In the paper "Walk versus Wait: The Lazy Mathematician Wins", the authors Chen and Kominers examined the following problem

A wants to get from the point 0 to the point d. A has two choices: either it can travel at the constant speed v₁ starting immediately, and arrive at point d after time d/v₁, or it can wait a random time t₀ and start travelling at the higher constant speed v₂, arriving at point d after time d/v₂ + t₀. the random time t₀ is chosen according to a probability density P. A prefers to arrive as early as possible, yet in the case of arriving at the same time, prefers to be able to travel at speed v₂. What should A choose?

The slighly complicated argument in the paper boils down to simply the following: what is the value of

P( [0, d/v₁ - d/v₂] )

If that probability is greater than 1/2, then the strategy to adopt the second option will give a higher chance of getting to the destination earlier than the first option.

This simple solution (which the paper didn't give, as the authors made some implicit assumptions regarding the profile of the probability density) is, as always, a result of a simplistic formulation of the problem.

The physical motivation for the problem is the following:

A mathematician is walking from point 0 to point d. There are n+1 bus stops along the way, situated at points d₀=0, d₁ ... d_n+1=d. Assume that the bus can travel at a uniform speed v_bus, and that the mathematician can travel at a uniform speed v_m that is slower than the bus. The mathematician wants to get to the destination as quickly as possible, but also want to walk as little as possible. Given a probability density P of when a bus will appear at the origin, where should the mathematician get on a bus?

The first observation is that it is always better for the mathematician to get on at the origin than at any other stop, since any bus that he catches at a subsequent stop can be caught at the origin as long as he waits. (The implicit assumption that buses do not skip stops is used here.) So the problem now becomes whether to walk the whole way or to wait for a bus, and the solution I've already described above.

However, the problem soon becomes more interesting if we make it more complex:

• For one, while we can assume the mathematician is fairly out of shape, we cannot assume that he is an immobile blob. In the case of a city bus, it is quite plausible that for short distances, the mathematician can in fact run, and possibly overtake a previous bus. And so, the assumption that "any bus that he can catch at a subsequent stop can be caught at the origin as long as he waits" is no longer true (and we have to consider a probability density P that extends to negative time as well, to consider the possibility that the bus has already arrived and left).
• Also, the problem gives explicit value preferences for the mathematician: rather be early, and rather be lazy. In real life, it is more likely that the preferences should be a sliding utility curve. For example, suppose the mathematician is heading to school to teach a class, then the utility function for arriving before the class time is 0, while the utility for arriving after the class starts is minus infinity. Or if the mathematician is going out on a date, then the utility function for arriving early is again 0, while the utility function for arriving late blows up exponentially. Similarly we can quantize the amount of laziness: riding the bus all the way would have 0 utility, while riding the bus for n-3 stops while running the first 3 stops would have a certain negative utility that is still better than walking all the way.

So let's make this into a problem of conditional minimization. Suppose the bus would show up at time t_bus at the origin (which can be negative) (for argument sake, let's assume the time interval between successively buses are sufficiently large that we can assume there to be only one bus running for all eternity. Multiple buses will change the result, but the basic method of argument will still be the same; we'll leave it as an exercise). Also suppose that there are infinite number of bus stops, so that the mathematician can hop on to the bus whenever the bus and the mathematician are at the same place.

We define a few things

• First we need a function for energy expense per unit distance measured by instantaneous speed. i.e., suppose the mathematician is traveling at speed v, let E(v)dx be the infinitesimal energy expanded at that instant (of space); in terms of infinitesimal time, the energy expenditure is vE(v)dt = E(v)dx. Assume traveling on the bus incurs no energy use.
• We also need a utility function for energy spent. It should be a decreasing function of energy (more energy spent, less utility; because the mathematician is lazy). Call it U_E.
• There is also a utility function U_T for tardiness. It should be a decreasing function of time of arrival.

Let v(x) be the velocity of the mathematician (depending on where he is). We can divide the trip up into two parts, the part he is running and the part he is on the bus. He catches the bus as point x₀ where

x₀/v_bus + t_bus = ₀∫^x0 1/v(x)dx

The total time of arrival then will be

T = d/v_bus + t_bus

if x₀ < d (the mathematician makes the bus) or

T = ₀∫^d 1/v(x)dx

if x₀ > d (the mathematician misses the bus). The total energy expended will be

E = ₀∫^x0 E(v(x))dx

where the integral goes up to d instead if d < x₀. Now, letting t_bus vary according to the probability density P, we have E and T are now random variables. So the proper question to consider is the following

maximize over all smooth velocity field v(x) which satisfies
₀∫^d E(v(x))dx < E₀
the quantity U_E(expectation value of E) + U_T(expectation value of T)

The integral condition over the allowed velocities is a finite energy condition: the mathematician does not have an infinite energy reserve; he will get tired and throw up at some point.

Of course, this becomes much more complicated and I won't attempt to solve it. The point is to illustrate that the simplistic conclusion drawn is undeserving of the rather grandiose title of the paper. When a problem gets simplified to a certain degree, it no longer applies to reality, and the solutions and intuitions drawn thereof should not be taken to be conventional wisdom. Furthermore, given the more complex formulation given here, it should be possible to write down a probability distribution P, and utility functionals U_E and U_T such that the preferred method of transit would be to walk or run for at least part of the way, which reflects well with empirical evidence of rush hour during the summer in New York City.