How to find cheap gas using game theory
[Update 2-14-09] I’m on vacation; articles will resume in a week or two, or as soon as I recover 
I usually find cheap gas based on experience. In my vicinity, I am aware of gas stations that sell for good prices. A few pennies here or there does not bother me as it is the average that matters. When I travel, I often ask a friend or I just suck it up since it’s a one-time expense.
But that is going to change. I learned of an ingenious solution from William Spaniel, who makes great instructional game theory videos.
Here is William’s story:
I ran into an interesting problem today, and I thought you might enjoy what I realized the solution was.
My friends and I were driving back from Las Vegas to San Diego. About half way through the trip, we needed gas. We weren’t worried about suddenly running out, but we definitely couldn’t make it to two towns ahead of us–we absolutely had to get gas in the town we were about to pass through. There were five exits to the freeway in this city. The goal was to get gas for as cheap as possible.The problem was that we knew virtually nothing about what the price of gas should be in this town, and we weren’t keen on going around in circles until we found the best one around.
Then, I realized that you had covered a similar problem about a year ago. Instead of cheap gas, the problem was to find true love.
The solution to maximize your probability of success was to reject the first few suitors and pick the first of the remainder who is better than those you had gone out with previously. I suggested we try a similar strategy with the gas stations, as it has the same property. We breezed by the first two (which were selling for almost $3.50 each) before stopping at one with a much more reasonable price of $2.95. As it turned out, $2.95 was the best station available.
I then explained to the other people in the car that we just solved a math problem. They may or may not think I am full of you-know-what.
I am definitely going to try this strategy! It is amazing that in statistics you can sample to find the best and save much time on your search.
Here is a quick cheat sheet on the rule (full details in how to find true love):
|
Number of gas stations in vicinity (N) |
Number of gas stations to reject (k) |
|
4 |
1 |
|
5 |
2 |
|
10 |
3 |
Note that the strategy works even better because, as game theory predicts, gas stations locate in clusters.
What about you guys? How do you find cheap gas at home or while traveling?
(And remember to subscribe to William’s game theory videos to show your thanks)
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12 Responses to “How to find cheap gas using game theory”
I’m only generally aware of game theory, so it’s possible I’m missing something key here, but I assume the idea is that if you believe you have N number of gas stations ahead of you, you’ll skip the first (however many) and then pull in. What if you can’t see how many there are? I know there’s *a* gas station; how lucky do I feel?
Not sliming the article, which *feels* right — just wondering how the logic works out.
By bill on Feb 9, 2010
Good question. I occasionally grab the first gas station in a rural area–who knows if there will be more? In a city, it is useful to make a guess. A heavily trafficked area may have 3 to 4 which is a strategy of skipping the first station. This works most of the time, but when it doesn’t, you may wish to circle back if the price difference is large enough.
By Presh Talwalkar on Feb 9, 2010
Thank you.
By bill on Feb 9, 2010
To confuse the issue a little, petrol stations adjust their prices according to many factors and some of those factors are relevant here.
I have noticed that the first petrol station (I’m from Australia) that you see after leaving the freeway is usually the most expensive in the whole town.
The second most expensive is usually the last one before returning to the freeway.
The psychology behind this is that if a motorist can see the freeway then they know this is the last one and that there will be no more petrol for dozens (if not hundreds) of miles. If the last petrol station is not within sight of the start of the freeway (or at least a large section of empty road leading to the freeway) then they won’t put their prices up quite as high.
The first one after leaving the freeway is capitalising on those people who simply go to the first one they see after the warning light comes on and those people who are so low on petrol that they aren’t sure if they’ll make it to the next petrol station at all.
This seems to bolster your strategy even more because by following it you will never visit the first and last petrol stations and these are nearly always the worst ones available.
As a final note, petrol stations that are within sight of each other are usually more competitive, as long as their price signs can be seen from any point along the road.
By Dave on Feb 9, 2010
A further complication — stations frequently don’t set their own price, if the station is owned by a conglomerate or the oil company. Two stations selling the same company’s oil can have a two or three cent per gallon difference. As I understand it, maximizing revenue is the name of the game.
By bill on Feb 9, 2010
Back when I first read the “Love” article, I was very intrigued by the concept. I was pondering situations I might find myself in where I’d be able to apply this strategy and one of those was this very idea: gas stations. Like Bill, I wondered about the best strategy when the number of options isn’t known. I also considered the points regarding the known pricing scheme of having the higher prices nearest the freeway, etc.
I do question what Bill meant by saying “A further complication…” as that, to me, seems to be a completely irrelevant comment. The reasons for a price being what it is doesn’t matter, unless that affects its placement (as it does in the near-exits case). The point is, there are a list of ‘unknown’ values and you have to pick the best.
A couple questions: I am wondering: when looking at values on the above table, should one round up or down for in-between values? I’m thinking I should round up, but want to know if that’s accurate. 5=2, 10=3, so…7=3?
Also, after reading the Love article, I tried creating a Monte Carlo simulation to approximate the above values, or even the 1/e solution, but didn’t get that results. Are there any other factors that are needed when doing such a thing?
I greatly love these articles and was particularly interested in this optimization stuff. Thank you.
By V Paul Smith Jr on Feb 9, 2010
It’s marginally relevant (no pun intended) because it means that the price isn’t being mandated by someone on the scene with local expectations of risk/reward. If the distant owner thinks it overall profitable, for whatever reason, that first price could well be the best, just to deny profits to all of the others.
I do admit, though, that the reverse would be more likely.
By bill on Feb 10, 2010
But you, not know which plan is in effect, have to take them as a pseudo-random sampling of prices and treat them as such.
If you have certain reason to believe that a given one is better than others, then yeah, go for it. If I, as cupid, line up a bunch of potential matches for you, in a certain specific order, and ask you, the game theorist to pick the best one, then using the above mentioned plan would net you the best odds. It doesn’t matter that I arranged them the way I wanted. If you don’t know that order then it doesn’t matter. If you do, then that is a whole different issue. Of course you take a specific one if it’s known to be best.
Or…I am completely misunderstanding your point. If that’s the case, sorry.
(I’d still be interested in somebody addressing any of the questions I raised in my original post.)
By V Paul Smith Jr on Feb 12, 2010
First, here is a Wikipedia article about this problem:
http://en.wikipedia.org/wiki/Secretary_problem
Unfortunately, gas stations don’t really work the same way. In particular, the pricing is distinctly nonrandom. Also, you don’t care only about picking the best one. You care about the actual cost. So a 90% chance of picking the second cheapest one is probably better than a 37% chance of picking the cheapest one.
By Andy on Mar 5, 2010
V Paul Smith: To answer some of your questions:
–Yes, one should round up in the love game. Solving the game discretely one has no trouble. If I recall correctly, the continuous approximation is a minimum so one needs to take the next integer.
–If you’re doing Monte Carlo simulation, you might need to do many trials! This is a case where there is pure noise so perhaps 10,000 simulations may be good? Would have to check to be sure.
By Presh Talwalkar on Mar 9, 2010
Andy: I would say the same is true of the secretary problem: it is better to have a 90 percent chance of picking the second best applicant than a 37 percent of picking the best.
I think the general principle helps though: one needs to have some sense of comparison shopping while not wasting too much time. So skipping the first or a couple (or delaying these hires) is a good way to gauge the market.
By Presh Talwalkar on Mar 9, 2010