Mind Your Decisions by Presh Talwalkar

Hotelling’s Game, or Why Gas Stations Have Competitors Nearby

25 March 2008 | by Presh Talwalkar

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Every Tuesday is a Game Theory article at Mind Your Decisions.

There are hundreds of gas stations around San Francisco in the California Bay area. One might think that gas stations would spread out to serve local neighborhoods. But this idea is contradicted by a common observation. Whenever you visit a gas station, there is almost always another in the vicinity, often just across the street.

Gas stations are highly clustered. To confirm the point, I made a map with the help of GasBuddy.com, a website that compares local gas prices. On the following map, the numbers indicate gas prices per gallon. More importantly, the prices are placeholders for the location of gas stations. Note the highly grouped gas stations. Many of the prices are right on top of each other, indicating locations in extreme proximity to one another:

<click for larger image>

I don’t think this characteristic is special to the California Bay Area. I’ve been across many parts in America, and time and again, whenever I see a gas station, there seems to be another one just across the corner.

The rare exceptions to the rule are the gas stations I’ve encountered while driving in remote areas, like when I drive through the cornfields to visit the University of Illinois at Urbana-Champaign.

The phenomenon is partly explainable because of population clustering. Gas stations will be more common where demand is high, like in a city, rather than in sparsely populated areas like cornfields.

But why do stations locate right across the street from each other? Why don’t they spread out?

There are clearly many factors at play. It’s an optimization involving demand factors, real estate, estimates of population growth, and supply considerations—like the ease of refueling a station. The answer is complex, and any explanation I offer will have its problems.

But we can gain valuable insight from a simple game. Today I’ll discuss a model about location competition. It’s a bare bones model that ignores many realistic considerations, but it is useful nonetheless. It’s interesting because of the individual firm strategy, the ultimate outcome, and the implication on social efficiency. It has also been used to explain why political candidates appear indistinguishable. I’ll briefly discuss that application near the end of the article.

Hotelling’s Game

There are two players in this game. Each player is a hot dog stand that competes for customers on a beach.

The hypothetical beach is made up of straight shoreline, in which customers are uniformly spread out. We will label the endpoints -1 and 1 for convenience.

The hot dog stands compete purely on location (we ignore brand quality). Each stand will pick a location.

Conditional on where the stands locate, customers will simply choose the stand closer to them. If the stands are in the same spot, customers will pick a stand randomly.

For instance, if a customer is at point 0.5, and the stands are located at -1 and 1, the customer will be closer to the stand at 1 and go there.

Here’s a graphical representation of the game. Note the endpoints of the shore and the placeholders for each stand’s location.

<image inspired by this one>

Where will each hot dog stand end up? That is, what is the Nash equilibrium of the game?

Finding the Solution (intuitive)

One way to approach the game is to ignore the competition. Assume you are the only hot dog stand on the beach. Where might you want to locate?

The answer is easy: any place you want. You are a monopoly so customers will have to walk to you no matter how far it is. If you choose to locate at one endpoint, customers will make the trek all the way from the other endpoint. It’s nice to be a monopoly.

But you are a paranoid monopoly, and any common sense would push you toward the center, labeled point 0. You worry that locating on the far left or far right end of the beach would leave you vulnerable to competition. If you favor the left side, for instance, an entrant could locate slightly to your right, closer to the center, and capture the majority of the market.

See this image where the market share of the monopoly (in blue) is overtaken by a new entrant (in orange).

Such a problem does not happen if you locate in the center. A new entrant to either your left or right side would gain less than half the market.

The logic shows that the center is favorable. If either party chooses it, the other will want to copy. Thus, both hot dog stands will choose the center point 0.

The above logic is correct though as a mathematicians would say, not “precise.” To get a better sense, it’s necessary to wade through the next mathematical proof. I highly encourage everyone to at least skim it since the thought process is important to solving many games.

But if you do choose to skip it, start reading at the section labeled “The Social Optimum.”

Finding the Solution (mathematical)

Each hot dog stand is simultaneously picking a location, a number between -1 and 1.

Each stand needs to take into account where the other might locate. That’s the key factor in game theory—decisions are interdependent.

I’ll break the problem down into two steps. This is a process you can use to solve other games.

Step 1: Think about Payoffs

Imagine the two stands locate at points 0.2 and 0.4. How much of the beach would each stand capture?

You can see it in the following diagram. It highlights the customers closer to the first stand (blue) and the second stand (orange).

Here’s how I came up with that picture: the first stand clearly gets anyone lower than 0.2 (the left), and the second gets anyone higher than 0.4 (the right). Customers are split in the middle region at the halfway point of 0.3.

The length of the blue and orange lines represents the market power of each hot dog stand. In this example, the first stand gets 65% of the line compared 35% for the second.

You can see this is location is not an equilibrium of the game. If the second stand inches closer to the middle, say, at the point 0.2, then both stands would be equally close and would split the market 50/50. Of course, the first stand would then retaliate by moving.

There’s a mathematical way to describe how each stand would react to the other’s choice.

Step 2: Writing the Best Responses

The best response is the location that one stand would choose optimally, subject to the other stand’s location. The other stand’s location is not known, but rather a hypothetical number from which to work.

To simplify matters, instead of dealing with market share percentages, let make the game win/lose. Imagine a hot dog stand “wins” the game by having a majority of the market.

Suppose the second stand chooses a location k. What is the best response for stand one? What are the locations that will capture over half of the market?

There is not just one correct answer. As we reasoned in the intuitive section, anything closer to the center will capture a majority of the market. These are points that are less than distance k from the center. These are exactly the numbers between the values –k and k on the line.

If k=0, and is the center, then the unique best response is to pick the center.

The best response is described by the following piece-wise defined correspondence (with i,j representing either stand one or two):

The Nash equilibrium is when both stands are both playing a best response. This only occurs when both stands choose k=0, exactly in the center, splitting the market. You can interpret this as either both winning or both losing.

The Social Optimum

Game theory tells us what players will likely do. But that’s unfortunately not always going to be the best for society.

The equilibrium is an annoying situation for many customers. The hot dog stands are located in the center of the beach instead of spreading out and being closer to beachgoers.

In fact, if the stands could be spaced out across the entire beach (at points -0.25 and 0.25) then everyone would be happier. Both stands would still share 50% of the market, but no customer would have to travel more than 0.25 hot dog stands.

Why isn’t this social optimum sustainable?

Lacking enforcement, there is an incentive to deviate. One of the stores could put itself to the middle and gain more customers. The other would retaliate of course, and the game would go on until both would end up exactly in the center once again.

Gas Stations and Other Examples

The model suggests why competitors always seem to locate so close to each other and compete on real estate. Think about big burger chains, supermarkets, and video stores. You will almost always see them clustered even though it would be nicer if they spread out.

The model also has been applied to political candidates. Imagine two candidates picking a platform on a political spectrum from -1 (very liberal) to 1 (very conservative). If voters pick the candidate closer to their views, and voters are spread out across the spectrum, then both candidates would converge to the middle. It’s no surprise that politicians seek the “average vote.” It also suggests why it’s so hard to tell the difference between candidates during the campaign trail.

I have another application. I think a similar concept occurs in TV news. These channels compete for attention, and they can choose to pick stories that appeal to any number of customers. It would be nice if they talked about different topics, but that doesn’t happen. We end up with the same story being reported in virtually all news outlets instead of having hundreds of different important stories being reported.

That’s why the web has been so liberating. Entry is cheap, so information stands set up all over the spectrum. You can get news stories on whatever topic you are interested in. It’s kind of neat.

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