This rule of thumb applies to the simplest everyday decision: which way should I go if I am in a hurry to reach my destination?
Take the first available turn.
Easiest example: You are on the corner of a city sidewalk, at an intersection controlled by a traffic signal, headed for some destination on the catty-corner block. To reach your destination the most quickly, you should cross the first way that becomes available.
There are corner cases where this isn’t the fastest way to your destination – trivial example, there is no crosswalk at the second crossing in one direction – but generally, this rule will get you to your destination the fastest.
Easy example: You are traveling, on foot, across town. Assuming your town is based on a rectangular grid, and that there are no known street closings. You should always take the first available crossing that is on your way.
There are a number of assumptions that are forced on the problem by the city streetscape. 1) The two turnings are equivalent – the streets are regular, whether I walk west on Walnut, or north on 12th, I can proceed at the same velocity toward my goal. 2) The rectangular grid of the city streets means that I travel the same distance whether I walk west first and north second, or vice-versa. 3) The traffic signals introduce a waiting period when I am not moving toward my destination.
The waiting time is the only difference between traveling north or west. I travel a constant distance and my speed while moving is fixed. To reach my goal the fastest, on average, I seek to minimize waiting. At each street corner, I take the first available turn.
Hard example: Remove the grid, remove the regularity, and consider only that I am constrained to move so that some component of my movement is toward my destination. (I can’t go past it or away from it. I know no particulars of what streets are closed, what streets are clogged with protesters, which road curves around the mountain, or which rises over it. To reach my goal the fastest, on average, I seek to minimize waiting. At each corner, I take the first available turn.
Hardest example: Remove the walking, remove the roads, and let the destination be anything – a hope, a dream, any goal at all. I know nothing except the direction from my current position. To reach my goal the fastest, on average, I seek to minimize waiting. At each decision point, I take the first available turn.
Can I prove this? Well, no. I think I could prove the easiest case, I might be able to work for a while and prove the easier case, I might be able to simulate the hard case, and the hardest case – no.
Still this is a pretty reasonable rule of thumb.
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