Hotelling’s Game, or Why Gas Stations Have Competitors Nearby
mindyourdecisions.com
mindyourdecisions.com
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.
This is incomplete. Negative advertising a) motivates the already-motivated, but more importantly it demotivates 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.
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...
This suggests that each race may have a seperate Nash equilibriam based on the canidates and the electrate.
Are there many "swing seats"/"marginals" in US politics, or does gerrymandering pretty much rule that out?
In MME/MMP it's a valid strategy to move away from the centre and to "capture" fringe votes before they reach a more centrist party.
Maybe the people who don't vote on the location of a hot dog stand simply are not interested in hot dogs, so their votes are irrelevant anyway.
Also, wouldn't "not everybody votes" only be a problem if the non-voters would happen to have a unique opinion? As long as their desires are reflected in the votes of the other voters (which seems statistically likely), it wouldn't make a difference?
The point is not to stop everyone from turning out, or to stop the merely apathetic from turning out, but to stop the other guy's voters from turning out.
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.
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.
After that, I noticed it everywhere I went.
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!
I have no idea how accurate this actually is, but it seems to have at least urban legend status among doctors.
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.
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.
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.
I agree that the starting move I proposed was sub-optimal. That was an arbitrary move I chose to get the idea in motion.
That algorithm is optimal in a greedy sense--going for the most immediate gains with no memory or foresight. I think this is where the game starts failing as a model for reality. The optimal game algorithm would probably have enough look-aheads to force the other players to the other side, and then take the 'majority' 1+ by moving to that same side, but closest to the 0, giving it 1 + an infinitesimal market share on the other side.
In reality, there are moving costs and one does not simply "win" when holding a majority market share. Perhaps a redefinition of the game is in order?
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 :-).
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)?
I wondered whether parity would be a big deal, but -- purely qualitatively and by eye -- it doesn't seem like it is.
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.
1) when squeezed, the middle player will move to just "outside" of the opponent further from a wall 2) otherwise, the player in the middle prefers to be in the middle of the middle (to hedge against new players or take advantage of stupid existing players) 3) an outside player in a good position will move to be directly next to the middle player 4) an outside player in a bad position will temporarily choose to be the middle player, and then /immediately/ choose between strategy #1 and #2
The only rule in this situation that causes player to not be directly next to another player is #2. The two outside players will then either a) invoke rule #3 to put all three players next to each other or b) invoke rule #4 to move all the players closer together. The middle player, when squeezed, just uses rule #1 to move next to another player.
In essence, it is my belief that there will be no stable configuration of the players for this game (unlike the two player case, where "next to each other in the center" is stable), but that at all times the players will be clustered together while they shuffle themselves.
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.
"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.
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'
So now I'm wondering why this is only obvious once mentioned, and to what degree this sort of market/real-world ignorance features in various decisions about stating a business. You often don't know what you don't know, so how does one avoid these sorts of blinders?
One answer: try stuff where you can fail fast and (hopefully) cheap.
Try "is 100% responsible for"; if (in CA, USA) I decided to defy or ignore the zoning bureaucracy and open a gas station (or any "bricks and mortar" business within city limits), I think the 'crats would apply "corrective measures" (up to and including jail time) within days (if not hours or minutes). Although in practice, they'd notice long before that I was missing the multitudinous appropriate permits, and bring my efforts to a crashing halt.
As to why this most obvious explanation was missed: maybe the authors have thus far in their lives avoided contact with such bureaucracies?
I don't often see consecutive stations on the same side of the highway/road --tho I do occasionally (19th ave in SF, for example).
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?
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.
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.
I also remember a radio station's competition where the petrol station that sold fuel at the lowest price won the competition. In the final hours of the competition, there were two or three stations selling petrol at 1c/litre... and they lost to the station that was paying customers 1c/litre...
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