After an emotionally challenging post last time, I hunted around for a more light-hearted puzzle for this week. When looking for puzzles (which is a puzzle itself), sometimes the neatest ones pop up in the most unexpected places. I found the idea for today’s discussion in the archives of the Wall Street Journal!
The puzzle involves how to share a community action. Suppose three people are leaving from a meeting to go home. They live in the same general direction and so decide to share a cab. (This is the example the author used, but the same issue could come up in many ways.) What is the fairest way to split the cab fare? Assume Alice lives closest and normally pays $1. (Hey, this is from the author. I know you cannot even get in a cab for $1!) Bill lives farther away and normally pays $5. Chris lives the farthest and normally pays $9. In addition to the ridiculously low rates, also assume that the two additional stops add nothing to the fare.
The combined fare then is only $9 instead of the $15 they would pay if they rode separately ($9 + $5 + $1 = $15). How should they split the net savings of $6? Naively you might say since there are three people, simply divide by 3 and give each $2. However, this means that Alice not only did not have to pay, but actually earned a dollar by riding. Is that fair?
From the various fares, we could estimate the relative distances and the equivalent time spent in the cab. Maybe we look at the hourly wages of the three (assume they are equal) and compensate each rider for the time wasted sitting in a cab. To compute this, I said there is a total of 15 units of time (or distance) that goes into the net of $6 saved. Dividing by 15 gives $0.40 per unit. This means that Alice gets $0.40 back for a payment of $0.60. Bill gets $2 back, which reduces his payment to $3. Chris gets the biggest share at $3.60, bringing the final to only $5.80. Is this a reasonable method?
But wait. Suppose Alice and Bill do not like Chris getting that much and decide to share a cab with only the two of them. Then Chris is a big loser in the sense of not saving anything while Alice and Bill still save something. Their combined saving is only $1. If they split it, then at least Alice had done better. This begins to have some of the complications of the pirate puzzle presented a few weeks ago.
Can you think of a better way to distribute the savings?
This type of problem can arise with roommates trying to decide on a fair split of rent when not all the accommodations are equal. A self-protection scheme is often employed when a group dines at a restaurant. You know that no matter what you order, or what your preferences are, there will be overwhelming social pressure to simply divide the total bill by the number of people and everyone pays a “fair” share. So you break your diet and order dessert to protect your interests, even though you do not want it.
Who do you ask for advice on the taxi problem? The author of the original article, Carl Bialik, did an obvious thing. He presented this puzzle to a selection of noted economists and asked them to analyze it. The result was predictable: he seems to have received more answers than economists. Their rationale for each method of splitting the savings ranges from ones similar to my attempts given above to an analysis based on the Talmud — and that one looks rather good! There is only one problem: that method only works well with two passengers. It gets really complicated to extend it to the three passenger problem. Those of you who like movies will be amused to see that one class of solutions uses theories from John Forbes Nash, Jr. (familiar to anyone who’s seen A Beautiful Mind).
This problem of splitting a cab fare goes back and forth between pure logic and human considerations. For instance, suppose you do not like riding with at least one of the other two; should you get a bit extra for your discomfort? Since other people are leaving the meeting at the same time, cabs might be hard to get, and they are more likely to pick up a group expecting higher fares. Therefore, can Alice claim a higher than usual split because the other two would not have caught a cab without her? What about a tip? Since tip is usually proportional to the fare plus a constant, each of the riders should be on the hook for different tips instead of sticking poor Chris with the disappointed driver.
Then there is the obvious question: How much effort is this worth? There is a reason the restaurant tab gets summed and divided by the number of people. It is easy, and since the enjoyment of the social gathering was part of the expense, the food was an extra. Of course, the waiter vastly prefers to write up and collect a single check instead of going to each person individually.
So what is your take on the best distribution of the savings? You can look up the original article and see what the economists said. My own feeling is that Alice should ride free and Bill and Chris share the remaining savings of $5 equally. So Bill only pays $2.50 and Chris pays $6.50. I could live with Bill only getting $2 and Chris getting $3. That has the advantage of not breaking up any bills. How did I come to this conclusion? I quickly saw several alternatives and decided that a single closed-form solution is not possible, so I started with Alice getting a free ride. She cannot complain about that. Then I could have applied the Talmud method to the two remaining passengers (and incidentally get the same answer), but I just put myself in the place of the people and asked what I felt would not feel unfair.
Finding a fair solution is sometimes more difficult than selecting one that is not obviously unfair. Finding the best solution is often not a well-defined concept.