The limits of Chinatown

The limits of Chinatown – Part one

 

Remember that time when you lost so badly, thinking to yourself that it was probably the worst defeat ever? Or when you won by epic proportions, wondering where this would place you if there was an official high score list for the game?

If video games can do it, why not board games?

 

Short of creating an official page where people enter their score, one thing we can do is simply to determine the lowest and highest possible score one can achieve in a game. This will give us at least the upper and lower boundaries.

 

But more than that, maybe we can simulate complete games and see what the score distribution should looks like. This could give us a rough idea of how really epic was your victory!

 

To explore this idea, I decided to start with a nice little board game that I discovered during the last year (via this magnificent review on Shut up and sit down).

Chinatown

 

If you ever dreamed of opening a small shop on a busy street corner and launch into a business empire of your own, engulfing real-estate, and recklessly trading with other local businessmen to control entire city blocks. This may be the game for you!

Chinatown is a game aiming to emulate the business euphoria of building your local empire of business in the booming New York’s chinatown of the fifties. 

It is an interestingly social game since, aside the randomness element introduced by drawing tiles and cards, most of the game has to do with trading. So forget about looking at your cards and elaborating tortuous strategy alone in your corner, this game is mostly about direct interactions with other players!

And if you look at it from a business perspective, everyone is making money!

You can trade shops and location, gambling on their future value, but here trading with other player means that both of you will improve their situation. So it can be a very positive experience!

 

Where are the limits of Chinatown ?

The point of this game is to make as much money as possible. But once you start playing, you realize that the maximum amount of money in the game is fixed, and can be easily determined.

So beside calculating a maximum for the sake of it, I thought this could help answer the following question:

How good a businessman are you?

But also:

How good at business as a group are you ?

Success depends on previous preparation, and without such previous preparation there is sure to be failure.

– Confucius

 

What to expect in this post

In this first part, I will:

  • Give a quick overview of the game, in order to give some context to the analysis.
  • Determine what are the Limits of Chinatown. Finding out what are maximum and minimum amount of money that can be made in this game
  • Then I will revisit the game, but from a cooperative point of view. How much money can be made, but as a group?

As an aside, I’ll try to give some insight on how I achieved this, and what are the difficulty from a computational point of view!

This will set the table for the follow-up of this post:

  • What is the expected score at Chinatown, based on simulating game for player using basic strategies
  • What luck should look like when playing Chinatown.
  • I’ll try to see how we can use probabilities to gain insight on tile selection.

The game

 

Chinatown has surprisingly few elements, so it is fast to teach.

 

The Chinatown board is composed of 85 shop locations divided in 6 city blocks.

Simplified version of the chinatown board

There is 12 different types of shops:

3 of max size 3     –    3 of max size 4    –    3 of max size 5    –    3 of max size 6

Chinatown shop types

 

Shops are built incrementally by placing store permits of the same type adjacently to each other. Here are some of the different legal shapes for any shop of size 4.

Different allowed shapes for a shop of Max Size 4

 

At the end of each turn you receive money according to the shops you own. The bigger the shop, the better, with a substantial bonus for shops that have reached their maximum size.

  • The maximum size of each shop is clearly indicated on each tile.
  • The number of tiles for a given shop is equal to its maximum size plus 3. (So 6 tiles for shop of max size 3, and so on).

This means that only one player can fully build a shop of a certain type, except for shops of maximum size 3 where there can be 2 fully built shops .

If a player put more shop of a type than the maximum allowed size, it will be considered as two distinct adjacent shops.

 

The core of the game

Chinatown is essentially a trading game.

Each turn, players receive store locations and shop permits that they have to trade with others to obtain contiguous spaces to build the larger and more paying stores.

When a player place a shop permit on one of his location, we consider this location built. The shop cannot be moved, or removed, but can be traded away (shop permit + associated location). The location in themselves do not make money, only built shop do. So you can keep any location unbuilt for future use, but at the expense of current return.

Aside from the tile distribution, nothing is left to chance. The mechanic to know how much each store will give you is straightforward. All is left to the players is to trade locations, permits and cash, gamble on future the values of those items, and to try to get the most money from others in their transactions.

It is the perfect game to see which of your friend is a shroud businessman!

 

How it plays

  1. Each turn the players receive n-emplacement cards, of which they have to keep m of them.
  2. Once they have made their selection, those new emplacement are revealed to others.
  3. Then each player draw k shop tiles that they place in front of them.
  4. Negotiation: The players can exchange locations,shop tiles, money or anything they want.
  5. Build phase: The player can decide to build shops on the emplacements they own, using their permits. (Shop ownership can be exchanged, but built shop permits cannot be removed)
  6. The payout. Each player receive cash according to their built shops.

Repeat this for 6 turns… and that’s it!

Let’s analyze the game superficially

 

Quick Chinatown facts

The Chinatown board is composed of 85 shop locations divided in 6 city blocks.

There is 12 different types of shop. For 90 permits, so more permits than locations.

If we were to completely fill the board with shops this would give us:

  • 6 completed shops of maximum size 3
  • 3 completed shops of maximum size 4
  • 3 completed shops of maximum size 5
  • 3 completed shops of maximum size 6
  • 7 incomplete shops of size 3 (using the 3 remaining tiles from completed shops)
  • 1 incomplete shop of size 1 (also using remaining tiles from completed shops)

With 5 extra tiles left not placed…

One such solution…

Filling the board with all the shops in Chinatown (board game)

But not everything is distributed in a real game

Locations and permits distributed during the game vary from turn to turn. But it also depends on how many players are in the game.

 Notation remark:   (7+4+4+4+4+4) should be read as:

7 tiles on the first turn,  4 tiles on the second turn,  4 tiles on the third turn, etc…

 

In a real game the following tiles are distributed

Shop permits given, according to the number of players:

  • 3 players: (7+4+4+4+4+4) for each player = 27 *3 = 81 shop permits
  • 4 players: (6+3+3+3+3+3) for each player = 21 * 4 = 84 shop permits
  • 5 players: (5+3+3+2+2+2) for each player = 17 * 5  = 85  shop permits

Store are built by placing the shop permits on the board. But to place a shop permit, you first need to own that location.

New locations kept for each turn in the game, according to the number of players:

  • 3 players: (5+4+4+4+4+4) for each player = 25 * 3 = 75 locations
  • 4 players: (4+3+3+3+3+3) for each player = 19 * 4   = 76 locations
  • 5 players: (3+3+3+2+2+2) for each player = 15 * 5    = 75 locations

The players actually receive a bit more locations each time, but they can only keep some of them.

The number shown represent the locations they are allowed to keep, so this gives us the real number of locations that will be in play at the end of the game.

 

In case of fire, keep calm, pay bill and run.

– Random fortune cookie message

 

The limits of chinatown

So with all this being said, what are the limits of money making in this chinatown game?

This is easy to determine, because as we just have seen, chance only affect which locations and permit types the players will end-up with, but the amount of location and permit drawn is fixed.

And since the payout is very straightforward, we can simply check what is the maximum payout, without having to simulate anything… Or can we ?

Payout for shops according to size (incomplete shops / shops at maximum size)

  • 1 location                        pays   10 000$ / – 
  • 2 adjacent locations pays   20 000$ / –
  • 3 adjacent locations pays   40 000$ /50 000$
  • 4 adjacent locations pays   60 000$ /80 000$
  • 5 adjacent locations pays   80 000$ /110 000$
  • 6 adjacent locations pays          –         / 140 000$

 

Minimum score

Yes, a player could choose not to build anything, and end the game with zero dollar.

But that would be a bit silly.

Instead, I’ll calculate the minimum amount of cash a player would make if he simply built everything he can, dumbly, without creating shops bigger than one or two adjacent locations (So he would only make 10 000$ per location, per turn).

For example, in a 3 players game, on his first turn, a player will draw 7 locations and decide which 5 of them he will keep, then he will draw 7 shop tiles. If he builds 5 shops on the five locations he own, at the end of the turn he will receive:

5 x 10 000$ = 50 000$ at the end of his first turn.

Those 5 shops will stay on the board the whole game. They can never be moved, but can be traded as a whole (location+shop).

If those shops do not expand during the game, they will simply pay 50 000$, each turn.

Since a game last 6 turns, for the entire game those shops will pay:

6 turns x 50 000$ = 300 000$

We can do the same for the shops built on the following turns (but since they are built later in the game, they only pay for the remaining turns)

 

For the full 3 players game we then have a minimum of:

  • 5 locations x 10 000$ x 6 turns = 300 000$
  • 4 locations x 10 000$ x 5 turns = 200 000$
  • 4 locations x 10 000$ x 4 turns = 160 000$
  • 4 locations x 10 000$ x 3 turns = 120 000$
  • 4 locations x 10 000$ x 2 turns = 80 000$
  • 4 locations x 10 000$ x 1 turns = 40 000$

For a Total of 900 000$. 

If the three players play similarly, the total cash in the game would be:

3 players x 900 000 $ = 2.7 millions

 

For 4 players we have we then have a minimum of:

  • 4 locations x 10 000$ x 6 turns = 240 000$
  • 3 locations x 10 000$ x 5 turns = 150 000$
  • 3 locations x 10 000$ x 4 turns = 120 000$
  • 3 locations x 10 000$ x 3 turns = 90 000$
  • 3 locations x 10 000$ x 2 turns = 60 000$
  • 3 locations x 10 000$ x 1 turns = 30 000$

For a Total of 690 000$  each.

Thus the total minimum cash for a 4 players game:

4 players x 690 000$ = 2.76 millions

For 5 players we have:

  • 3 locations x 10 000$ x 6 turns = 180 000$
  • 3 locations x 10 000$ x 5 turns = 150 000$
  • 3 locations x 10 000$ x 4 turns = 120 000$
  • 2 locations x 10 000$ x 3 turns = 60 000$
  • 2 locations x 10 000$ x 2 turns = 40 000$
  • 2 locations x 10 000$ x 1 turns = 20 000$

For a Total of 570 000$  each.

Or in total:  5 players x 570 000$ = 2.85 millions

 

The maximum score

The maximum score a player can make is a bit more complicated, conceptually not that much, but in practice quite a bit.

Here I will assume that the player stays with the same number of location than he receives during his turns. I assume the player do not trade, or only exchange his location in one for one transactions, keeping the same number of location than received during the game.

 

To attain the maximum, we will consider that the player is the luckiest person in the world, or at least has a such a good combination of luck and trading skills that he can always place his tiles in adjacent location when needed.

 

This of course is for the sake of determining the absolute maximum limit. I intend to simulate games and finds out how it compares to normally expected score. But this will be for the part two of this article.

 

 Notation: To ease the reading, I will denote each shop as a fraction:

current size / maximum size

 Eg:

  • A fully built shop of size 5 will be: 5/5
  • 2 adjacent tiles of a shop of maximum size 4 will be:  2/4
  • One tile from a shop of maximum size 3 will be: 1/3
  • etc…

 

And now, the moment you’ll all have been waiting for:

The way to build the shops to maximize your income.

Starting with the 3 players game:

In the first turn, a player receive 7 locations, 2 of which will have to be returned. To simplify, we can say the player receive 5 locations. Then he draw 7 shop tiles, that he keeps, he will always have enough shop tiles to fully build on his locations, so once again the only useful number is the number of new location he can keep each turn.

 

The most rewarding way of placing the shop permits on the board is the following:

  • On the first turn, fully build a shop of maximum size 5 (on five adjacent locations). According to the payout chart, such a 5/5 shop will pays 110 000$ at the end of the turn
  • On the second turn, you will receive 4 new tiles. You then should build a 4/6 shop with those tiles (adding 60 000$).
  • On the third turn,  with 4 new tiles:
    • Complete the 4/6 shop to a 6/6 shop, so the payout increase from 60 000$ to 140 000$.
    • Start another 2/6 shop with the rest of the tiles (+20 000$).
  • On the fourth turn, with 4 new tiles, complete the 2/6 shop to a 6/6 shop (to get it at 140 000$).
  • On the Fifth turn, with 4 new tiles, build a 4/4 shop (+80 000$).
  • On the Sixth turn, with 4 new tiles, build a 4/4 shop again (another 80 000$).

For a grand total of 1 960 000$ dollars!

 

The full description, written in a more concise way is the following:

How to reach the maximum score in a 3 players game

Turn New tiles Ownership Turn Payout
1 5 1 x 5/5 x 110 000$ 110 000$
2 5 1 x 5/5 x 110 000 + 1 x 4/6 x 60 000$ 170 000$
3 5 1 x 5/5 x 110 000 + 1 x 6/6 x 140 000$  + 2/6 x 20 000$ 270 000$
4 5 1 x 5/5 x 110 000 + 2 x 6/6 x 140 000$ 390 000$
5 5 1 x 5/5 x 110 000 + 2 x 6/6 x 140 000$ + 1 x 4/4 x 80 000$ 470 000$
6 5 1 x 5/5 x 110 000 + 2 x 6/6 x 140 000$ + 2 x 4/4 x 80 000$ 550 000$

For a maximum score of 1.960 millions!

 

Maximum in a 4 and 5 players game

Unsurprisingly, we can proceed in a similar fashion for the 4 and 5 player games.

 

How to reach the maximum score in a 4 players game

Turn New tiles Ownership Turn Payout
1 4 1 x 4/6 x 60 000$ 60 000$
2 3 1 x 6/6 x 140 000$ + 1/6 x 10 000$ 150 000$
3 3 1 x 6/6 x 140 000$ + 4/6 x 60 000$ 200 000$
4 3 2 x 6/6 x 140 000$ + 1/4 x 10 000$ 290 000$
5 3 2 x 6/6 x 140 000$ + 4/4 x 80 000$ 360 000$
6 3 2 x 6/6 x 140 000$ + 4/4 x 80 000$ + 3/3 x 50 000$ 410 000$

Which give a maximum score for a player of 1.470 millions!

 

How to reach the maximum score in a 5 players game.

Turn New tiles Ownership Turn Payout
1 3 1 x 3/6 x 40 000$ 40 000$
2 3 1x  6/6 x 140 000$ 140 000$
3 3 1 x 6/6 x 140 000$  + 3/5 x 40 000$ 180 000$
4 2 1 x 6/6 x 140 000$  + 1x 5/5 x 110 000$ 250 000$
5 2 2 x 6/6 x 140 000$ + 1x 5/5 x 110 000$ + 2/4 x 20 000$ 270 000$
6 2 2 x 6/6 x 140 000$ + 1x 5/5 x 110 000$ + 4/4 x 80 000$ 330 000$

So the maximum score for a player is 1.210 millions! 

 

How is this a useful ?

 

In an ideal world where you are omnipotent, that is what you should build.

When playing the game, it is not that clear you will be able to complete the shops you want… And others are aiming for the same shops as you, so it will necessarily be a land of compromise…

This is what make this game interesting after all!

 

So the interesting thing here is the maximum amount achievable for a fixed amount of tiles. While this is just a part of the picture, having an upper limit allows to place a personal score in the context of what is achievable. And having the minimum should inform you on how bad the decision to wait too much before starting to build could be!

 

To get a better picture of how good or bad your game is. We’ll need to have an idea on what an average score should be for this game.

I’ll get to this next time.

(Sorry, but I had to establish the limits first to put everything in context!)

 

What shop size you should be aiming for in a real game, given the probabilities of completing them is another thing. I’ll also try to gain some insights into this in my next posts. But it will be more complicated (and probably more fun 🙂 ).

 

Why obtaining the maximum was complicated ?

Some context before continuing with the group minimum and maximum.

I know, this does not look like much! But it was not that obvious to come up with! At first, I hand calculated quick solutions, but since I needed something automated for exploring the cooperative hypothesis (down below), I used the individual maximum problem as a starting point.

This was a good thing since, beside some dumb errors, I would also have missed the most paying solution! I had compared two approach, and convinced myself it was best to gun for the biggest shops you can. 

But this was an error since not one of the maximum solutions use the three available shops of maximum size 6!

My solutions were getting almost as much cash, but if the point is to set an upper limit, close is not good enough!

This was not a painless approach, since the exploration tree is much larger than the one I searched when counting Catan solutions. So much that I had to work quite a bit to write a program that was fast enough to produce an answer in a reasonable amount of time (Here I’m talking about my lifetime) !

 

Click the following to get some insight on the challenge I encountered in finding the optimal solutions.

 

 

How to win, as a group!

 

Chinatown is a competitive game, but it has a feeling of a friendly competition.

Exchanges made during the game are always win-win since there is no point in making an exchange that won’t benefit you!

Well, sometime you feel like the other is taking advantage of your situation… But hey, you will still make money out of the deal!

 

All this made me wonder: if instead of a fierce businessman competition, we were to look at this game from a cooperative point of view. A group of friendly businessmen doing their best to make sure everyone makes as much money as possible. Or at least as a group?

 

This is interesting for two reasons.

 

First. The maximum amount of cash one player can make directly depends on the other players making less optimal choices or unbalanced trades. So the more your opponents are unskilled, the best you can do. A personal maximum score ending-up measuring mostly how bad your opponents are is not a proxy for how good you are yourself!

 

Second. Because of the assumptions I made when calculating the maximum payout for a player, it is only a soft upper limit. A very good trader who manage to receive more tiles, or large sum of cash from other players during the game could end up with more cash than the previously calculated maximum limit. This feel a bit anti-climactic if we are aiming to calculate a real maximum!

 

You cannot fool the group

If we examine what is the maximum score as a group, we can determine a hard limit on how much cash can be earned in a game. The rules are strict and there is no way to exceed this amount. So this is an objective measure and this will allow us to give context on how players fare as a group, even in a competitive game!

For this, you can simply play normally, and see at the end how much cash all of you made, together. This can be an indication on how good you are at spotting beneficial trade and acting on it!

 

Minimum as a group

For the sake of being thorough, let’s start with the minimum amount of cash that can be made.

This is simple, since the minimum was simply based on how many tiles received, we just have to go back to the minimum for individual players, and multiply it by the number of players. This gives us a lower bound for the cooperative gameplay:

3 players x 900 000$ = 2.7 millions

4 players x 690 000$ = 2.76 millions

5 players x 570 000$ = 2.85 millions

 

Maximum as a group!

 

For this, we just have to imagine that all players are pooling their locations and shops to build the highest paying shops, and that they redistribute the money/shops/tiles as they go along.

So for a 3 player games, instead of receiving 5 tiles on the first turn, we will pick 15 tiles (3 players x 5 tiles)…. And so on.

 

Optimal way of building shops for a cooperative play between 3 players

Turn New tiles Shop placed per turn Added Payout Total Turn Payout
1 15 3 x 5/5 x 110 000$ 330 000$ 330 000$
2 12 2 x 6/6 x 140 000$ 280 000$ 610 000$
3 12 3 x 4/4 x 80 000$ 240 000$ 850 000$
4 12 2 x 3/3 x 50 000$ + 1 x 6/6 x 140 000 240 000$ 1 090 000$
5 12 4 x 3/3 x 50 000$ 200 000$ 1 290 000$
6 12 4 x 3/incomplete x 40 000$ 160 000$ 1 450 000$

For a group total of 5 620 000$

 

Since there is a limited number of complete shops that can be built, the group has to build incomplete shops at the end. Whether they are ¾, 3/5, or 3/6 shops, this does not matter since the payout will be the same for each (incomplete) shop.

 

So for three players the maximum is 5.62 millions, or divided equally 1.8733 millions. A bit lower than the theoretical maximum of 1.96 millions a player could make on his own.

 

 

For a 4 player game

Turn New tiles Shop placed per turn Added Payout Total Turn Payout
1 16 1 x 6/6 x 140 000$+ 2 x 5/5 x 110 000$ 360 000$ 360 000$
2 12 2 x 6/6 x 140 000$ 280 000$ 640 000$
3 12 3 x 4/4 x 80 000$ 240 000$ 880 000$
4 12 2 x 3/3 x 50 000$+ 1 x 5/5 x 110 000$
+ 1 x 1/incomplete x 10 000$
220 000$ 1 100 000$
5 12 4 x 3/3 x 50 000$ 200 000$ 1 300 000$
6 12 4 x 3/incomplete x 40 000$ 160 000$ 1 460 000$

For a game of 4 Group maximum is: 5.74 Millions

Or 1.435 million each (the individual max was : 1.470 million).

 

For a 5 player game

Turn New tiles Shop placed per turn Added Payout Total Turn Payout
1 15 1 x 6/6 x 140 000$ + 1 x 5/5 x 110 000$
+ 1 x 4/4 x 80 000$
330 000$ 330 000$
2 15 1 x 6/6 x 140 000$ + 1 x 5/5 x 110 000$
+ 1 x 4/4 x 80 000$
330 000$ 660 000$
3 15 1 x 6/6 x 140 000$ + 1 x 5/5 x 110 000$
+ 1 x 4/4 x 80 000$
330 000$ 990 000$
4 10 3 x 3/3 x 50 000$ + 1 x 1/3 x 10 000$ 160 000$ 1 150 000$
5 10 2 x 3/3 x 50 000$ + (1/3 to 3/3) x 40 000$
+ 1 x 2/incomplete x 20 000$
160 000$ 1 310 000$
6 10 2/incomplete to 3/incomplete x 20 000$
+ 3 x 3/incomplete x 40 000$
140 000$ 1 450 000$

For a game of 5 Group maximum is: 5.89 Millions

 Or 1.178 million each.(the individual max was : 1.21 million).

The whole picture

 

The limits of Chinatown

Nmb Of Players Minimum individually Maximum individually Minimum as a group Maximum as a group
3 900 000$ 1.96 million 2.7 millions 5.62 millions
4 690 000$ 1.47 million 2.76 millions 5.74 millions
5 570 000$ 1.21 million 2.85 millions 5.89 millions

Or to compare the best individual to the equally divided share for the best as a group:

Nmb Of Players Maximum individually Maximum share in a group
3 1.96 million 1.873 million
4 1.47 million  1.435 million
5 1.21 million 1.178 million

 

Visually, you can see here the limit on cash per player. With the horizontal lines showing how much would be made per player if it was played cooperatively.

This show the limits on cash per player in Chinatown.

As you can see, this does not represent an enormous advantage!

 

Final word

So there it is. Next time you play, you can compare yourself to those limits!

I understand that this is only partly interesting since the absolute maximum are unlikely to be reached in practice… The odds of picking out adjacent tiles every time being way too small!

But at least, make sure that you make more than the bare minimum!

 

To get a better idea of where you fit in the grand scheme of things, we need to simulate actual games, with proper tiles drawing selection, and include some trading. This should gives us an idea of how much one (or a group) could expect to make!

 

I already started the work for this, but, as always, there is quite a bit of things to consider! Here is a quick 4-player game simulation, to whet your appetite.

Here a simulation, only showing the tiles the players choose to keep at each turn (the first turn, they receive 6, and keep 4, after that is 5 -3). And we can see that this simple advantage of tile selection is enough to get some shops of interesting size… And there is no trading implemented yet! (And no consideration of the shops themselves)

Chinatown - Location selection for 4 players. Teaser animation

 

 

Hopefully you had fun with this quick introduction to Chinatown Limits…

Next time: Looking at realistic score distribution in The limits of Chinatown – Part two

 

 

Let me know if I missed something.

Or if you would like that I explore some other details (or game) in my coming post!

What other game do you think could benefit from a limit analysis?

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