Fairness of the random setup

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ypaul
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Joined: 10 January 2020, 14:15

Fairness of the random setup

Post by ypaul »

Hi guys.

I just discovered this game and I’ve been enjoying it a lot so far. In the game help, it was stated very clearly that
Note that this random setup does not give any player a noticeable advantage.
I was wondering how true this is, especially for a two-player game. My intuition tells me that having your high-numbered tiles grouped together with other tiles of your own colour near the middle will give you a much better chance compared to if they are isolated and near the corners/edges.

Of course, in the end, your skills will be the main deciding factor of who will win the game, but given two players of equal skill, an unfair setup can really skew the outcome of a game. Perhaps given the number of tiles on the board, the number of fair setups greatly outnumber the number of unfair setups, but this may still be an issue in a competitive environment.

I have only played a couple of games so far, so I’m not speaking from experience, and this may very well not be an issue at all. What do some of the more experienced players think about this? It will not surprise me if there is actually a natural balancing mechanism in place such that having your high-numbered tiles in seemingly useless positions would turn out to actually give you an advantage at another part of the board, but this outcome isn’t immediately obvious to me.

Thanks!
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BobTheTurtle
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Re: Fairness of the random setup

Post by BobTheTurtle »

This is an interesting topic. From my experience, the random map set-up puts very little luck into the game, and even then only if both players are at a very high level. This is because you can easily divide the map with your borders, so much so that in a few turns the game changes entirely. This makes it so that the map set-ups are not ever as unbalanced as they may appear. For example, if you have a lot of your high-number tiles on the edge then you can easily divide them with your borders so that they will not be in the same group. Thus, the initial distribution of tiles affects who wins the game very little. From what I have seen, initial tile distribution almost never determines the outcome of the game.

The only way that the initial map set-up can really give one player an advantage or a disadvantage is based on where the swing district tiles are located, since these tiles determine the tie-break. For example, if the neutral district and one of the 1-point districts are surrounded by a certain player's tiles, then that player will have a slight advantage over the other player. This won't make a difference unless both of the players are high level. In my games, when two high level players play, the ending score will almost always be 3-3, 4-3, or sometimes 4-4 or 5-3, as compared to when I play a lower level player, in which case sometimes the game is close but other times there will be a larger difference, often 5-2, 5-1, or even (my best score ever) 6-1.

This means that who has more of the swing districts will often win the game if the game is between two higher level players. In a very close game, the game will often end with the player with less swing districts trying to carve up the map so that they win with a one point advantage and the player with more swing districts trying to make the game have an equal number of districts so that they win on tie-break. This makes going for swing districts very important early on, and means that where the swing districts are located can give one player a slight advantage. However, if you do not win more swing districts you still have a very good chance to win the game if you play well and end up with one more district than your opponent at the end, which can happen often.

Thus, the initial map set-up can at most give one player a very miniscule advantage. I have never seen a map distribution that gave one player or the other a particularly large advantage, and do not think that any such set-ups exist. Because of all this, I can say confidently that the initial map set-up puts a very, very small amount of luck into the game, and I agree that "this random setup does not give any player a noticeable advantage".

**Note: I only play this game in 2 player mode, because otherwise I think that it is unfair how players might place borders that give another player an easy advantage, even if they are doing it unintentionally, since if two players are competing with border placement then the other players will do much better. Thus, I cannot say how the random map set up affects games with more than two players, but my guess would be that it does not give any player a large advantage in most cases.
Last edited by BobTheTurtle on 30 January 2021, 20:12, edited 1 time in total.
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ypaul
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Re: Fairness of the random setup

Post by ypaul »

Thank you for taking your time to share your experience!

Your thoughts on the swing districts make a lot of sense. In my games so far, I've been trying very hard to win the swing districts too, because to me it's essentially an extra point.
I have never seen a map distribution that gave one player or the other a particularly large advantage, and do not think that any such set-ups exist.
This is questionable to me though. I think I can come up with a hypothetical scenario where one side is clearly at a disadvantage.

https://imgur.com/a/6kkanxT

Granted, this may be an extreme example, and it probably doesn't really happen in real games, but it feels too bold a statement to claim that every setup is fair.
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BobTheTurtle
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Re: Fairness of the random setup

Post by BobTheTurtle »

Yes, in your example, blue probably has a decent advantage, larger than with almost any map set-up. However, even this position I wouldn't call completely winning for blue. The game would have an interesting playout, with some fighting over the border at first, followed by a very interesting phase of each player dividing up their own territory and the other player's territory to try to make smaller districts in their own and larger districts in the opponent's territory. If red was able to take a bit of extra territory at the border (even just all of its own tiles, and maybe one of blue's) and was able to make it so that blue had large districts in their own territory (7, 8, 9 tile districts) while red had small districts (4 or 5 tiles each) then red might be able to win the game with a one-point advantage.

However, the most likely scenario would be for blue to win with an equal number of districts and one extra swing district to win the tiebreak. I would actually find this starting situation very interesting to play out. If both players did the best moves it would probably be a win for blue on tiebreak, but of course since the game has no randomness after the initial set-up all positions are hypothetically won or lost for each player with perfect play on both sides. It would definitely be an interesting set-up to play out though, since it is completely unique from all other set-ups I have seen. An even less balanced set-up would be if a few of red's lower number districts were in the blue territory, so that blue has more territory and can make more districts. Of course, all of these set-ups are highly improbable, and from what I have seen it is by far the most likely to get a set-up that is very close to balanced for both players.

They should add an option to set up starting positions in training mode games, and then we could play situations like this out and see if we can get further on this. Also, in order to ensure that the starting position is as close to balanced as possible, they could make it so that starting positions equal in number to one more than the number of players are created, and then have each player choose one set-up to NOT play with. This way, if, although unlikely, a set-up that has a noticeable advantage for one player or the other is created, then the player whom it disadvantages will choose not to play with it. For example, if one set-up gave blue an advantage and one set-up gave red an advantage, then blue would choose not to play with the set-up that gave red an advantage and red would choose not to play with the set-up that gave blue an advantage and the more neutral set-up would be chosen. This would actually be really fun to try because you would then have to evaluate each starting position to determine which one is worst for you, and which one is best for you. Again, these advantage and disadvantages would most likely be minimal, but at the higher levels even a minimal advantage can make a difference. This would create a new challenge in evaluating different starting positions.
Last edited by BobTheTurtle on 30 January 2021, 20:13, edited 1 time in total.
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ypaul
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Re: Fairness of the random setup

Post by ypaul »

That was very insightful! This is the kind of analysis that reminds me exactly why I love this genre of games. The fact that people of different skill levels can see the same thing, yet have different levels of understanding can never cease to amaze me. I’d love to see a commentated game between two skilled players, for both a random setup and also one of these ‘unfair’ setups.

Your comment on what would make the initial setup even more unfair is also really interesting. This is potentially a challenge puzzle for people: to come up with the most unfair setup possible for a game between two seasoned human players.

And yes, I that your suggestion probably works to balance things out even more for top level play, if this game ever turns really competitive. Or perhaps even some sort of pie rule may help, but I’m not sure.

[strikethrough]The pie that I have in mind is: player A places one red divider, player B places two blue dividers, player A places three red dividers, player B chooses to play as red or blue, and the game proceeds from there, with blue going next.[/strikethrough] (EDIT: this probably doesn’t work that well if the blue has the advantage. Nevermind! The pie rule only works when you have either the first or second player that has an advantage, and it wouldn’t work if the advantage changes between setups. Your suggestion works best!)
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ypaul
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Re: Fairness of the random setup

Post by ypaul »

The only issue with the number of players plus one setup rule is that in the event that both players chose the same setup not to play, then there’s no good way to decide which of the two leftover setups to use. Flipping a coin to decide doesn’t feel right to me.
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Romain672
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Re: Fairness of the random setup

Post by Romain672 »

BobTheTurtle wrote: 30 January 2021, 10:05
Please create some space between your ideas. It would be way better to read :)
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BobTheTurtle
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Re: Fairness of the random setup

Post by BobTheTurtle »

I thought it might be helpful to add in some numbers, so here we go:

First, how many possible different starting positions are there? We can determine this using permutations. There are 37 tiles total, 2 red of each number 2-9, 2 blue of each number 2-9, 1 red 1 and 1 red 10, 1 blue 1 and 1 blue 10, and 1 neutral district. To determine the number of ways to arrange these tiles, we can do 37 factorial, which is about 1.37 * 10^43. Then we have to divide by 2 for each repeated number since swapping them does not create a new position, so we divide by 2^8 since there are 8 pairs of 2 identical tiles. This gives us approximately 5.38 * 10^40. Finally, we have to divide by 6 because, since the board is a hexagon, there are 6 ways to rotate it that will all really be the same position. That means that there are 8.96 * 10^39 starting positions. This is still a VERY BIG number.

Next, what percentage of these positions are balanced and what percentage are unbalanced. Here are my estimates, based on my experience playing the game:

-Balanced (Equal game or extremely small advantage for one side) = about 85% of the time:
These games will have the tiles of both players approximately evenly distributed across the board, with the swing districts in equal positions, that is each swing district being adjacent to an approximately equal number of red and blue tiles, or one swing district being in an equal position, one being surrounded by one player's tiles, and one being surrounded by the other player's tiles. Set-ups like this give neither player an advantage.

-Close to Balanced (Small advantage for one side) = about 15% of the time:
In these games the tile distribution will be about equal for each player, but the distribution of the swing districts will give one player a small advantage, because 2 or 3 swing districts will be almost completely surrounded by the same player's tiles. This gives this player a small advantage, but this advantage will not often affect the outcome of the game; the other player will just have to play to win with a one district advantage (at least). Note that if there are an odd number of districts total in a two player game then which player has more swing districts will not matter at all, since there is no way for the two players to have the same number of districts. The player who does not win the swing districts can sometimes use this to their advantage.

-Unbalanced (Large advantage for one side) = about 0.001% of the time (or once in 100,000 games):
These games have a tile distribution in which it is very easy for one player to earn more districts than the other player, based on the tile distribution. All of one player's highest point districts will be in the same location, with their lower point districts surrounded by the other player's tiles. Even positions like this I would not consider a guaranteed win for the side with the advantage, if the other player plays much better. I have never had a starting set-up of this category in one of my games, although I have only played about 100 games.

The reason that I give the unbalanced games such a low chance of occurring is because of how easy it is to change the map by placing borders, to your advantage. If the map gives you lots of your high point tiles right next to each other, then that shouldn't be too much of a problem because you can place borders in between them, and then you will win at least two districts with this set of tiles. If the map places some of your districts in an area that your opponent dominates, then you can place borders that force these tiles into the same district. Of course, your opponent will be trying to stop you, but who is more effective in this will win the game, which is why this is a game of strategy. Also, note that there is a sort of natural balancing mechanism in the initial tile distribution: if all of your tiles are in the same area of the map, then all of your opponent's tiles must also be in the same area of the map (the area in which your tiles are not). This doesn't mean that it is impossible for an unbalanced set-up to occur, it is just very, very unlikely.
Last edited by BobTheTurtle on 30 January 2021, 20:55, edited 1 time in total.
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BobTheTurtle
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Re: Fairness of the random setup

Post by BobTheTurtle »

ypaul wrote: 30 January 2021, 11:03 The only issue with the number of players plus one setup rule is that in the event that both players chose the same setup not to play, then there’s no good way to decide which of the two leftover setups to use. Flipping a coin to decide doesn’t feel right to me.
If both players choose the same set-up not to play, then that must mean that all three set-ups are balanced, since if any of the set-ups gave one player an advantage then the other player would have chosen not to play with that set-up, so it does not really matter which of the remaining set-ups is chosen.

Additionally, there is a really easy way to make sure that it never happens that one set-up is never chosen: have the players choose which set-up not to play with in order, rather than all at once. This way, with three set-ups to choose from, one player chooses one of the three not to play with, and the other player chooses which one of the remaining two not to play with.

If the players are both good at evaluating starting positions (which might actually be really difficult) then you should end up with the most balanced of the three set-ups. If one player is better than the other at evaluating starting positions, then that player has a greater chance of ending up with a slight advantage, which would just add another dimension to the strategy of the game.
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BobTheTurtle
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Re: Fairness of the random setup

Post by BobTheTurtle »

Romain672 wrote: 30 January 2021, 15:39
BobTheTurtle wrote: 30 January 2021, 10:05
Please create some space between your ideas. It would be way better to read :)
Sorry about that. I tried to improve it. Is it easier to read now?
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