Splitting the bill at restaurants using game theory

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“We’re going to split the bill,” said the organizer at my friend’s ninth grade birthday party. I didn’t think much of it until I ended up paying $40 for a $10 entree. I felt cheated because I didn’t order a drink like most others. I was afraid to ruin the party mood, so I concealed my own anger, and that ended up ruining the night for me.

If you’re like people I know, I’m sure you’ve had similar horror stories. The overall feeling is that everyone seems to pay too much and no one is happy.

Should you just pay up? Do we need to teach better etiquette? Is better technology required?

No, no, and no.

It’s hard to change how people respond to incentives. But the group dining situation is screaming for better outcomes. As I’ve suggested before, if you don’t like the outcomes, don’t hate people, change the game. And the best tool is the aptly named field of mechanism design (brief intro here).

I’ll frame the dining problem by reviewing the three common payment systems–splitting evenly, pay for what you ordered, and separate checks. I’ll then suggest a solution for those seemingly intractable birthday parties.

Splitting Evenly

This is probably the most common solution. But three economics professors have shown that it’s a very inefficient system (pdf).

The system is problematic because it distorts the incentives of diners. At a table of 6, each item you order contributes 1/6 to your cost and 5/6 to everyone else. You have reason to order more, and so does everyone else. The outcome is that everyone wastefully orders and everyone ends up paying too much. It’s a version of the tragedy of the commons. Restaurants are well aware that diners rack up more expenses, so that’s one reason they favor keeping a common tab.

Furthermore, splitting the bill is unfair to people who prefer less expensive items. If you’re a non-drinker, or if you’re a vegetarian, you’re probably going to end up subsidizing the meals of the meat-eating alcoholics. Or, if you are a saver, you’ll end up subsidizing the people who splurge on ridiculous items.

It’s considered impolite to point out large price differences. It makes gluttons feel guilty and ruins the mood. Besides, is there a better solution?

Yes, you might say. Just pay for what you ordered.

Pay for What You Ordered

I always say, when you collect more money than is needed, it’s a sign that you’re in good company.

Hence, pay for what you ordered is the method I use with close friends. We all trust each other to calculate tax and tip, and not screw each other over. Actually, it’s not so much that we trust each other as friends, but that we are made to trust each other in a game theory sense because we can punish cheapskates the next time we go out.

Unfortunately, this system has its flaws too. How do you split a bottle of wine? Do you go as far as calculating the cost of each glass? What about when you share food, or finish off someone’s meal because they are too full? The problem is there’s a small difference between what you order and what you eat. The system favors people who order cheaper items but then try to sample as much as possible of other people’s food.

There’s one more issue. It’s often not possible to say who put in what. There’s an incentive to pay just a little bit less hoping someone else will be generous and cover. If the pot is short, often everyone is asked to put in an extra dollar or two. So the system still favors the cheapskates–especially the people who pay less and then sneak away to the bathroom when everyone is asked to pitch in more.

If you don’t trust your company, why not just separate checks?

Separate Checks

This is the most efficient system, in terms of incentives. People will only order what they really want because they ultimately have to pay for it. Perhaps this is why researchers found that 80% of subjects favored separate checks.

So the economics are pretty favorable, but there are a few practical problems.

First, it takes longer to pay as your party size gets bigger. As a matter of practice, waiters don’t enjoy this.

Second, separate checks are impractical when you’re sharing appetizers or bottles of wine. What if you share an appetizer with one friend but then take a glass of wine that was split with four other people? It’s a hassle to sort it all out.

Third, in parties, people usually treat the guest of honor. Would a restaurant actually split one item across everyone else’s bill? I am not sure, as I’ve ever seen this.

Perhaps the biggest problem is that separate checks are not even allowed at some restaurants–particularly nicer ones. It’s probably because they know people order more on a common bill, and it’s somehow seen as less “classy” to offer separate checks. I don’t totally understand it, but I have been rejected by at least one snobby waiter (Oh, no, no, monsieur, we don’t do that).

Thinking about an ideal outcome

What’s an ideal outcome? I think it would have three elements: that each person orders properly, each person pays what is owed, and the money is collected quickly.

How do the three methods compare on these criteria?

Splitting the bill –>Over order, over pay, quick collection
Pay what you ordered–>Order correctly, a few overpay, moderately fast collection
Separate checks–>Inconvenient for sharing, pay correctly, slow collection

So each method is the best at exactly one trait. Perhaps that’s why there’s such disagreement over what to do–each method has its unique merit.

Let’s abstract the problem a little more to consider why the dining problem is so complicated. I see six main characteristics that cause trouble:

1. Hidden preferences on food
2. Hidden willingness to pay
3. Selfish desire to pay less
4. Desire to look fair
5. Desire to maintain party atmosphere
6. Hidden payments on a common bill

I’ve mulled over several solutions, and I think I’ve come up with a good start.

I’ll deal with the situation of a birthday party with a large number of people. Other situations can be handled similarly–they might even be easier because you don’t have to cover the guest of honor’s meal.

The Suggested Answer

Such a simple situation demands an answer! But how might we proceed? There are two big steps that can make a drastic difference.

The first step has to do with the problem of disparity. Often, people end up ordering items that cost vastly different amounts. While it’s nice to have choice, it’s worse that everyone ends up with a wildly different amount that’s owed. This confusion is what allows cheaters to conceal their motives and pay less.

Thus, the first step is to make everyone happier by offering a reasonable menu. The most practical way is to call the restaurant and see if you can make a special fixed menu. Some restaurants will print you small menu cards to make the occasion special. But don’t worry if they don’t–just make your own menu. Select a few reasonably priced items and design your own menu to hand out to guests.

The second step is to assign one person as the money collector. Preferably this takes place before any one eats food. This guarantees guests have money, and creates an incentive for someone to collect fairly. In large parties, you could even offer to compensate this person a few bucks from the group. I am sure everyone would be willing to willing to spend an extra dollar to ensure the party atmosphere is good. If your party does not have an organizer, volunteer! People will be happy.

Here’s how the solution might work in practice.

An example calculation

Let’s say you have 10 guests, plus one birthday boy. You want to go to Gordon Biersch, a brewery restaurant.

They have lots of food on the menu. I count 13 starters, 12 salads, 10+ pizzas, 5 steaks, 6 seafood dishes, 5 specialties, 9 pastas, 14 sandwiches, and 5 desserts.

There is a large difference between a $10 burger and a $27 steak, so you can already see that a free-for-all “split-the-meal” is going to be a disaster. And I didn’t even mention the drink menu.

Here’s one way to proceed.

Step One: Create a reasonable fixed menu

Choose 3 garlic fries starters for everyone to share.
For drinks, allow people to order a soda or beer. Avoid complications like sharing wine.
Limit the menu to pizzas (except do it yourself) and sandwiches. The price range is $9-$12.25.
Choose 5 desserts for the table to share.

Step Two: Estimate costs and collect the money in advance

I will round some costs up to be safe.

3 garlic fries 3 x $6 = $18.00
11 entrees11 x $12.50 = $137.50
5 desserts5 x $6.25 = $31.25
2 beers for birthday boy 2 x $5 = $10.00
Total common = $196.75
Tax + Tip (8% +18%) = $247.91

Per 10 people (treat the birthday boy) = $24.79–>round to $25

The only difference in what people pay is the choice of drinks. I think sodas cost $2 and beers $5. I’ll add tax and tip and do some rounding.

Soda: Contribute $28
1 beer: Contribute $31.50
2 beers: Contribute $38
n beers: Contribute $25 + $6.50 n

That’s it! Collect money in advance, and you’ll have every thing covered.

Step Three: Send everyone an email

The dining problem is about expectations and information. Get every one on the same page before the party starts. Here’s an email you could send to all your attendees:

Hi everybody,

We’re going to be celebrating Dan’s birthday party next Friday at Gordon Biersch.

Because we have a large group, I want to coordinate the food and cost in advance so the party runs smoothly.

For the entree, we request you order a pizza (except “make your own”) or a sandwich.

For the appetizers and desserts, I will order a few for the whole table to share.

As for paying, we’re going to split the costs (and pay for Dan’s meal). This should be fair because the pizza and sandwich entrees are similarly priced. It’s not perfect, but it will make collection easy.

I’ve estimated the costs and have a very good idea of what you’ll need to pay. The only difference is what you decide to drink. Here are the estimated costs:

No drink: Contribute $25
Soda: Contribute $28
1 Beer: Contribute $31.50
2 Beers: Contribute $38
n Beers: Contribute $25 + $6.50 n

Please pay me in cash before you start eating. In the unlikely event we’re short, I will ask everyone to chip in an extra dollar or two. Please don’t run to the bathroom when it comes time to pay the bill

Let me know if you have any questions!

Under this solution, people will roughly pay for what they order, it’s easy to collect and enforce payment, and a good time should be had by all. No one will be surprised by the $25 minimum cost. Big spenders can order more beers for themselves.

Some of you might be thinking, “What, no wine? No steak?” I agree the choice is limited. But I think people are willing to give up a little choice in exchange for everyone having a good party. This is because parties are not about getting exactly what you want–you can do that on your own time. It’s about coordinating a large group.

Do you have another solution to the dining problem? Please share in the comments and I’ll post the best ideas.

Great Reader Ideas:

In the comments, there are a bunch of good ideas. Here is a sampling.

Anna suggests a much simpler email with the same message:

Why not just say “we have a prix fixe options arranged with the restaurant including a chicken dish, pizza, and vegetables and one alcholic beer/wine or non for $25 including tip. Family style. Extra drinks are on your own.

Eric Wu finds my email too commanding, and offers a way to allow for wine purchases:

It seems a bit too commanding and even a bit irrational to limit the choices to a select few within a certain price range. To me, the basic problem is the ordering if shared items such as bottles of wine or appetizers, which are the only items that need to be limited. Here is my solution:

1) Pick someone who is honest and very good at math, adding 25% to your items ordered is not difficult. He will divvy up the bill for everyone but himself.
2) Pick another person to collect the cash and divvy up the first person’s bill All excess collected funds go to the server. This is so no one has incentive to over-collect.
3) Limit the choices for appetizers to group/table appetizers and then order wine by the glass.

Definitely will not be as quick to collect if the person in charge is not good at math, but it allows everyone to order what they want, and everyone ends up paying the proper amount. Though the appetizers still cause problems because of the incentive to eat a lot of the appetizers and then eat a smaller meal. A solution would be to have the server divide the appetizers evenly.

Nick Eastop discusses a very cool mathematical method that seems to work out quickly:

The “Chris Dicken” method:
Everybody estimates on a scale from 1 to 10 what level of consumption they had. 10 being somebody who ate and drank a lot and had expensive dishes and 1 being somebody who only had a starter and a glass of water. It can be quite amusing discussing the different levels, but surprising in how quickly a consensus can be reached.
All the “level” numbers are added up and the bill is divided by the figure reached. This give a “unit”.
Each person pays the unit multiplied by his “level” plus a small percentage for whatever tip is appropriate.
I travel a lot with an orchestra and frequently eat in restaurants with groups of all different sizes. After trying this method for a while, everybody who tries it agrees that it is easy to use and a lot fairer than simply splitting the bill as we had done for years”!

Links:

In researching this article, I came across a lot of interesting web articles. The first New York Times article is where I got the idea to limit the menu. The rest of the articles share some popular opinions.

New York Times: You Do the Math
If it’s a fancy place, consider limiting the menu choices ahead of time to several reasonably priced alternatives and house wines, perhaps to be printed on a keepsake placard. If you have piles of money, consider paying for everybody. If you don’t, consider disclosing a rough price of entry ahead of time. And if that is exceeded, suck it up, because the alternative is just unpleasant.

The Onion: Caltech Physicists Successfully Split The Bill
“Before the arrival of the check, several early bill-splitting theories were proposed, including a simple process of dividing it into eight identical fragments, the Random Contribution Model, and a theory posited by Newbury himself–who insisted that he was bound to treat everyone–which was widely rejected on the basis that it would undermine the whole objective of the evening.

“When the check came, we all immediately agreed that the total of $284.57 could be defined as an irrational number of dollars for a party of eight to spend at a chain restaurant,” said Dreyfuss.

Money-Grubbing: An interview with Uri Gneezy, UCSD Professor
Also, culture can matter. In Germany, for example, they count down to the last penny. In Israel, if you do this, it’s not a good way to go around. You’ll get a very bad reputation for calculating up to the last cent. The cultural issue is very important.

But I Only Drank Water! Do you have to split the bill?
If everyone pays only for what he or she orders, then when the check comes, guests get tangled in petty calculations. Inevitably, someone miscounts how many beers he had, or doesn’t put in enough for the tip, or forgets to chip in for the host’s share. Worse yet, fussing over the details of the bill makes you look stingy. Dan Rubin, a spin instructor in San Francisco, says: “Why spoil a great dinner quibbling over awkward minutiae like who had three beers and who had one?” Rubin isn’t wealthy, but he finds this so distasteful, he says, “It’s worth me spending the extra $25 or whatever to save my reputation.”

But I Only Got the Soup!
Personally, I am with you. Establish a grand total, including the tip. (You would be surprised how a group is all too happy to have someone be the designated bookkeeper at times like this might at well be you.) Split the check by the number of people. Couples pay the two-fer rate, singles pay the single rate. Easy. Except when seven credit cards get tossed at the server. “Put $50 on this Amex, $22.50 on the two Visas,” and so on. Such a pain.

New York Times: A Measure of Guidance: Dividing the Bill
If one person or one couple has ordered significantly more expensive food or had much more to drink, it’s ideal if he, she or they bring up the subject of bill division and volunteer to pay a bigger share of it.

And if the discrepancy between that higher share and the lower share really isn’t much, it’s sweet of the lesser spenders to reject the offer, stave off the arithmetic and suggest a straightforward division.

These are social situations, after all. They’re not algebra classes.