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Where do you build your lemonade stand?

There is a 1000 meter stretch of pristine beach.
It is serviced by 3 lemonade stands, of which you are the owner of one of them.
Each morning, you decide at which point you want to set up your stand, and it has to stay fixed for the rest of the day.
The beach patrons are equally distributed along the beach. They will favor the lemonade stand that is closest to them.

  1. If you are the first to arrive, where would you want to set up your stand?
  2. Is it the best to be the first to set up your stand?
  3. Does the above answers change if there is more competition for business?

Note by Calvin Lin
2 years, 5 months ago

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So this place is in Helsinki, Finland, with google maps coordinates

I'm going to use the blue line to show that the area between the beach and the blue line is only feasible.

Since the beach petrons are uniformly, and I assumed tourist are from other countries, I'm going to serve Simas

Ok, the answers::

1) GREEN CIRCLE. I will either pick the orange region or the green region or somewhere in between those two. Zooming in the green region: we see

So there's plenty of cars, which means the there's highest traffic of people there despite having a uniform amount of beach patrons as people would either be exiting or arriving to the beach, either way they will be thirsty for some Simas. Not to mention I can obtain customers below the GREEN REGION who rides boat. And also, if I open my stand next to the roadside (see map), and customers don't have to walk down to buy them.

If I'm in the ORANGE CIRCLE, I would only get tourists who stay put for a long time (sunbathing) to buy my Sima. Since the number of patreons is uniform everywhere, it's less likely for the same customer to quench their thirst with another Sima, unless it's that good (Diminishing marginal utility).

Business should be the best in GREEN CIRCLE!

2) Yes, you get the customers who can't get Simas from anywhere else. Win-Win!

3) Definitely. because (I think) the locals who go there by car (thus arriving at the GREEN CIRCLE) regularly would probably already know who sells the best Simas. And without competiting with other stalls, I would move to ORANGE CIRCLE or closer to Ourit and sell them at a higher price for the tourist, and still making profit from the tourist, whereas the GREEN CIRCLE still has plenty of competition between stands, locals will be definitely prejudice towards one stall. Either way, I'm not taking the risk (risk averse) to be under prejudice, so I will be in ORANGE CIRCLE. If I'm a risk taker, I would probably still stay in GREEN CIRCLE. Pi Han Goh · 2 years, 5 months ago

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@Pi Han Goh Its amazing !I will never open a business with you( @Pi Han Goh ) around. Kalash Verma · 2 years, 5 months ago

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@Pi Han Goh Nononononono. I was done trolling for the day ... The picture was supposed to inspire interest ... Calvin Lin Staff · 2 years, 5 months ago

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@Pi Han Goh Not only will you get he most money, you will cause more people to congregate around you which results in more money. This causes more people to trust you for good lemonade, compared to others. Your method influences the human psychology. I hope this thinking doesn't bend the rules. Sharky Kesa · 2 years, 5 months ago

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@Pi Han Goh Yep. Even I searched for the pic in Google Maps ! +1 Azhaghu Roopesh M · 2 years, 5 months ago

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This is also a game theory problem, in which the problem is underspecified: are the other two people attempt to minimize our profit, or attempt to maximize their own profits without regard to how the other two are doing, or some sort of mixture of both, or any other objective? Ivan Koswara · 2 years, 5 months ago

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@Ivan Koswara You never know what other people will do when playing a game .... Calvin Lin Staff · 2 years, 5 months ago

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@Calvin Lin Game theory needs assumptions on how the other players react. (It's also assumed that each person's objective is common knowledge; a player's strategy is adapted to what the other players' objectives are.)

If you don't specify any, then I assume worst case: conspiring to make you fail. In which the answers are trivial...

Question 1: Nowhere; you will never be profitable. If you set your stand on \(x\), the other two can set up their stands on \(x-\epsilon\) and \(x+\epsilon\), guaranteeing your loss.

Question 2: Likewise, you should never choose to be the first to set up your stand. In fact, it can be shown that you will never be able to get more than \(250\) meters:

  • If you set up your stand first, see Question 1. You're not guaranteed anything at all.
  • If you set up your stand second, first player plays at \(500\). WLOG you play on \(a < 500\) (if you play on \(> 500\), flip the beach). Second player plays on either \(a - \epsilon\) or \(a + \epsilon\), whichever cuts your area more. Since you have an area of at most \(500\) meters, and this can be cut in half, your guaranteed score is \(250\) meters.
  • If you set up your stand last, the other players play on \(250\) and \(750\). You can verify that you can't get more than \(250\) meters.

Question 3: As Question 1 states, you should never play early enough; you have to be one of the last two people to set up your stand.

On the other hand, here's a comparison to if both other players are maximizing their own scores, without regard to what the other players get.

Question 1: Even as first player, you can actually guarantee \(250 - \epsilon\) meters. (I think you can get more when moving latter, but this is enough to show how different assumptions make different results.) Set up a stand on \(250 - \epsilon\). The second player won't play anywhere below \(250 - \epsilon\), for the simple reason that playing on \(750\) is strictly better: if the third player then plays below \(750\), the second player is guaranteed \(250\) (the part to the right), while if the third player plays above \(750\), the second player is also guaranteed \(250\) (at least \((500,750)\)). Ivan Koswara · 2 years, 5 months ago

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@Ivan Koswara Actually, you don't even need the strong assumption that they are conspiring to make you fail. All that is required is the (reasonable) assumption that everyone is rational and out to maximize their profits.

The irony is that if you're the first person to place a stand, and you place it at (near) the center, then when the 2nd person comes along, he will choose to place his stand right next to yours (choosing the smaller side) and then the 3rd person will come along and place his stand next to yours on the larger side.

This is a game which benefits from "backward induction analysis", where we try to understand how the third person will act under any placement of 2 stands, so as to determine how the second person will act under any placement of 1 stand.

What other strategies are there? Is the first player doomed to not make any money at all?

Hint: Remember the game of "100 jewels amongst 5 pirates", how much can the first pirate get? Calvin Lin Staff · 2 years, 5 months ago

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@Calvin Lin And of course I throw it to another mathematical community...

Assuming players only maximize their profits, the response was that everyone can guarantee anything lower than \(0.25\), but I'm not confident each player can guarantee \(0.25\). Ivan Koswara · 2 years, 5 months ago

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@Calvin Lin Speaking of which, regarding the pirates problem, note that in the pirates problem the assumed objectives are clearly indicated: pirates' objectives are to survive; if they guarantee survival, they want to win as many gold as they can; between outcomes of the same amount of gold, they want to see as many dead pirates. If the objectives are not stipulated, how would you know what the best strategy is? Maybe all pirates' objectives are to obtain strictly everything without regards to their own safety (in which everyone but the last two pirates are doomed); you'll never know, if not stated. Ivan Koswara · 2 years, 5 months ago

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@Calvin Lin

All that is required is the (reasonable) assumption that everyone is rational and out to maximize their profits.

This is what I mentioned at the end of my comment, stating that it's possible to get at least \(250 - \epsilon\) when you assume they are only maximizing their profits and not minimizing yours. (For two-player zero-sum games, the two concepts are identical, but not so for more than two players, hence why I asked.) Ivan Koswara · 2 years, 5 months ago

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