Sharing a Taxi

Two passengers P1 and P2 wish to share a taxi from a common origin A to two different destinations B and C, to save on the cost of two separate journeys from A to B and A to C. Passengers P1 and P2 both travel together from A to B, then passenger P2 continues from B to C. Assume that the fares a, b, and c for the various journeys are proportional to the distances between the points A, B, and C, as shown in this diagram:-

A B C b c a

The fare saved from sharing is (b + c) - (a + c) = b - a.

How should this saving be shared between the two passengers?

The equitable division is to share the saving equally, as this division would be preferred by both passengers to all others.

The fares f1 and f2 to be paid by passengers P1 and P2 are therefore

f1 = c - (b - a) ⁄ 2
f2 = b - (b - a) ⁄ 2

The shares s1 and s2 of the total fare to be paid by passengers P1 and P2 are
s1 =
2c + a - b / 2(a + c)
s2 =
a + b / 2(a + c)
.

These sharing formulas give appropriate results when applied to some specific examples:-

A B C b c a

When a + c = b:-

f1 = c ⁄ 2
f2 = c ⁄ 2 + a

s1 =
c / 2b
s2 =
a + b / 2b
.

A B C b c a

When c = b:-

f1 = (c + a) ⁄ 2
f2 = (c + a) ⁄ 2

s1 =
a + c / 2(a + c)
= 50%
s2 =
a + c / 2(a + c)
= 50%

In a typical example, when a = 7.5 km, b = 23.2 km, and c = 16.2 km:-

f1 = 16.2 - (23.2 - 7.5) ⁄ 2 = 8.35
f2 = 23.2 - (23.2 - 7.5) ⁄ 2 = 15.35

s1 =
2×16.2 + 7.5 - 23.2 / 2(7.5 + 16.2)
≅ 35%
s2 =
7.5 + 23.2 / 2(7.5 + 16.2)
≅ 65%.

The distances a, b, and c are the actual distances travelled by the taxi, not the Euclidean distances ("as the crow flies").
The variation of s2 can, however, be visualised using Euclidian distances. In the following diagram, a and c are varied while b is fixed, and each contour indicates a constant value of s2.
If c + a > b + a, it's better to travel from A to C to B instead of A to B to C. So only values of c ≤ b need to be plotted. The 50% contour for c = b is a circle. The other contours are Cartesian ovals. Each contour is the locus of points satisfying c + a(2s2 - 1) ⁄ (2s2) = b ⁄ (2s2).
If c + a > c + b, it costs more to share the taxi than making two separate journeys. This is avoided when a ≤ b and B is inside the red arc, where a = b. If only one taxi is available, however, the two passengers can still share it and split the increased fare using the same formulas.

Christopher B. Jones, Sydney, January 2018

Note that this article does not apply to all taxi fare schemes. The NSW Taxi Council, for example, allows multiple hiring, where dropped-off passengers are charged, and the meter reset, after each leg of the journey. It does apply, however, to the NSW Taxi Council's shared ride.