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Machi Koro» Forums » Strategy

Subject: The Probability and Math of Machi Koro rss

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Curtis Rollo
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Hey Guys,

I was just wondering if there are any math/ probability people out there who can help me out.

I'm trying to think about this game mathematically, ultimately creating a math formula for choosing the best cards.

Here's what I have so far:

Each card basically adds to the Expected Value (EV) of your overall hand.

So, if you choose to buy a Wheat field (Cost 1, Gives you 1 on a roll of 1), then the marginal EV of your hand on your own roll increases by:

(1)*(1/6) / 1 = 1/6

Return * Probability / Cost


Considering that this card costs 1 coin, the cost of one coin raises your hand's EV by 0.18 for every one of your own rolls.

Similarly, choosing the Forest card becomes this:

1 * (1/6) / 3 = 1/18


But, since the Wheat field gives you a coin on anyone's roll, you have to factor that in and think about the return for every round after all players roll.

Return * Probability * Players / Cost

So the more players, the more valuable Blue cards are, and similarly Red cards.

So I perceive the point of the game is to maximize the EV of your overall hand this way.

However, here is where I get stuck. It seems that a well-balanced hand is more advantageous than a lop-sided hand. And I'm not sure how to weight the probabilities to reflect this.

I'm assuming, any roll that nets you 0 coins should be a negative weight. So while choosing between a Wheat field and a Cattle farm both have the same marginal EV, it would be better to choose the cattle farm because the removes the chance a roll netting you nothing, compared to a roll of 1 netting you 2 coins.

Anyone have anything thoughts on how to calculate these weighted EV?

The next question is when does one start to invest in higher value property.

Thanks guys!
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Luke Halvorsen
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This is quite the interesting dilemma, and its certainly intriguing to think about.

A couple of considerations to take into account:

1. The addition of the second die for you or other players might make the Wheat field irrelevant and add then change the EV of the Forest.

2. Second, you also have to account for cards that feed off other cards, so the Forest for example gains EV from the Furniture Factory, but at an increased cost. That cost is then distributed between the Forest (and Mine) cards that you already own or may buy in the future, and so the EV changes every game depending on how much of a mix you have.

3. Additionally, players get to choose each turn how many dice to roll, so calculating the actual EV for a game is essentially impossible, unless you meticulously tracked all of your games data (each roll, player tendencies, etc.) and created a post-game EV that you MIGHT use in future games against the same players. However, if that player played a different strategy, then your numbers are no longer relevant.

4. I guess it might be possible to "solve" the game by creating a massive spreadsheet for each possible combination:

-i.e. 2-player game, 4th round, you own a Wheat Field, 2 Bakeries, and a Ranch, your opponent owns a Wheat Field, a Bakery, and a Cafe, and you each have 2 coins. You could find that in your spreadsheet, and find out which card had the best EV at that time...

But that sounds like a whole different game to me?

Thoughts?
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Jake Staines
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curtisrollo wrote:
It seems that a well-balanced hand is more advantageous than a lop-sided hand. And I'm not sure how to weight the probabilities to reflect this.


Does it? Plenty of games I've seen have been won when a player with an imbalance towards X gets that roll, pulls in a load of money and buys an expensive landmark, giving him a significant advantage for future turns... if he hasn't already won.

In terms of probability, a 1/1 chance of getting 1 coin is the same, over enough turns, as a 1/6th chance of getting 6 coins. So having a balanced hand is meaningless in those terms.

In Machi Koro, however, you have the added advantage that getting all your resources in one go helps a bit against cafés and restaurants, because a) it means that when you roll them you often don't have any money to pay and thus the owner misses out on the income, and b) you get money from your cards after you paid the restaurateur, so - say - if you own five mines, start your hand with nothing and roll a nine... you'll pay out nothing to the restaurant owners then it's hello, Radio Tower.

The disadvantage is that - particularly early on in the game - missing out on income because you've over-specialised means that you can't buy things on many of your turns so you miss the opportunity to build your town up so quickly as your opponents.





It seems to me that generally, you want a balanced tableau at the beginning of the game and then to switch to a specialised one later in the game... and the trick (c.f. skill) in that is that which specialisation makes the most sense and when to switch from general to special are both functions of what your tableau looks like, what your opponents' tableaus look like, and what cards are still available.

Not to say that on any given turn there isn't a mathematically 'best' card to pick; there quite possibly is. However, the problem space is large and complex enough that I don't think it's really worth trying to work it out, even leaving aside that if you did 'solve' the game you'd essentially be ruining it for yourself and anyone else who reads your work forever!
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Chris Okasaki
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Bichatse wrote:
curtisrollo wrote:
It seems that a well-balanced hand is more advantageous than a lop-sided hand. And I'm not sure how to weight the probabilities to reflect this.


Does it? Plenty of games I've seen have been won when a player with an imbalance towards X gets that roll, pulls in a load of money and buys an expensive landmark, giving him a significant advantage for future turns... if he hasn't already won.


What you've just pointed out is a pretty common phenomenon. Maximizing expected value may be a good path to maximimizing your expected total score, but in many games that is NOT the same as maximizing your chance of winning.

The catch is that maximizing your chance of getting a GOOD score often actually decreases your chance of getting a GREAT score. With the kind of specialized approach you're talking about, you increase your chance of getting a GREAT score, while simultaneously increasing your chance of getting a TERRIBLE score. And for many games, especially as the number of players increases, your chance of winning without a great score is small.

It's an old dilemma in the gaming world--is winning really the only thing that matters? If so, you want to maximize your chance of winning even if that increases your chance doing horribly. On the other hand, if you value doing well even if you don't win (for example, coming in 2nd in a 4 player game), then maximizing your chance of getting a good score is probably the way to go, even if that decreases your chance of actually winning.
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Darcy Dueck
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Cokasaki is exactly on the mark.

Like any individual race game with a random distribution, think a golf tournament, the winner will be the one who had the greatest positive variance from their expected value (EV).

In a PGA golf tournament, you could shoot par every round for week after week (ie a consistently good score around 60th percentile) but since there are 144 players in the tournament you would have zero chance at winning. There the optimal strategy is high risk, high reward because the payout for first place is nearly 90 times higher than the payoff for 58th place.

Similarly, in Machi Koro the optimal strategy for more players becomes more specialized with higher variance, especially the 5 and 9 cards because they have the greatest variance. With more players, the 5 and 9 also have the benefit that you get paid on other player's turns.

Playing for max EV is not optimal for winning because the variance swings far out weigh the EV.

Now if you don't care about winning and want to avoid losing at all costs, then the max EV / lowest variance strategy is optimal.
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Kevin Steffler
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As an added subtlety to this high variance strategy, it is also very important not to duplicate another players strategy if they are ahead of you (assuming shared outcome like mines and forests). A common mistake would be, in the face of a player who seems to be making lots of money off their 2 mines, to buy one for yourself. By doing so you are linking your fates, both of you relying on a roll of 9 to advance. However, since your opponent has more mines than you, there is almost no way that you will be able to beat him this way unless you have alternative strategies in place. But since you are already dependent on those alternative strategies, you are better off doubling down on them and hoping your opponent's luck changes and no more 9s come up.
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