Justice

Suppose I take a taxi with a friend. She gets out when the fare is $3 and I get out when the fare is at $6. How should we distribute the total cost fairly? One idea is that I should pay double what my friend pays. $6 = X + 2X where X is the amount paid by my friend. So I would pay $4. But another idea is that we should share the fare up to her exit, then I should pay alone after. So X = $3/2 where X is the amount paid by my friend.

Comments

Here's another idea. Figure out how much would it have cost each of you to take the taxi separately. Let's say it's $3 for her (it went straight to her place), and $5 for you (the taxi had to go a bit out of its way, from your point of view). Then each of you should pay the appropriate proportion of what you would have paid, had you taken separate taxis.

Let her be A, and let you be C. Let AJ be her fare for your joint ride. Let AS be her fare had you ridden separately; in practice, AJ=AS, since the fact that you are in the cab won't change the route (unless you ask the driver to take the scenic route, for reasons we will not discuss). Let CJ and CS be yours jointly and separately, too. Normally, CJ >= CS (going together didn't make the fare to your place less than it would have been), though we don't have to assume that, either.

Anyway, consider now the ratio CJ/(AS+CS). This is the proportion of the actual cost to what it would have cost to get you home separately. My proposal is that she should pay AS*CJ/(AS+CS) and that you should pay CS*CJ/(AC+CS). Note that if we add these, we get (AS+CS)*CJ/(AS+CS) = CJ, which means we have the right amount of money in the end.

This will save you both money, obviously, so long as CJ/(AS+CS) < 1, that is, so long as the total fare for the joint trip is less than it would have cost to get you home separately. (I.e., you don't live in completely opposite directions.) If it would have cost more, then it would not have made financial sense anyway for you to share a cab, though you might have had other reasons, even good ones, to do so. (These probably had to do with taking the scenic route.) In that case, however, it does seem as if you ought to share the additional cost, which this formula delivers.

Either way, you each save (or spend) the same proportion of what you would have spent, so this seems fair.

This cleanly extends to the case of more than two people. It also applies to any circumstance in which people share an expense and thereby reduce (in the normal case) their joint expense. This makes me suspect that this idea of proportionality is well-known and may even have a name. Perhaps something involving "proportionality".

In the taxi case, it may be hard to calculate everything in practice, of course. So now we need to consider how to get a decent approximation to the formula we had, and to do that we'll have to make an idealizing assumption. Let's suppose that your friend's place is on the way to yours, so that dropping her off makes no difference to your fare. That is, CJ=CS and AJ=AS, since we'll also and that no one is taking the scenic route. Then we can calculate your cost and hers knowing only the actual fares. In practice, you end up paying a bit more than if CS<CJ, but, so long as her place isn't too far out of the way, the discrepancy will not be large. And you'll pay less than you would have paid on your own, anyway.

So, in your example, we'd get that you pay 6*6/6+3 = $4 and she pay 3*6/6+3 = $2, which was your first proposal. So, there you go.

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