# Best strategy to find a parking spot

New Bounty Edit (2 days remaining on the Bounty):
To point out that the only answer given at this time cannot be considered an answer, because it simply gives a hint on how to formally model the problem, which is not what I was looking for, considering I wrote it informally on purpose. Still looking forward to some analyses of this problem!

I was wondering about the following problem.

Assume the following.

• you have to find a parking spot for your car in a very busy saturday night to go in a restaurant;
• you search for this parking spot by basically going around (literally) in the hope to get a spot;
• of course, (the saturday night is really busy) other people are in the same situation as you are and they are running in circle like you are;
• the direction of the movement is only one (again, you literally go around);
• the time frame of the problem lies between 20:00 and 00:00. Finally (of course!);
• when you start your search at 20:00 there are no free parking spots.

Question:
What is the best strategy you can use to find a parking spot?

1. Should you stop in a place and wait until one of the cars that you can cover with your eyesight leaves?

2. Or is it better to move around in the hope to find a free parking spot?

I was thinking about the following few variables that I think should essentially change the nature of the problem:

• Cardinality of the set of parking spots (countable vs. uncountable);

• Cardinality of the set of agents involved in this situation (countable vs. uncountable);

• Probability of having a car that already occupies a parking spot leaving that lot in function of time (normally distributed, uniformly distributed, etc);

• Farsightedness of the agents (extreme cases: one place ahead of you, whole circle ahead of you)

Hence, a solution should be explicit about what is assumed concerning those variables.

[Notice that the in general I assume that the space where you are looking for a spot is homeomorphic to a circle]

Any feedback as always is most welcome.

PS: As you can guess, where I live it is very (very!) difficult to find a parking spot on Saturday nights...

Bounty Edit:
As in the bounty text, I would like to know what are reasonable answers to this question (without considering as options using the bus, the tram, a bicycle or an helicopter...).

• Take the bus, instead. – Gerry Myerson Oct 7 '15 at 11:23
• By a parking lot do you mean a parking spot (that is, a place for an individual car)? ¶ @GerryMyerson: Haha. – Brian Tung Oct 8 '15 at 17:57
• Ops, indeed. I am going to edit it. – Kolmin Oct 8 '15 at 18:00
• You need to be much more concrete about the model if you want to get any good answers. The question at the moment basically says: "make up a model and solve it". – Winther Oct 13 '15 at 21:58
• @Winther: I do not really agree. What I am saying is that there are various points that should/could be relevant. A complete answer is one that takes into account all this points (if they are all relevant afterall). – Kolmin Oct 13 '15 at 22:32

Here's an attempt at a reasonable mathematical model for this. We'll suppose you're on a circular road with $N$ parking spots (numbered $0$ to $N-1$) and $N$ corresponding waiting positions (also $0$ to $N-1$) for your car. At each waiting position $x$ , you can see $m$ parking spots ahead (positions $x$ to $x+m-1 \mod N$). Yours is one of $p$ cars waiting for a spot. Parking spots become available one-by-one in random order. If one of the $m$ spots you can see (say position $y$) becomes available, you can get that spot unless one of your competitors ahead of you (in position $x+1$ to $y$) can get it. You can also move to the next available position. Your competitors have the same abilities.