
Hotelling’s Game, or Why Gas Stations Have Competitors Nearby - madiator
http://mindyourdecisions.com/blog/2008/03/25/game-theory-tuesdays-hotelling%E2%80%99s-game-or-why-gas-stations-have-competitors-nearby/
======
jacques_chester
The application of Hotelling's Law to public choice theory breaks down in
voluntary voting scenarios because the full spectrum of voters is no longer
present.

It _does_ work, roughly, when voting is made compulsory.

Australia, where voting is compulsory, has quite a quite pedestrian, quite
retail sort of politics. There's sloganeering and accusations of skullduggery,
but most of the pitch is usually quite unrhetorical in its format. Policy
debates are closely aligned on the median voter, and both major parties work
tirelessly to position themselves in that centrist position.

The USA, where voting is voluntary, has a mix of soaring rhetoric and
absolutely maximised negativity.

The difference is that in Australia, you're looking at the people who are "on
the beach". In the USA, the goal is to _deter the other guy's customers from
turning up at all_ , while ensuring that yours do. Hence the mix of beauty and
bile.

Edit: removed surplus apostrophe. The unutterable shame.

~~~
orky56
Let's say the US population breaks down to 50% Democrat and 50% Republican. If
I am a D candidate, I want the Ds to show up and the Rs to stay at home and
same if I were R. If I defend the Ds too much, the Rs take it offensively and
vice versa. Assuming they both use equivalent tactics to harm the other while
improving their own popularity, how does that not lead to a Nash equilibrium?
BOTH populations are riled enough to support their chosen candidate (and
prevent the other from winning) or be apathetic enough to stay at home (and
let the other candidate win). With voting, candidates cannot be satisfied with
just 50% of those who voted i.e. become a commodity and split it. They need to
use the fact that belittling someone has a more visible reaction that just
praising themselves. The balancing act that BOTH candidates need to play leads
to the Nash equilibrium. But really the fact is that the population is not
50/50 and there are regional/demographic differences which just makes
candidates' strategies that much more complicated.

~~~
jacques_chester
> Rs take it offensively

This is incomplete. Negative advertising a) motivates the already-motivated,
but more importantly it _de_ motivates the less motivated.

Essentially, your goal as the Democrat is to make the Republican moderate say
"a pox on both their houses!"

And yes, it's more complicated than a simple straight line and a fifty-fifty
split. We're talking about median positions floating in hyperdimensional issue
spaces (the most interesting times in politics are the catastrophic jumps from
one local minima to another). But the simplified model has surprising
explanatory power.

Edit:

I didn't properly address your argument which, if I read it correctly, is that
perfectly symmetrical strategies will cancel out in a perfectly symmetrical
race.

That's true, as far as it goes. But the implementation varies, the candidates
vary, the electorates vary (especially in the USA where you have
gerrymandering -- over here electorates are carved out by an independent
commission).

Nevertheless, the beauty-and-bile strategy fits the circumstances better than
beauty or bile by itself. It's a minimum viable strategy.

~~~
showerst
One interesting quirk of the US is that the two main parties have strongly
differing strategies with respect to turnout at the state/national level.

The Republicans have a stronger 'base' that is more likely to turn out, while
Democrats traditionally capture more undecided voters, but have a smaller
base.

What this means is that in elections with lower overall turnout (Non-
presidential year house races, for example) tend to favor Republicans, and
their game-theoretic response is to generally 'Go after the Base' and swing
more conservative (E.g. Reagan, Bush II, adding Palin to the McCain ticket,
and the US House races in 2010 and 1994).

Democrats tend to have much more impetus to 'get out the vote' and get
undecided voters to show up to the polls, even if they risk adding Republicans
to the ranks, so you tend to comparatively more centrist Democrat Candidates
(Clinton, Kerry, Obama). They also tend to do better nationally in
Presidential Election years with high turnout (Ceteris Paribus).

If you have any interest in how this plays out in the actual nuts & bolts of
voter targeting, Hal Malchow's "Political Targeting" (2nd Ed) is the Bible for
beginners.

[http://www.amazon.com/Political-Targeting-Second-Hal-
Malchow...](http://www.amazon.com/Political-Targeting-Second-Hal-
Malchow/dp/0615184618/)

~~~
jacques_chester
Thanks for elaborating, I'm (obviously) less familiar with US politics than
Australian politics.

------
flomo
As the story goes, McDonalds had a very sophisticated system for finding
locations. They were constantly doing geographical analysis, looking at
development plans, and so on. Meanwhile Burger King did not. Burger King would
look where they were opening a McDonalds and try to place a store nearby.

Eventually Burger King figured out that the McDonalds' locations were
generally more prominent and easily accessible than their own. For example, a
McDonalds might be convenient to rush hour traffic, while the nearby Burger
King was on the wrong side of the freeway or required a U turn to access. They
may have also realized that, when given the choice, more people prefer
McDonalds. Their copycat approach was hurting sales.

Eventually, Burger King built up their own location-finding capabilities and
started locating stores in places where McDonalds was not.

~~~
jacques_chester
A simple rule of thumb for fast food, petrol stations and other "drive by"
businesses is that they will position themselves on the side of the road
dominated by "homeward" traffic.

Most folk in the morning are anxious to get to work on time.

But on the way home, you can catch them on an impulse.

It's not an ironclad rule, but look around and you'll see what I mean.

~~~
nowarninglabel
Having worked closely with a Mcdonald's franchisee for some time, I'm pretty
sure your observations are merely anecdotal. If you look at Mcdonald's
revenues by average franchise, and adjust for number of hours served,
breakfast & dinner revenues are roughly equal (in the simplest case just
measuring rush hour breakfast revenues of 7-9am with rush hour dinner revenues
of 5:30-7:30pm).

~~~
jacques_chester
I'm willing to be wrong on this. It's based on my observations in a few
different cities I've lived in.

~~~
Flenser
Were those observations mostly done while you were driving home hungry?

~~~
jacques_chester
I was driving towards a CBD one day low on fuel and noticing that all the
petrol stations were on the other side of the road.

After that, I noticed it everywhere I went.

~~~
westicle
Anecdotally the petrol station thing aligns with my experience, and my
preference for filling up on the way home.

Slightly different situation for Mcdonalds though - they're trying to market
convenient, fast breakfast food to morning commuters. If the location is most
convenient to city-bound traffic, I think it would have a bigger positive
impact on morning revenue than the corresponding negative impact on evening
revenue. People are in a hurry in the morning!

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smeatish
OK, now add a third hot dog stand. What's the nash equilibrium now?

With 3 hot dog stands, they have an incentive to spread out - if you're in the
middle of the other two, then you move to the outer side of one of the other
two to capture everyone on that side of the beach. This repeats - there is no
stable equilibrium.

~~~
eric-hu
Assuming the 3 hot dog stands start out with spacing to allow for equal
coverage, their positions are -2/3, 0, 2/3. Everyone gets 2/3, 1/3rd on each
side.

The two on the outside have an incentive to move towards 0 because they can
take more market share without losing any. The one in the middle does not want
to move to either outside area until moving means he can have more market
share than what he currently has.

If the rightmost player decided to move inward from 2/3 to 0.5, he'd have 0.75
market share (out of 2). The middle player would have 1/3 + 1/4 = 7/12 or
0.58. It still wouldn't be in his interest to become the rightmost player, as
his upper market share limit would be 0.5 (a little less).

However, there's already an incentive to move _to_ the right player's
location, causing the locations to be (-2/3, 0.5, 0.5). The left player gets
1/3 + (2/3 + 0.5)/2 = 1/3 + 1/3 + 1/4 = 11/12 or 0.91. The remainder, 1.09, is
split equally among the other two to 0.545 each. Only the left player has
incentive to move at this point, since each of the right players stand to lose
the right-side market by moving inward, or losing half the left-side market by
moving outward.

The leftmost player has incentive to move inward now, and can do so until he
takes enough market share from the other two that one of them can move to him
and gain more. If they all did this and ended up at 0, they'd again have an
equal 2/3rds. However, anyone can move slightly to one side or the other and
increase his market share to nearly 1.

Looks like you're right. I would say there's "equilibrium behavior", though--
the 3 players will oscillate between the boundaries [-2/3, 2/3], with someone
frequently taking the same position as another.

~~~
saurik
> If the rightmost player decided to move inward from 2/3 to 0.5, he'd have
> 0.75 market share (out of 2).

This move would not have been made by theoretically optimal players: if he had
moved from 2/3 to 0+ instead he would have had 1 market share (out of 2).

> Only the left player has incentive to move at this point, since each of the
> right players stand to lose the right-side market by moving inward, or
> losing half the left-side market by moving outward.

The middle player actually has an incentive to move to the center of the board
for the same reason that the original player in the two player case had an
incentive to choose the middle of the board (despite having monopoly).

> If they all did this and ended up at 0, they'd again have an equal 2/3rds.
> However, anyone can move slightly to one side or the other and increase his
> market share to nearly 1.

In this configuration the two players on the outside would actually each have
~1, leaving the player in the middle ~0. If we call these positions using
formats like 0-, 0, and 0+, the 0 player will move to 0++, causing either the
0- player to move to 0 and the 0+ player to move to 0--.

Now we have 0--, 0, and 0++. Now, the 0++ player will move to 0+, and the 0
player will move to 0++, leaving the 0-- player to move to 0-. The 0+ player
will move to 0--, the 0++ player will move to 0, and the 0- player will move
to 0+. "Finally", the 0-- player will move to 0-.

I am pretty certain that this algorithm does not terminate.

~~~
gjm11
I did a little simulation. Start the vendors off in random places. Each tick,
pick a random vendor and move it to its optimal place (either just left of the
left-hand other one, or just right of the right-hand other one, or half-way in
between; I assume for convenience that some second-order effect makes the
midpoint best when you're between two others, and when putting a vendor in a
new "outside" position I put it 1% of the way from the old "outside" vendor to
the edge).

The resulting evolution settles down quickly to a situation in which all three
are very near 0.5, and every time the middle one gets the chance to move it
does so (to be just on the other side of whichever of its rivals has more
space on the other side). If you plot a graph of this, you get a sort of
braided effect, with the braid never getting very wide or moving very far. So
it's kinda-sorta stable even though (necessarily) the vendors keep changing
places.

With n>3, all sorts of more interesting things happen. For instance, for n=4
the state of the world is usually as follows. You have a cluster of (usually)
2 vendors near a (<1/2) and another near b (>1/2). The outermost vendor in
each cluster is happy where it is; the inner ones will jump to just beyond the
outermost of whichever cluster is further from 1/2. This means that the
clusters tend to be equally far from 1/2 (because the nearer-in one is
preferentially jumped to) and tend to move outwards (because a jumping vendor
always jumps to just outside a cluster). But this situation can break down in
two ways. (1) A vendor jumps from one cluster to the other, so we have 3+1
instead of 2+2. Then the now-isolated vendor, on its next move, will join the
other cluster; we now have 4+0. Now what happens is that cluster moves en bloc
towards 1/2, at which point it typically splits in two. (2) The two clusters
get far enough apart that a leap into the middle becomes favoured. Actually,
in my simulation #2 never happens because #1 always happens first. For larger
numbers of vendors, though, you get a kinda-similar situation (two clusters,
one on each side, slowly moving outward) but the number of vendors in each
cluster is large enough that #2 can happen before #1 does, so sometimes you
get one leaping into the middle. (As the number of vendors gets big, this
becomes the dominant kind of transition, and the overall effect is that
generally the vendors are quite evenly spread.)

A more realistic simulation might give quite different results, but I'm at
work right now :-).

~~~
eric-hu
Interesting simulation. The braided behavior sounds like the 'equilibrium
behavior' I mentioned earlier. Does this simulation also allow for taking
another player's _exact_ location, forcing a split of their market share on
both sides? I imagine that's a subtle but game-changing move.

For the higher vendor numbers (3+), did you notice similar behavior when the
vendor parity was the same (i.e. all even or all odd)?

~~~
gjm11
No, it doesn't allow two vendors to be in the exact same spot. The amount of
trade that gets you is always exactly half-way between being just to the left
and being just to the right, and can therefore only be the best option when
all three of those are exactly equal, so I don't think it makes much
difference.

I wondered whether parity would be a big deal, but -- purely qualitatively and
by eye -- it doesn't seem like it is.

~~~
eric-hu
Very interesting observation.

In the 2 player case, not allowing for parity means that equilibrium would
never be reached. The simulation as you programmed it would approach
equilibrium as time goes to infinity, though, so I guess your model works
about as well as one with parity, and is simpler. Nice.

------
vacri
Game theory makes the mistake of assuming its the only variable. Another
reason for clustering is the congregations of similar stores attract more
business. Think of a shopping centre. Now take one of the clothes stores and
stick it on a suburban street. It's generally going to do less well by itself
because you have to know about it before you go - you have to decide to go to
that shop, rather than just show up and see what's on offer, as it were.

Perhaps a better example: I'm about to go to Vietnam. Hoi An is "the city with
all the tailors". Everyone says "spend a couple of days in Hoi An and get lots
of clothes made up". Now, there are tailors all throughout Vietnam of course,
but they cluster in Hoi An - and tourists specifically wait until they get
there to purchase clothes.

~~~
saurik
While your explanation is generally interesting, and while I'm also not
claiming that game theory is the only (or even the primary) variable in this
situation, I feel the need to point out that one does not go from gas station
to gas station and find that they came home with 30 gallons of gas after an
accidental "shopping spree" at the gas mall.

~~~
vacri
While that is true, if you have equidistant from you two locations, one with
one station, and one with four, then all other things being equal, you're
going to go to the location with four stations - more price competition, less
likelihood of queueing and the like.

~~~
saurik
Unfortunately, it could easily be the case that the mentioned price
competition outweighs the value of increased traffic due to decreased queuing,
given how incredibly small the margin on gas stations is (from what I've heard
from friends that have managed them).

------
chaz
NPR Planet Money did a similar story a little while ago: "Why Clusters Of Like
Businesses Thrive."
[http://www.npr.org/templates/story/story.php?storyId=1213048...](http://www.npr.org/templates/story/story.php?storyId=121304873)

------
goodside
The conclusion here has an unfounded moral lesson:

"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."

How do you know it would be nicer if they spread out? You'd waste money (and
carbon emissions) shipping resources to remote businesses that aren't
profitable, or at least not as profitable as they could be if they were closer
and easier to ship to. As a reductio ad absurdum, you can't build and maintain
a gas station in Antarctica just so that it would "nicer" if someone happened
to be there on vacation with their snowmobile. The reason there's no gas
stations in Antarctica is the same reason there's so few in Wyoming. It's also
the reason they're all clustered around high-traffic areas immediately outside
of major cities.

If you can't calculate from empirically established methods a "socially
optimal equilibrium" that doesn't _directly imply that we should be building
gas stations in Antarctica_ , you don't know that the current distribution is
suboptimal. More generally, there's a lesson for policymakers here: you
shouldn't endeavor to destroy established equilibria that you don't
understand.

Otherwise, kudos to the author for a pretty neat example and visualization.
People who do stuff like this are awesome.

~~~
showerst
My reading of this is that the 'socially' optimal equilibrium (that is to say,
the equilibrium for customers) is to build stations in a way that minimizes
people's effort to get to them (assuming an even distribution, it would be at
.3~ and .6~ on that line he uses), whereas the equilibrium for station owners
in light of competition is to cluster roughly in the middle.

Social optimality in this case is all about population density (finding the
minimum of the function that represents the total effort of everyone involved,
_assuming station owner profits are constant_ which is a big assumption), and
has nothing to do with forcing stations into Wyoming (unless a few million
people suddenly move to Cheyenne).

I could be reading this totally wrong though, you can define social
equilibrium an infinite number of ways depending on what you're trying to
maximize, and what you simplify to 'ceteris paribus'

------
VladRussian
zoning.

~~~
9999
I'm surprised your brief rebuttal is not floating higher, as I think it is a
far superior explanation for the phenomenon. In rural areas where zoning laws
do not account for clustering of services like gas stations and fast food
restaurants, there are many other factors that can confound a game theory
based solution to this sort of question (availability of utilities like
communications, electricity and water for example).

~~~
wisty
In China, shops try to open up near competitors. Most cities have an "electric
bike street", where all the shops in an area are electric bike shops (or
restaurants / convenience stores). Customers like choice, and hate to go to a
place where they can't compare prices. That might be zoning (I can imagine
Chinese authorities saying "OK, that street can sell product X), but I think
it's also a long standing tradition.

Here, shops _hate_ to be near competitors, and it's not unknown for them to
lobby the local council over zoning infringements that their competitors are
making. This is despite their competitors pulling in a lot of business. After
all, shops will pay many times the rent to be in a mall, proximately because
that's where the customers are, but ultimately because that's where the
competition is.

OK, a small hamburger joint needs to worry if McDonalds sets up next door. But
if another small hamburger joint sets up nearby, it will increase traffic. The
"competition" isn't the guy next door. It's the guy next door, the food strip
in the mall, and home-cooked food; and only the guy next door is pulling in
foot-traffic to your area. But guess who most business owners will try to put
out of business?

------
Eliezer
I've yet to see a Lucky next to a Safeway. Why are supermarkets different from
gas stations?

~~~
Simucal
Right next to my house (Ofallon, MO) there are two grocery stores within 200ft
of each other. This is just ancedotal but any time I've thought about this
topic I've always thought about it in relation to those grocery stores.

~~~
sesqu
I do some of my shopping in the corner store, and some in the nearby business
cluster that has, among others, 4 supermarkets. I'd guess real estate, loyalty
programs and product offerings outweigh proximity, in the case of foodstuffs.

------
impendia
The map looks cool but is misleading. There are many more gas stations in San
Francisco than that (and the prices are more variable).

------
aresant
Game theory is a well worn chapter in the internal "best-practices" conversion
voodoo guide.

EG - The OPTIMAL variant for conversion rate is actually at least TWO
variants.

This isn't perfectly in line with the gas-station example (variants aren't
dissipated, they're stacked) but it follows the same logic.

Damn hard to test and balance with off the shelf tools, but if you're at scale
this is a truth.

~~~
andrenotgiant
Game Theory + MVT? That is tantalizingly interesting, you can't just leave us
hanging! I would love to hear, (or be directed to) a more detailed explanation
of this!

~~~
aresant
Awesome - it's one of my favorite "discoveries". The day true mathematicians
enter the CRO field the rest of us are toast.

I will 100% write up a blog post on this some day - hit me up via my user name
contact details and I'll point you in right direction in the meantime.

------
latch
Anyone else think of the Price is Right when he's explaining the optimal
position?

------
icefox
Sidenote: One has to wonder why doesn't gas buddy make their map interface
better. They must realize that no matter where you look the prices overlap.

------
asifjamil
you would just hope that the competition between two adjacent gas stations
would drive the cost of gas down!

~~~
icegreentea
Gas of price has relatively little to do with the business actually operating
the gas station [1]. You'll see that on average, you have something like 20
cents going to "Distribution Costs, Marketing Costs, and Profits" at 4 dollars
a gallon. You're not going to be able to squeeze much out of that.

Unfortunately, I can't get a source to it, but from what I remember, the
margins on gas are already so thin that most profit at gas stations come from
the random other stuff they sell. I mean think about it... 20 cents (maybe)
for a gallon of gas, or like 80 cents (or more) for a can of pop.

Also, a side effect of the razor thin margins, as well as the relatively fixed
consumption pattern means that while price at the pump tracks crude prices
going up almost instantly, on the way down, every single gas station wants to
be able to enjoy their extra margins for as long as possible, so they reduce
their prices slower. And this clustering phenomenon just encourages that even
more, since the moment you undercut your competitor, they'll know in like 5
minutes or something and match you, and now both of you just missed out a
bunch of extra cash.

Edit: silly me, linky here: [1]
<http://energyalmanac.ca.gov/gasoline/margins/index.php>

~~~
joshmlewis
I've also heard this from a guy who works as a tech at a gas station. They
make way more off of lottery tickets, alcohol, and other various things than
actual gas. And he also went on about car washes, how freaking well they do
and how little maintenance is required for the things to make a lot of money.

