# Sharing Cabs With Economists

#### 24 January 2016

#### An everyday conundrum. Simple economics to ponder on when you share a cab. Join the movement #splityourfare #hopsee

Three economists get into a cab. They’re each getting off at different places along the route. How should they split the bill?

It’s not a joke but an everyday numbers dilemma, and it highlights some important economic principles. I asked several economists to solve the problem, and they came up with some unique approaches. One called on the work of game-theory pioneer John Nash (the inspiration for “A Beautiful Mind”) to divide up the bill. Another referenced the ancient Jewish legal text, the Talmud.

The scenario poses a fundamental question of fairness: How do you best allocate costs and benefits between parties who have come together for mutual benefit? Assuming that this isn’t rush hour in midtown Manhattan and cabs are plentiful, our economists would each get home at least equally fast by riding in separate cars. Presumably they have come together because the total fare is cheaper than the sum of what they would each pay if traveling separately. How much of that savings should each rider get?

“Economists tend to be pretty good at this question of, ‘Should we share?’ “ says Aaron M. Swoboda, assistant professor at the University of Pittsburgh. “But then, how we distribute this surplus becomes a harder question.”

Lest you detect big-city bias, know that even if you prefer to sit behind the wheel, you could face similar questions. The same problems are posed by the case of several car passengers being dropped off in separate locations who must divvy up the gas bill.

For the sake of this column, we’ll stick with the case of a cab running on a meter, and we’ll assume that all of the passengers are traveling in the same general direction – some just live farther from the starting point than others. It’s easy to apply the sharing schemes we devise to more complex scenarios.

Let’s say that passenger A’s usual fare would be $1, passenger B’s is $5 and passenger C’s is $9. If all three share a cab (and assuming A and B are allowed to hop out on the way to C’s destination, without incurring any special fees), the total bill would be $9 – rather than the $15 they’d have to pay, total, to ride alone. How should they divide up the cost of the shared $9 ride? Or, put another way, how do they share the $6 of total savings?

Before consulting with economists, my sense was that all three passengers should evenly split the first leg of the trip to A’s house, because each one needed to go that far anyway. Then B and C should split the leg from A’s house to B’s, and C should pay for the rest. That would leave A paying 33 cents, B paying $2.33, and C paying $6.33 (someone will have to volunteer to pay the remaining penny for the bill). An advantage of this method compared with the ones that follow is that each passenger knows when he gets out what to contribute. Other strategies would likely require passengers to settle up later.

Dr. Swoboda, from Pittsburgh, advocated splitting up the $6 savings proportionally, based on how much each person would have paid to travel alone. That total cost would have been $15. So A, who would have paid $1 alone, gets 1/15 of the $6 savings, or 40 cents; B gets 5/15, or $2; and C gets the rest: $3.60.

That works out to a split of the $9 shared-ride cost of 60 cents, $3 and $5.40, respectively. Another way of thinking about this is that every passenger pays an amount proportional to what he would have paid without the savings. Proportional splitting of surplus and debts is a common approach in U.S. law, including bankruptcy cases.

Harvard University economist David Laibson suggests looking to Prof. Nash’s work for a solution. The Nash bargaining strategy – an approach based on game theory in which each cab passenger is seen as a party to the deal and is negotiating his best outcome – would have the passengers split the savings equally, so that A, B and C each gets $2 knocked off his bill. Why share the savings equally? Think of the shared cab ride as a contract being struck to yield savings: Any party could walk away from the deal and kill it, so each should share equally in the fruits of the deal. With the numbers I’ve chosen, this method yields the highly unlikely scenario that B pays $3, C pays $7 and A makes a profit of $1 for his troubles.

Prof. Laibson suggests eliminating that troublesome possibility by adding an exception: If splitting the savings equally means one passenger is getting paid, give him a freebie and split up the rest of the savings equally. So A would get a free ride, B pays $2.50, and C pays $6.50.

Glenn Ellison, a professor of economics at the Massachusetts Institute of Technology, pointed out that the bargaining model can get more complicated if individuals can form coalitions and bargain jointly. For example, if B and C traveled together, without A, they could still save $5. So they could argue to A that they should get to split $5 of the $6 total savings, and that only $1 is up for the equal split. That hard bargaining would yield an outcome of A paying 67 cents, B paying $2.17 and C paying $6.17. (Again, there’s an extra penny for the tip.)

The Talmud guides Barry Nalebuff, professor at the Yale School of Management, in his explanation, though taxi meters were scarce in ancient times. He uses the principle of the divided garment, used in the Talmud to settle disputes over estates and other financial matters. Say a father has an estate valued at $1,000 upon his death, but he has mistakenly left instructions for one son to receive $800 and another to receive $400. The Talmud would suggest that each son should first concede to the other the amount he is clearly not entitled to. For Son One, that means conceding $200 to Son Two ($1,000-$800), and Son Two would concede $600 to Son One. What’s in dispute, then, is that remaining $200, and the Talmud argues that the two should split that portion equally. Thus, Son One ends up with $700 instead of $800, and Son Two gets $300 instead of $400.

We can apply that approach to a two-person cab ride, which is like the divided estate problem flipped on its head. Each rider would have to concede he’ll pay the amount he would have paid for riding separately, minus the total savings. So if B and C were splitting a cab and paying $9 total instead of $14 for riding separately, C would concede he’d pay at least $4 ($9 minus his $5 fare) and B would concede nothing. The $5 savings in dispute would then be split equally – the same answer as the bargaining model.

But for three or more people, it gets a bit more complicated. To solve the problem, you need to consider all three possible pairs of two riders from the group of three, imagine them haggling over the savings, and come up with an overall solution – modeled after the Talmud’s teaching on the divided estate – that works for all three negotiations. The math was formalized in a Nobel Prize-winning paper by Robert Aumann and Michael Maschler 20 years ago, and is too involved for our purposes here, but the result is this: If the surplus is less than half the total cost, split the savings equally until one of the riders gets back half his original fare. Then he caps out, and the fare is split equally among the rest until the next rider reaches half his original fare, and so on. In our case, with fares of $1, $5 and $9, A would end up paying 50 cents, B pays $2.50 and C pays $6. (An earlier version of this article incorrectly stated the amount each rider would pay in this example.)

Prof. Nalebuff says this solution is preferable to one that divides the excess money proportionally. “We only have one notion of fairness in this world, and that notion is equality. The normal notion of proportional division is treating each dollar equally. This treats each person equally.”

Of course, economic models usually need some tweaking to apply to reality. One complicating factor with the shared cab ride is time: All riders after the first one likely will have at least a slightly longer trip than they would have had they ridden alone. How, if at all, should they be compensated for their time? One approach would be to calculate how much their lost time is worth, and subtract that from the total savings as well as from the amount each person is expected to pay. If you take that approach, don’t share a ride with Bill Gates unless you’re going to drop him off first.

Conversely, perhaps your fellow passengers’ company is so good that it’s worth letting them take a free ride with you.

Jonathan Gruber, a professor of economics at the Massachusetts Institute of Technology, didn’t suggest a specific solution but instead pointed out the difficulty of arriving at a single answer because of individual preferences. “The key question is whether the person going the farthest would have made a different choice in the absence of his co-riders,” Prof. Gruber wrote me in an email. “Suppose that the guy would have taken the bus if he was by himself, but with co-riders to share costs he might rent a car. Then it is a bargaining situation: each person is better off if they share the costs of the car, but they would each like to pay as little as possible. In that situation once again there is no ‘right’ answer – it depends on social norms and bargaining strength.” In other words, if Economist A is good at pretending she didn’t really want to come along for the ride, she could get away with paying less; and if Economist C is Economist B’s boss, then B may pay more than his share.

The biggest complicating factor about trying to apply a scheme to this shared cab ride arises when some of the passengers aren’t economists, and might prefer to pay more to spare themselves the complexity and social awkwardness of seeking the ideal solution.

“Talking about money amongst friends – that may not be camaraderie,” says Dr. Swoboda. “Several people would probably rather pay the extra cab fare and not have the economist next to them figuring out how much extra everyone should pay.”

He recalls when he was a graduate student in Berkeley, Calif., living in a house with four people. Some of the bedrooms were much nicer than others, and the roommates puzzled over how to adjust rent shares based on which bedroom each one got. “It was something I thought was interesting to talk about for hours, but everyone else was just like, ‘Hey, let’s just figure out our rooms,’ “ Dr. Swoboda says.