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Why Ice Cream Sellers Stand Next to Each Other Instead of Spreading Out (sjsu.edu)
101 points by lionhearted on Oct 16, 2009 | hide | past | web | favorite | 72 comments

Three men went off on a sailboat together, a physicist, a chemist, and an economist. Unfortunately they ran into a storm and the boat was wrecked on an uninhabited island. The only food they were able to rescue from the wreckage was a case of baked beans. As they got hungry, they began to wrestle with the problem of how to open the bean cans. The physicist said "I'll climb a tree and throw a can onto a rock and it'll split open." The others didn't much like this idea because they thought the beans would just splatter everywhere. The chemist said "We can soak the cans in salt water and they'll rust through." The others didn't much like this idea because it would take too long. Then the economist said "Hay--no problem, we'll just assume a can opener."

Economics should always be applied to real life with many caveats. It's been a while since I took my semesters of game theory but here's some brief problems with real life applications.

One dangerous assumption in the hoteling model is that there are no costs to changing your position. That is, the two parties will keep relocating until they're at the Nash equilibrium (neither party can benefit from relocating). This is just plain wrong in real life. Political candidates lose a lot of credibility when they reverse stances on something. It is true that after primaries, candidates then to become more moderate but most other applications of hoteling are dangerous. (Edit: same goes for changing marketing stances, product positioning, pricing... etc.)

Removing relocation/repositioning costs illuminates another unrealistic assumption: there are only two players. If you have huge costs to relocating then you don't want to allow a competitor to come in and take your half of the market because he positioned himself right next to you (i.e. M, N, O). If you add in these two assumptions, players might actually want to have an agreement to separate such that any new entrant would not be able to gain as much profit as they are currently making (while still each controlling 1/2, (i.e. at .33 and .66 on the spectrum or something).

A third assumption is that it is a one dimensional world or that consumers judge choices only along one spectrum. If you add in that another competitors could compete not just on the "beach"... Or if a competitor could differentiate itself on a topic that consumers care about more (not all spectrums made equal)... Or spectrums change, the median consumer could shift from one year to another.

I could go on but the point is to not assume the can opener. In entrepreneurial terms, maybe all the competitors out there are "hoteling," could that be justification to be different? Purple Cow? Or maybe it is better to make a few consumers love you than many to just like you.

Actually, the first major assumption in this particular model is that the demand will walk to the nearest outlet. Everything else - adding a second dimension, relocation costs, possibility of entry, first-mover advantages, prices, product quality, etc. - can be easily added to the model - just add a bunch of ratios and coefficients* to your equations.

*The second major problem which this model does not suffer though is that when your model gets closer to reality, its predictions become highly dependent on many coefficients the true values of which you do not know most of the time.

Science is made of small reproducible pieces. Yes, real world is more complicated, but what the article gives is a pertinent explanation to a certain case. When we have enough such "certain cases" completely understood we go to combining them, and then we have something useful. If you always work with the real world case you may get quicker results, but it will be statistics, not science.

>> One dangerous assumption in the hoteling model is that there are no costs to changing your position

location cost is not really necessary. think of this as a game that the ice cream sellers play before they locate. they figure, where do i place my hut to get the most business? it's at the center, no matter how many operators there are.

>> another unrealistic assumption: there are only two players

doesn't matter how many players there are. suppose there's one ice cream guy. where is he going to put his ice cream hut? in the center. let's say another one shows up. where is he going to put his hut? in the center. the logic repeats.

>> A third assumption is that it is a one dimensional world or that consumers judge choices only along one spectrum

yeah, this is a model for a perfectly competitive market. ice cream is a good example, as well as gas stations. this model is probably bad for products with lots of differentiation.

>>> doesn't matter how many players there are. suppose there's one ice cream guy. where is he going to put his ice cream hut? in the center. let's say another one shows up. where is he going to put his hut? in the center. the logic repeats.

No it doesn't. With three sellers in the center, the one in the middle is close to very little people. He's in the worst possible spot and he'd better move anywhere else. So that's no longer a stable layout.

By the time you have walked half way along the beach to the icecream stalls, the distance between them will be negligible. If I walk up to three icecream stalls, I'll be right infront of three icecream stalls, so location won't matter.

I'm not sure how this aspect applies to the political senario, which is the more interesting one.

In politics what it usually means that while there can be advantages for two parties to tend towards the political centre, they always tend to keep enough distance between themselves to be clearly differentiated. The same logic still applies more or less - although the further towards the extremes the parties get, the larger the gap between them will tend to have to be.

> I'm not sure how this aspect applies to the political senario, which is the more interesting one.

That's more difficult. For in most voting systems you have to take strategic voting into account. E.g. in a winner-takes-all systems people don't want to waste their votes, and thus the system tends towards two parties.

How about moving to the edge in the center? Then he'll capture the whole side until someone else claims his place, rinse and repeat.

> doesn't matter how many players there are. suppose there's one ice cream guy. where is he going to put his ice cream hut? in the center. let's say another one shows up. where is he going to put his hut? in the center. the logic repeats.

That's not true. Let me explain. In this model (1 dimensional spectrum, be it distance or whatever), if a third player entered the market and the other two players are located at the center then the new players rational choice is to be on either side of the incumbents. He therefore sandwiched the middle player and stole all of his customers, the middle player will then move to the outside and again a player will have been sandwiched. The cycle repeats.

But if you add in costs associated with relocating (time or revenue lost or rebranding or moving costs) then this no longer is an effective strategy. A player might realize that this would happen and purposely not locate in the middle in order to avoid a fight. I don't quite remember the solution to a three player model but that was also based on an assumption that there are only three players.

It gets highly complicated when there is the possibility of new entrants. The lesson could be watch out for new entrants and stay nimble (so it is cheap for you to react when there are new entrants).

To me the most striking/exciting way to extend this model and what assumption to question is the zero sum assumption.

If the equilibrium is such that both ice cream sellers stay near the median, we might also imagine that the # of customers available to both sellers depends on how far they have to walk. I don't think it's unrealistic to imagine that the # of willing customers decreases as distance to the nearest seller increases. So in the case described by the article, some of the people on both ends of the beach might decide agaist buying ice cream at all, given they have to walk half way across the beach.

This is exciting to me because it might indicate that if both sellers were to cooperate (@ .33 and .66 distances along the dimension) both sellers could be better off in the aggregate because the potential number of customers they are splittig could be larger than at the nash equilibrium described in the article.

Expading the scope beyond even the sellers, we might also imagine that as a result the entire system is better off because the average customer saves time walking by having more evenly spaced ice cream sellers. That means even if we were to imagine the demand function was completely inelastic with respect to distance to nearest seller, at the very least the customers save time on average walking to the stands. This time could more productively be used to dream, plan, create the next startup, come up with the next big idea for society, etc.

The value of cooperation, expading the pie, and not assuming a zero sum game...as an optimist, these are the themes I take away from something like this.

As a meta-comment, I liked your points but please hold the "humourous" story opener - we shouldn't need tricks to be engaged or convinced by your argument.

Not all rhetorical tools are tricks. Analogies can be informative when taken with a grain of salt. Presenting an analogy a joke is especially honest because invites the reader to not take the analogy too seriously. So I don't think the opener was a trick in a negative sense at all.

While the walking distance is not optimal, there is an upside. The ice cream vendors will be driven to differentiate their ice cream lines, and the beach dwellers will be able to comparison shop without walking halfway across the beach.

Hm, while reading the article I noticed an interesting aside of this theory, similarly to yours: Assume it is correct, then your need to compete will make all the difference it takes: Have double the amount of helpers so as to reduce waiting times (in the political analogy one could double the supporters to increase "fishing" returns). Differentiating the ice cream lines is in the same league. What this effectively means is that given an existing ecosystem of companies, a new competitor needs to combine both: near spatial placement within the ecosystem w.r.t. to the competitors with the advantage of having superior products/services.

What I am asking myself how this affects the computer science industry, i.e., in our business spatial-nearness is not necessary and for our customers superior technology is usually judgemental; any ideas?

If you take web business for instance, all businesses a certain person knows about are near each other (in the address bar), so you have to actually differentiate your product/service to sell it.

If only they developed an on-line store and a delivery mechanism. It would be the optimal solution from the customer perspective (price, feature and distance). Thanks in part to the ubiqutious internet.

You might be making a joke, but you also might have something here. If you can get GPS coordinates localized down to 5 meters, then you can have a mobile app that will let you place orders in places like public parks and on the beach. The app would look up vendors in the same location who are currently "open", and present virtual storefronts for them. The app would send the name, the precise GPS coordinates and, at the option of the user, a photograph to aid in finding the customer.

On the vendor side of the app, you'd have a list of orders and a map of the customers. Done properly, this could be a big win for vendors, since ordering would be easier, increased volume would be likely, and they would be better able to plan production/delivery.

Basically, it's like Loopt but for hyper-local b2c. Instead of hooking up singles or friends, you're hooking up cart vendors and their customers.

I remember watching a program about venture capitalists and the internet bubble and some experienced VC talking about how many millions he had blown trying to get a scheme somewhat along these lines up and running. Maybe the increased prevalanced and usability of mobiles might make it more feasible, but it a massively non-trivial undertaking - certainly trying to do it from on top. Some sort of organically growing website that vendors/customers use might work, but it is one of those chicken/egg problems - unless you have customers coming to the site it is a waste of time for the vendors to have to deal with the trickle of orders from it, and until a large number of vendors are present customers will get little use from it.

One way I could see working is a widespread existing business collection/franchise spreading out into more and more products over time - so companies/organisations as diverse as interflora or starbucks might start by offering products in their core competancy online, maybe start allowing other companies to join the scheme to cover gaps in their coverage, and eventually spread out to cover a wide range of goods so the website/app grows beyond the original founder.

Start in a single concentrated location, like Central Park or so.

yes, that was entirely my point.

The application to politics is especially relevant: "Anthony Downs noted that Hoteling's model could explain political competition. ... As with the ice cream sellers on the beach the political candidates will choose a political position that is virtually the same as their opponent's. Furthermore the candidates will be driven to select the political position of the median voter."

This jives with gut intuition, that in the end the differences between Republican/Democrat aren't really that great.

Would love to see this as an interactive applet or widget where you could adjust the locations and quantity of sellers (or politicians).

It depends on the voting system too, though. This doesn't happen so much in systems that are not "winner takes all". In Italy, you can happily vote for the hammer-and-sickle communists (or the extremists on the other end) and they'll pick up a few seats in parliament; it's not like their representatives need to win in any one district - it's enough to get a percentage of the national vote. This has both benefits and its own problems, as well.

But to rule the country you need half of the votes, and no matter how many parties a country has, alliances form between them as to present themselves to the population as a viable alternative to the parties currently in power. People who dislike the current government do not want to throw their vote away on parties outside those two main alternatives - changing the alliance in power takes precedence.

We have 7 political parties represented in our parliament("riksdag"), but in effect you can only choose between two alliances who each has joint policies. Which have now become about the same by the process described in the article. So the choice we have is between the guys who want to lower the tax by 1-2%, or the ones who want to raise the tax by 1-2%. The rest of the difference is rhetoric.

In Italy (but not just Italy), there are very real philosophical differences between the parties in the coalitions. On one side you still have real, unrepentant communists, on the other side, you have Alessandra Mussolini active in politics.

What often occurs is that the big center left/right party has, say, 45%, and needs a minor party (that polled, say, 6%) to push them over 50%. The minor party then gets to hold the bigger party hostage: we want this, this, and this or else there is no deal. This gives the minor party a share of power all out of proportion to their actual percentage of the vote.

Not only are the differences between Republican/Democrat not that great, but in the last 50 years they have largely swapped constituencies!

It started when JFK pushed civil rights. This made Democrats popular with Northern civil rights activists. Nixon took advantage of this with his "Southern strategy" that used the issue as a wedge to get Southerners to vote Republican for the first time ever. (Previously the South wouldn't vote Republican because they held the Republicans responsible for the Civil War.) The swap cascaded from there as each successful candidate got there by appealing to what had once been the other party's base. (Reagan took a particularly chunk, hence the phrase "Reagan democrat".)

The swap completed with George Bush.

This is already well studied and modeled as the median voter theorem.


Nice. Also looks like Probabilistic Voting Theory works in N-dimensional space: http://en.wikipedia.org/wiki/Probabilistic_voting_theory

Many browsers and OS combinators suffer problems when Java applets are automatically launched.

Please add a [JAVA] tag in future.

Huh, you're right, there is Java in there - I had no idea, it looked like simple image files. Couple questions for more savvy people -

1. Why would the author do that for static images?

2. Is there any way I can see that going forwards to warn people? I'm running Chrome/Win7, and didn't even notice.

I looked up the source and there it is -



But it really makes no sense why the author would do that - it's just a straightforward, somewhat boring image. Thoughts?

Perhaps it was an attempt to stop people from "stealing" his images.. or perhaps the author just doesn't know any better...

We should probably apply Hanlon's razor "Never attribute to malice that which can be adequately explained by stupidity." and assume he doesn't understand progressive enhancement


They're quite valuable images. I'm going to steal them for my myspace page.

Perhaps it's supposed to be interactive .. interestingly I did try and move the spots but then conccluded that it was just an image. Maybe it's a public beta ;0)>

Many browsers and OS combinators suffer problems when Flash objects are on a website.

Please add a [FLASH] tag in future.

There's some discussion of this in Starbucked, i.e. why Starbucks stores open across the road from each other. Part of it is if people are already going to that location for coffee, having an extra Starbucks and shorter lines will incite them to go more often; the illusion of choice. I don't really recall the rest of the reasoning as it seemed somewhat spurious, but the fact quoted in the book, that when a Starbucks opens across from another they both do better business than the solo shop, is quite remarkable.

Downs has done some remarkable work in systems theory. He famously observed that the equilibrium between motorists and transit users (and between peak and off-peak drivers) adjusts when you add lane capacity. The Downs-Thomson Paradox states that adding more peak driving capacity draws a corresponding number of people out of transit and into personal vehicles, causing overall traffic to remain the same or worsen.

Related question: why do big hardware stores, like Home Depot and Lowes, always seem to be next to each other? Same with Walmart, Target, and KMart. Same with gas stations.

It seems like it'd be better to space them out, especially if you're the inferior business. They can't just get up and move like the ice cream scenario. Why do they do it that way?

Well - there are also things like zoning laws that come into play in that case.

But what happens when there are three ice-cream sellers?

There's no stable equilibrium. The ones on the edges are constantly incentivized to move towards the center, but at some point the guy in the middle is better off moving anywhere else. Kind of like the "divide the dollar" game.

I doubt that position is important if the 3 are directly adjacent one another, or at least not as important as say the appearance of your store front or your prices or your quality or your reputation or your display of well known logos ....

If I walk all the way down the beach 3m further to get to a nicer looking icecream shop is nothing. Indeed I think I'd look at the price of the icecreams first, then the ones people are leaving with and make a cost analysis, then probably choose the one selling pistachio flavoured icecream.

3 miles or 3 meter?

There should probably have been some punctuation: if I walk down the [whole length of the] beach, 3m is not much further.

Metres, not "mi" miles.

Thanks. I am not too familiar with the non-SI systems.

Actually, the ones on the edges should spread out a bit, they will reduce the competition for those coming from the ends of the beach who will now have to walk further to get to another shack, and they still split the difference between themselves and the next shack. How MUCH they should spread out and what more competition will do is unstable.

Sure there is equilibrium. Differentiation on price points, quality, service and length of lines will balance things out.

That said, the difference between a third ice cream cart and a 30th ice cream cart in the area is great. But then, the market should correct itself so that the optimal number of ice cream sellers are available.

I've been thinking about that. I don't know what the eventual distribution would look like, there doesn't appear to be a stable solution. But at the beginning, he should locate at a point one-third of the way from the center to an end - at that point 41% of the beach is closer to him than to either of the original competitors.

The three get together, but the middle one get no clients.

And the middle one moves to the end of the line, grabbing half the clients. Now, the problem is the same but someone else is in the middle, who then must relocate himself, ad infinitum.

Two of them get together and manage to keep the likely thirds marginal?

The one in the middle forecloses on his house.

In the pg essay "It's Charisma, Stupid" http://www.paulgraham.com/charisma.html he says Opinions seem to be divided about the charisma theory. Some say it's impossible, others say it's obvious.

I was in the "it's obvious" camp, but I had a sense of deja vu as I was reading the essay. I have the vague feeling that I found it obvious not because I had figured out on my own, but because I'd read about it elsewhere.

This article brought it back to me. Back when I was in high school, some popular math book that I read (don't remember which one) had this as an example of how simple mathematical reasoning can explain questions of great social importance. I remember being impressed.

The other half of it is that many commentators* have noticed that charisma is an important factor in presidential elections, even if they didn't say it was the only one. So I guess at some point I'd put two and two together in my brain, which was why everything in the pg essay looked familiar.

Anyway, that was a long-winded way of saying "thank you for posting this!" :-)

*Scott Adams is one. I recommend reading his blog if you don't already; guised as humor you will regularly find deep wisdom.

There is a game theory course on itunes U that covers this more in depth during one of the first lectures - worth taking a look at if anyone is interested in delving deeper into this sort of thing.

Ah, yes! that's the course, thank you!


I don't have itunes on my work machine or I would list the link. I remember it being the only course specifically on game theory at the time.

If your competition is selling ice cream, current market pressure forces you to innovate in a way that allows you to stand out.

Therefore having balloons or a fireworks show may attract people to your stand. Carry "healthy" only 8 brand frozen yogurt.

You need to create a market black hole that sucks people's attention to your product and it's gotta be a rich enough experience to keep their attention, for as long as you can.

Dense neutron star hot fudge should help pull folks in.

Our local town has about 6 Mobile phone shops all in a row, in the same street - something that's quite common to towns.

I guess the other point is that it encourages customers to shop around. If they can see 6 shops in a row, they'll likely go into a few of them and see if they're getting the deal they want.

If those 6 shops were all over town, each one would get far less custom, since a visitor would likely only go in one of them.

They'd get fewer customers through their doors but not necessarily less customers buying.

Sure, so I guess there's 2 alternatives:

1. Be all over the town, and rely on marketing to get people to come to your shop and not the others.

2. Be all in the same street, get more walk in customers, and rely on good salespeople and good deals to get the sales.

I'm guessing that they've found 2 to work better than 1. Also means they can take advantage of each others marketing.

Also I expect once a street becomes 'known' for something, it'd be silly to set up a shop anywhere else for that thing. For example Tottenham Court Road is where you go for Computing/tech shops in London, so that's where most people go if they want tech.

Does this explain the disproportionate prestige of .com?

> Does this explain the disproportionate prestige of .com?

It's part of it. Here's a few other reasons:

1. It's a quality indicator: It means it's more likely that the company has been around a long time and that it's not a scam.

2. People's default surfing habits had them looking for the .com. I had a .net domain for a company I ran and we bought the .com for $3000 after a number of clients told us they forgot our site, and worse - sent emails to the .com address that we never got.

3. This is a more recent development, but with the iPhone having a ".com" button on the keyboard, that makes it even more advantageous going forwards.

Summary: position is, er, a positional good.


I wonder if it's related to the following phenomena:

An antique seller has a a shop downtown. A second antique shop next door, and his business drops. Then a third antique shop opens across the street, the neighboorhood becomes 'the antique district', and business booms for all three.

No. Two different conclusions. The only similarity is that at some point there were two vendors.

Otherwise known as The Price is Right Asshole Bidding Theory: the same bid plus one.

Care to elaborate?

In TPIR, people have to try and guess the price of an item, and whoever comes closest without going over wins. So if I bid after another player, I can bid one dollar more than him, effectively stealing his bid (unless he happened to guess exactly right). People do this quite frequently; it's not uncommon to see the fourth and final bidder bid one dollar higher than the highest of the three other bids.

Of course this works at the top and bottom - so you should also bid one dollar lower than the lowest if you think it is lower than everyone thought. This is all only true if you are bidding last, any time before that you should be honest* to give the later players the toughest choice of going higher or lower than you.

If you go last and think it is between two other bids you chose 1 higher than the lower bid (because of the too high bids being ignored).

*I think in fact because of the high bids rule there is always an incentive to shave your bid to err on the low side, but it is difficult to quantify what size this effect is.

Actually, if you think everyone's bid was too high, you bid $1, since the goal is to get as close as possible without going over. Like blackjack.

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