the bill equally avoids the hassle of figuring out who got what but is still pretty annoying, making change and whatnot. And of course that’s unfair to the people who ordered less, not to mention that it distorts what people order.

If you’re not a stickler for fairness and social efficiency, the hands-down fastest solution is credit card roulette: put everyone’s credit card in a hat and pick one. That person pays the whole bill. If everyone happened to have spent the same amount, then credit card roulette is also perfectly fair. [1]

The question, then, is how to get as close as possible to the convenience of credit card roulette but with perfect fairness: everyone pays, in expectation, exactly what they owe. Fortunately, and perhaps surprisingly, we don’t need to figure out who ordered what — for the most part — in order to achieve this.

Here’s what you do! Start with any item on the bill. The person who ordered that item pays the bill with probability equal to the cost of that item divided by the subtotal. Flip the appropriately biased coin [2]; if that person is it, then you’re done. If not, then subtract that item from the subtotal and repeat, recursively, with another arbitrary item. If you start with expensive items then you’ll probably find the person who’s paying after a handful of items, but it doesn’t matter for fairness what order you pick things in. You won’t have to figure out all the confusing drinks and appetizers (yet the outcome is as fair as if you had!).

Expectorant

David Pennock is the one who first thought of this algorithm. We’ll leave it as an exercise for the reader to prove that it works. Proving that it’s the fastest fair algorithm should also be straightforward, once you pin down what “fastest” means. But enough theory. Bethany Soule and I wrote a little Android app that makes this algorithm quite convenient. We call it Expectorant (“exquisite fairness in expectation”). Here’s an example of how it implements the above procedure:

Say the subtotal is $100 and the items on the bill are $5, $25, $60, and $10. Enter 100:5 and have the person who ordered the $5 item pick a number from 1 to 20. If their number is lit up (a 5% chance) they get to pay the whole bill! If not, amend the expression as 100:5,25 and repeat for the person who got the $25 item. They’ll “win” with probability 25/(100-5). If they’re off the hook, amend again to 100:5,25,60. This time most likely — p = 60/(100-5-25) — the $60 person will win the honor of paying the bill. If not, notice that 100:5,25,60,10 yields 10/(100-5-25-60) = 1. So if the process makes it to the last item on the bill then whoever got that item is it. Mathemagically, it doesn’t matter what order you put the items in — each person “wins” (pays the whole bill) with probability equal to their own fair share of the bill. In other words, you pay in expectation exactly your fair share. Including tax and tip, even though we never entered those. Pretty slick! Speaking of tips, remember you can minimize the hassle by starting with the most expensive items. Then you don’t have to figure out who most of the items belong to. Oh, and if 3 people split the $10 pickled monkey balls just treat it as 3 items, $10/3 each (expressions instead of numbers are allowed).

I bet I can game this by…

You really can’t! It doesn’t matter if you get two cheap appetizers instead of one entree or if you arrange to go first or last. Your probability of paying will be the sum of the costs of your dishes divided by the subtotal. That’s true even if none of your dishes were ever identified on the bill as yours. Another common question about this algorithm is what happens if no one gets chosen. The answer is that that can’t happen. If the process makes it to the last item on the bill then whoever got that item is it. But it rarely gets to the last item, which is the beauty of it. On average you’ll only traverse half the bill — and that’s half in terms of dollar value, not number of items. So much less than half the items if most of the bill is concentrated in a few expensive entrees.

To help see that the order doesn’t matter, that being up first is no better or worse than being up last, here’s a dirt simple example: Suppose three of us each ordered $1 items. The first person will pay with probability 1/3, just as they should. The second person pays with probability 1/2 if the first person is off the hook, which happens with probability 2/3. So the second person’s probability is 2/3⋅1/2 = 1/3. Perfect. And the third person will definitely pay if the first two people are off the hook, which happens with probability 2/3⋅1/2 = 1/3.

In general, the key to why this works, fairly, even without computing what everyone ordered, is this: Imagine a pie chart where everyone has slices of a unit pie in proportion to the amounts they owe. The total probability that you should pay is equal to the total area of your slices. But we can decompose that starting with one slice of size x, where your other slices total y. The probability you should pay is x + y which is x + y(1-x)/(1-x). That second term is what you get by renormalizing the rest of the pie after subtracting x. In other words, you first pay with probability x and then the rest of your probability (and everyone else’s) is covered in the recursive step, if you get to it.

Bonus Expectorizing

Expectorant is actually a more general tool. You can enter a probability (between 0 and 1) or an arithmetic expression that evaluates to a probability. A subset of the numbers 1-20 will light up such that any given number will be lit up with the given probability.

Here’s another way that’s useful. Say you owe me $7 for lunch but only have a twenty. If you give me the twenty with probability 7/20 then in expectation you’ve paid me $7! So type in 7/20 and tell me to pick a number from 1 to 20. Hit “Expectorize” — or have me do it so I know you didn’t cheat — and if my number is lit up then I lucked out and get the $20. (If you trust Expectorant to randomize properly — you can, we promise — then you can just always choose 1, or any number. [3])

We’ve even added some handy syntactic sugar to generalize the case of needing to pay someone with one of two amounts, one of which is too small and one which is too big. Entering something like 7@5,20 is a shortcut for (7-5)/(20-5). That’s the probability p such that (1-p)⋅5 + p⋅20 = 7. WTF? Here’s TF: If you have a five and a twenty and owe me $7, then give me the twenty with that probability and the five otherwise. Exquisitely fair (in expectation)!

Addendum: A Faster Fair Method!

There’s been a lively discussion of Expectorant on Hacker News since this went to press. The best comment was by Michael Donaghy who pointed out a faster version of the mechanism: Pick a uniform random number between 0 and the subtotal and walk down the bill adding up the costs (oblivious to who ordered what) until you hit the item that pushes you over the chosen number. Whoever ordered that item pays. One complication: if that item is something that occurs more than once on the bill then you need a way to disambiguate. You could randomize again or you could use a pre-decided ordering of the diners. Either way, you have to make sure to identify all the people who ordered that item. That’s perhaps a disadvantage compared to our version of Expectorant where items are always considered one at a time in isolation. But other than that, this version is certainly faster, and just as fair.

Related reading

• “You Do the Math” in the New York Times
• Academic paper that concludes that splitting evenly is inefficient
• Interview with the author of the above paper
• Splitting the bill at restaurants using game theory
• Advice columnists weigh in: “But I Only Drank Water”, “Dining out, I eat veggie … but pay like I ordered steak”, “A Measure of Guidance”, “But I Only Got the Soup”

Footnotes

[1] A common complaint about stochastic schemes is that they’re “only fair if you do it repeatedly with the same group of people”. That’s true if you insist on ex post fairness. We’re usually happy with ex ante fairness. Consider selling me a (perfectly fairly priced) lottery ticket for a dollar. That’s guaranteed to be unfair, ex post. Either you sold me a worthless piece of paper for a dollar, or I got a million dollars and only paid a dollar for it. But the fact that none of us knew which would happen made the one dollar price fair. Same story with venture capital investment, for example. You may need a gambling mentality to be down with it, but it’s quite fair even if only done once. The fact that it averages out in the long term to be perfectly fair ex post is icing on the cake.

[2] Here’s a way two people can flip a biased coin using nothing but their brains. It’s even quite robust to the notorious inability of humans to generate plausible random numbers. Person one writes down a fraction p of the numbers from 1 to 20. Person two guesses a number from 1 to 20. If their guess was one of the written down numbers, call the biased coin flip heads. We haven’t tested this rigorously but our sense from doing this a lot amongst ourselves is that the two people’s lack of randomness pretty much cancels out, particularly if they’re trying for opposite outcomes, and yields reasonably random Bernoulli outcomes. Of course with numbers 1 to 20 you’re limited to a granularity of 5% on the probabilities. You could use, say, 1 to 100 but that’s a bit unwieldy.

[3] There’s a very small way in which you do have to trust Expectorant: If the probability percentage is not a multiple of 5. For example, to get a 1% probability you’d need to light up a fifth of one of the numbers (1-20). That’s not allowed so Expectorant lights up one number with 1/5 probability. It always does this fairly for probabilities that don’t work out to an integer number of numbers lit up. In theory it could cheat and round in the wrong direction. But it doesn’t, so you’re actually getting probabilities to many decimal places of fairness, despite the discrete grid.

Illustration: Kelly Savage
The ingenious restaurant bill splitting algorithm was devised by Dave Pennock of, appropriately, oddhead.com. Thanks to Sharad Goel and Kelly Savage for expectorizing the bill with us every time we go out to dinner.

Your team is down by a couple of baskets going into the final minutes of the game. Is it time to panic or is it still anyone’s to win? Plenty have certainly come back from apparently dire situations. Just last month the Lakers beat the Thunder in double overtime after being down by 11 points with four minutes on the clock. In fact, such comebacks have led the renowned sport statistician Bill James to conclude that with four minutes remaining it’s not really over unless the gap is at least 19 points.^[1] Spectacular reversals of fortune may be rare, but conventional wisdom holds that a lot can and does happen in that last quarter.

“An NBA team leading by twice the square root of minutes left in the game has an 80% chance of winning.”

As explained by Moneyball author Michael Lewis, “one statistical rule of thumb in basketball is that a team leading by more points than there are minutes left near the end of the game has an 80% chance of winning. If your team is down by more than 6 points halfway through the final quarter, and you’re anxious to beat the traffic, you can leave knowing that there is slightly less than a 20 percent chance you’ll miss a victory.” While that’s a compelling heuristic, an analysis of over 7,000 games over the last six seasons reveals that it only holds for a few short minutes, losing accuracy even by the six-minute mark in Lewis’s example. A better rule of thumb is that a team has an 80% chance of winning if they lead by twice the square root of minutes left. Going into the final quarter (12 minutes) of the game, for example, that 80% threshold is achieved at \(2\sqrt{12} \approx 7\) points. By comparison, the standard heuristic suggests a substantially larger gap of 13 points is required to achieve an 80% chance of success.

Why does the square root rule work? Basketball games can be reasonably well modeled as a series of independent one-minute rounds consisting of approximately one possession for each team. In each interval, the score differential between the teams changes by about two points, with each team roughly equally likely to win that round. In statistical terms, the intervals are approximately mean 0 with standard deviation 2. The central limit theorem then shows that with \(t\) minutes remaining, the score gap changes approximately according to a normal random variable with mean 0 and standard deviation \(2\sqrt{t}\). (By contrast, the rule of thumb Lewis cites implicitly — and incorrectly — assumes randomness increases linearly in time.) Since it is unlikely a team will make up a one standard deviation deficit (i.e., \(2\sqrt{t}\) points), a lead of at least that much is relatively safe.

It’s perhaps surprising that a seemingly modest seven-point deficit is difficult to recover from with a full quarter of play remaining. Granted, coming back 20% of the time is not exactly a snowball’s chance in hell (more like the chance of sunshine in Juneau, in December), but it’s certainly not an enviable position to be in. The data further show that even starting the fourth quarter down just five points does not bode well, with only a 1 in 4 chance of success.

At least anecdotally, there appears to be a general sense that entering the final stretch down a few points is not a big deal; it’s still a “close” game and anything can happen. Understanding the effects of such deceptively minor deficits, however, could lead to better strategic play. For example, a team that correctly recognizes their low likelihood of success may attempt riskier plays so as to increase the randomness of the game and in turn boost their chance of winning. Alternatively, realizing the point gap amassed in the first three quarters is in fact quite consequential, teams may increase their intensity of play earlier on. But who really knows if such armchair statistical strategies would actually work in practice? Certainly not me — I’ve never even played a proper game of basketball — but that hasn’t stopped me from blogging about it before!

Footnotes

[1] Strictly speaking, James’s rule was intended for college — not professional — basketball, but I suspect he would arrive at a similar estimate for NBA games.

Illustration by Kelly Savage. Data obtained from BasketballValue.com. Thanks to Kiran Limaye, Dave Pennock, Matt Salganik, and Sid Suri for teaching me that basketball is the one in which you try to put the 2-sphere through the 1-sphere.

I’m very deadline driven. So much so that I go to truly ridiculous lengths to impose deadlines on myself. The truly ridiculous lengths are needed, of course, because self-imposed deadlines are notoriously prone to getting weaseled out of. There’s another kind of deadline that is also notoriously prone to getting weaseled out of: instructor-imposed deadlines. It’s a sad fact of academia: Students with too much self-respect to ask for special treatment are penalized relative to those who do. In other words, you can get away with murder by whining to the instructor. As an instructor it’s very hard to neutralize the whining strategy without feeling like a heartless jerk.

Why Deadlines? [1]

You may well ask why instructors should penalize late submissions at all. First, it’s just a practical issue for the instructor: you don’t want to grade everything at the end of the semester. But more importantly, it induces the students to spread their effort out over the whole term. The best instructors I know maximize the number of intermediate deadlines. Here’s how David Reiley put it:

When my students have a term paper, I give them a deadline to turn in a proposal, a deadline to turn in a first draft and share with their classmates, and a deadline for their final version. They learn a lot more that way. Same thing with giving them nearly weekly problem sets instead of just a midterm and final.

And the best students understand this, too. Dan Ariely describes in his book, Predictably Irrational, an experiment with his students relating to deadlines and commitment devices that drives home the point. The setup is that he let his students pick their project deadlines for the whole semester. Once they were picked the students would fail if they didn’t meet them. So rationally they should choose all the deadlines to be on the last day, to maximize flexibility. But you won’t be surprised by the punchline: Students who precommitted to spaced-out deadlines did better.

The key is to reduce the burstiness of effort. Especially enterprising students can overcome their procrastinating natures on their own, given the right tools.

Late Day Budgets

Now that we’ve established that deadlines are good, let’s get back to the problem of being sensitive to the need to occasionally miss them without having to stray from a universal policy. A common solution is to deduct points for each late day. Such a policy, however, requires rather harsh penalties (e.g., 1/2 grade off per day late) in order to ensure assignments are not turned in weeks late. An alternative is to give every student a budget of free late days to be used throughout the course. With this built in flexibility, students can use their allocation of late days without offering up any excuses. But that solution only goes so far. You’ll occasionally have students who run out of late days but still feel they have a good excuse, or who miss the deadline by just a little and argue that it shouldn’t count as one of their late days. Just being day-based means you’ll end up having to make judgment calls (or be draconian) about when exactly the transition from n to n+1 days late happens.

“It’s socially inefficient!”

And there’s an even bigger problem with the late-day budget. It’s socially inefficient! Students are incentivized to be conservative with their late days early in the semester and then have late days to burn at the end. What you really want is efficient late submitting: students should take only as much extra time as they need (to a point).

A Graduated Penalty Function [2]

Here was my solution to this problem, from the syllabus of the last class I taught:

If an assignment is late by t seconds then the score will be adjusted by multiplying by $$1-\left(\frac{t}{7\cdot 24\cdot 60\cdot 60}\right)^4$$ which is just \(1-t^4\) if \(t\) is measured in weeks. This policy is applied totally automatically based on the submission timestamp. Note that within two days of the deadline less than 1% — \((2/7)^4\) — of the points are lost so the late policy can be treated as having essentially a 2-day grace period for every assignment. Because of that built-in leeway, the only exceptions will be for serious emergencies. Most excuses will not warrant a manual override of the late policy. Once the nominal deadline has passed, no resubmissions will be allowed. So when submitting during the variable penalty period, check over your submission carefully.

Note that the multiplier is zero after exactly one week. The reason for not allowing resubmissions after the nominal deadline is that otherwise you couldn’t start grading until a week later, when all the points are gone.

The beauty of this late policy is that you never have to make judgment calls — where you end up rewarding whining — yet you don’t have to feel like you’re being anal and inflexible. There’s more automated flexibility built in than with a late-day budget. And by eliminating the discontinuities in the penalty at the day boundaries, there’s no such thing as quibbling about what fluke technical difficulty delayed submission at some critical moment. Most importantly, with a late penalty that starts out negligible and gradually, smoothly transitions to severe, students are incentivized to target the nominal deadline and take as much additional time as they truly need, subject to the constraint that you don’t have to worry about stragglers beyond a week. And despite what is essentially a 2-day grace period on every assignment, I found that most students met the nominal deadline.

Footnotes

[1] Some pedagogy nerds may object that late penalties, by reducing grades due to lateness, make grades less reflective of what was truly learned. That’s true, but the failure to spread work out harms actual learning so harming the meaningfulness of the grades is a small price to pay.

[2] The sort of Messy Matters readers who read our footnotes may be wondering why Beeminder doesn’t use a graduated penalty function. In some sense it does, in that you go from the green zone to the blue zone (right lane of the yellow brick road) to the orange zone (wrong lane of the yellow brick road) to the red zone (emergency day) before finally getting charged for going off track. But it’s true that the penalty incurred for derailing happens all at once at a certain time. That may be draconian but it’s exactly what Beeminder users are signing up for. Still, it’s not a bad idea! I hadn’t actually considered it till writing this blog post, which I intended to be entirely unrelated to Beeminder!

Ironically fitting illustration by Kelly Savage.
Thanks to David Reiley, Sharad Goel, Bethany Soule, Nicci Nunes, and Kevin Lochner for comments.

This article is cross-posted on the Beeminder Blog.

The problem of self-control may be a ridiculous first world problem but it’s the granddaddy of first world problems and I want to solve it. We live amidst a deluge of opportunities for instant gratification, especially in the form of food and entertainment, and most of us don’t handle it well. The general problem, known as akrasia, is this: you understand your own best interests when you consider them dispassionately, but in the moment your decision-making is distorted. The best time for, say, a workout is always “tomorrow”. And the best time to start working is after “just one more” clip of [redacted; save yourself!] on YouTube. Knowing that the problem boils down to time scales (hyperbolic discounting) implies the fundamental solution: commitment devices. For example, many people buy cost-inefficient individual-sized packs of cigarettes or candy, paying a premium to throttle their future consumption. Deleting games from your computer or going somewhere without internet access to get work done is another common example. You need to lock yourself in to your chosen course of action, like maintaining a healthy weight, eating right, or getting blog posts written every month.

The last time I wrote about this topic on Messy Matters, 1.4 years ago, we included a straw poll with a slew of commonly used commitment devices. As I pointed out then, the tricks and lifehacks on that list are surprisingly toothless — they’re mostly band-aid solutions. Economists have been wondering for at least half a century why we don’t see more serious use of commitment devices in the real world. But they’ve had an answer for equally long. As Robert Strotz speculated in 1955 [1], the reason is risk and uncertainty, “both as to future tastes and future opportunities.” [2] This should be obvious to Messy Matters readers. We sometimes call it the Principle of Delayed Commitment: the simple decision-theoretic fact that you shouldn’t commit to a course of action until you have to. Flexibility is valuable; laziness is a virtue. For example, you shouldn’t toggle the state of the toilet seat after you use it because you don’t know what state the next person will want it in. What if you expend the effort toggling only to have the next person to use the bathroom, perhaps you, toggle it right back? Such waste! I’m sure I’m preaching to the choir here. [3] Or take going to the gym. You shouldn’t pay for a membership if you can, for a similar price, pay as you go. After all, in the future you may decide not to go to the gym, making pay-as-you-go cheaper, in expectation.

“Two forms of irrationality can cancel each other out”

But you know where this is going. Yes, you might well decide not to go to the gym, but it probably won’t be a rational decision. The solution, ironically, is more irrationality. Commitment devices blatantly violate the principle of delayed commitment, removing future flexibility for no reason other than to thwart your future self. But if that impetuous version of yourself will undermine your own goals (such as getting in shape) then that’s quite rational from your current perspective. Two forms of irrationality can cancel each other out. Which is to say that if you irrationally over-weight immediate consequences — if you’re akratic — then you should “irrationally” pre-commit.

A gym membership is a literal commitment contract, albeit a weak one. It turns the gym fees into a sunk cost, committing you to pay whether you visit the gym or not. Commitment devices may be an old idea but they’re enjoying a technology-fueled renaissance in the form of a crop of new web services. GymPact had the clever idea to give the standard gym membership commitment contract more teeth, by charging an additional fee for missing workouts. StickK.com — a pioneer in bringing commitment devices to the web — generalized the idea and offers commitment contracts on anything you can (pre)specify. Fighting irrationality with irrationality is easier than ever.

But we’d like to make those two irrationalities cancel each other out while minimizing the collateral damage. In other words, bind yourself with a commitment device that thwarts your akrasia while retaining maximum flexibility as to future tastes and future opportunities. [4]

The Akrasia Horizon

Here’s a now well-tested way to do that, by focusing on data tracking:

Akrasia Horizon: The time horizon beyond which you can make rational decisions, undistorted by akrasia

1. Retain the flexibility to change your contract in light of new information [5] (like, 40 hours of actual focused work per week is damn hard!). That sounds like it defeats the point of a commitment contract, but recall the fundamental problem of akrasia: over-weighting immediate consequences. To beat akrasia you only need to bind yourself for whatever the horizon on “immediate” is. Based on a study on grocery-buying habits — when buying groceries online for delivery tomorrow people buy a lot more ice cream and a lot fewer vegetables than when they’re ordering for delivery next week — and our own accumulating evidence, we’re taking that akrasia horizon to be one week.

2. Just commit to progress. You don’t have to know what you’re committing to when you commit, which also sounds (oxy)moronic but what I mean is this: Commit to keeping your data points on a path to your goal (the “yellow brick road”) that you have control over as you go, subject to the akrasia horizon. That is, commit to something like “work out more” or “lose weight” and then decide as you go what that means based on your data.

So a maximally flexible self-control tool is one that commits you to keeping all your data points on a path to your goal that you specify and can change the steepness of at any time. The only part of the path that’s fixed is the upcoming week. [6]

Easy Street

You may be wondering how anyone could ever fail to stick to a commitment that’s this flexible. Here’s how: if you’re highly akratic. Such a person may well find it a daily struggle to stay on track. Yeah, you can always choose to wuss out and flatten the yellow brick road, but only starting in a week, which you don’t want to do. You want to wuss out Right Now, dammit! I mean, just for now, while you eat this pie, and then you’ll behave again. No such luck though.

The daily struggle to stay on the road does not induce you to touch that road dial. You always want to make it easier “just for today” — which the akrasia horizon doesn’t allow — and you always think you’ll get your act together by next week.

Eight Hundred Dollar Postscript

For three years now we’ve scrambled to make sure a Messy Matters post got out the door at least once in every calendar month. Such is the powerful psychology of “don’t break the chain”, also known as the Seinfeld hack. But the Seinfeld hack’s greatest strength is also its fatal flaw: Once you do break the chain, all the motivation it provided bursts like a bubble. You’ve got to somehow motivate yourself to build up another long chain to not break. Until then you’re on a “one more day won’t matter” slippery slope of sloth. (Notice how we missed our end-of-February deadline and how many days into March it now is!)

Well, Beeminder, as we say, is safety rope for slippery slopes. So it’s time to do this right! We shall henceforth publish on Messy Matters as often as the following yellow brick road dictates, or give one of you $800. We’ll get the added flexibility, of course, to change the rate of posts as needed, subject to the akrasia horizon. I think it can also add a different kind of flexibility and help prevent end-of-month all-nighters: If we publicize a draft of a post (to our hard-core readers who follow our twitter feed) that will count as half a post (publishing a draft will count as the other half).

I’m quite serious about this $800 offer, subject to some fine print [7]. As the dozen or so people who have collected on our $20 typo bounties will attest, we’re good for it. [8]

 
Illustration by Kelly Savage. Thanks to Bethany Soule, Sharad Goel, and David Yang for comments.
 

Footnotes

[1] This is the earliest treatment of akrasia and time inconsistency that I know of in the economics literature: Myopia and Inconsistency in Dynamic Utility Maximization, by Robert H. Strotz in Review of Economic Studies, 1955-1956.