A point-value model for dominoes
Posted: 09 September 2026, 00:45
A point-value model for dominoes
I have been experimenting with a systematic way of assigning values to dominoes.
The main difficulty is that the value of a domino is not simply the number of points it immediately gives when placed. Several things interact.
I will use the following notation throughout:
W = Wheat
F = Forest
L = Lake
G = Grass
S = Swamp
M = Mines
* = crown
I will call each half of a domino a square.
For example:
WM*** = Wheat/Mines with three crowns
FF = double Forest
Section 1: Scoring in the last round
The last round is the easiest place to start because, under certain conditions, the result can be calculated exactly.
Even here, however, there can be complicated interactions.
For example, there may be two forest crowns, but only one can still be placed in a high-scoring position. A crown may only be placeable in the desired location if the player avoids taking another domino that would block that location. There can be many similar dependencies.
What I want to analyze first is a more exact situation.
Both players can place every relevant domino in the desired location, and there are no conditional dependencies between the values of the individual dominoes.
In such a position, at the beginning of the round, both players can determine exactly how many points each domino will ultimately be worth to them, regardless of the combination in which they receive them.
Section 2: The value of a domino
If a domino can be placed by both players and there are no complicated dependencies, we can define its value as:
Value of the domino = points gained when placed by Player 1 + points gained when placed by Player 2
We can see this as follows: if we pick a domino, we score points from it, while the other player loses out on the points they would have scored from it. For the point difference these two effects should be considered with the same weight.
The order of the dominoes does not matter in the last round. Since there is no future round, the only thing that matters is who gets which domino. We can therefore compare the total value of the available dominoes directly.
Consider an FF domino in the last round. Assume that both players have all their Forest crowns connected, both players can place the domino, and all Forest crowns have already been taken.
Depending on the Forest crown distribution between the two players, the points gained by the players are:
Forest crown distribution (Player 1 / Player 2)
0/6: Player 1 = 0, Player 2 = 12, Total = 12
1/5: Player 1 = 2, Player 2 = 10, Total = 12
2/4: Player 1 = 4, Player 2 = 8, Total = 12
3/3: Player 1 = 6, Player 2 = 6, Total = 12
4/2: Player 1 = 8, Player 2 = 4, Total = 12
5/1: Player 1 = 10, Player 2 = 2, Total = 12
6/0: Player 1 = 12, Player 2 = 0, Total = 12
The key observation is that the domino has a fixed total value of 12, regardless of the crown distribution. If we take a domino for 0 points in order to prevent the opponent from getting 12, that is equivalent, in terms of total domino value, to taking a domino worth 12 points for ourselves which is worth 0 points for the opponent.
The scoring outcome in terms of total points will be different, but the point delta between the players is identical. The domino is worth 12 points between the two players regardless of whether those points come from scoring ourselves or denying points to the opponent.
Section 3: The same principle in the early/mid-game
The same principle also applies earlier in the game. The crowns do not have to be placed yet for us to assign a value to a domino.
Suppose we take an FF domino in the first round. At that moment, neither player may receive any immediate points from it. However, if we assume that whenever a player later takes a Forest crown, that crown can eventually be connected to this Forest square, then the domino already has its eventual value at the moment it is drafted.
In other words, if we know that the Forest squares and Forest crowns will remain connected, an FF domino can already be treated as a 12-point domino when it is taken, even though the points will only be realized later.
Section 4: Normalizing the square values
The value of a domino consists of the value of its two squares plus the value of any crowns present on them.
At this point, we have not yet discussed the value of different types of crowns. First, we can determine the base value of each square.
A Forest, Lake, Grass, or Swamp square has a value of 6 because of the 6 respective crowns present in the deck.
A Wheat square has a value of 5.
A Mines square has a value of 10.
We can normalize these values by subtracting 6 from every square:
W = -1
F = 0
L = 0
G = 0
S = 0
M = +4
This means that an ordinary domino without crowns can be evaluated by adding the normalized values of its two squares. For example:
FF = 0
WF = -1
The purpose of this normalization is simply to make the calculations easier. Most of the dominoes can be treated as zero-value dominoes. The actual scoring values have not changed.
The remaining part of the domino value comes from crowns.
Section 5: How much is a crown worth?
We can use similar reasoning to determine the value of crowns. Assume that, for a given type, both players have a single connected group of that type. In particular, no square of that type is disconnected from the corresponding crowns of its player. Then the value of one crown of that type is equal to the total number of squares of that type belonging to both players.
The important point is that this total does not depend on how those squares are distributed between the two players.
For example, there are 22 Forest squares in the game. If Player 1 eventually has 18 Forest squares and Player 2 has 4, then one Forest crown is worth 18 points to Player 1 and 4 points to Player 2, assuming both players managed to connect all of their Forest squares.
18 + 4 = 22
The distribution could instead be 11/11, 5/17, or 0/22.
11 + 11 = 22
5 + 17 = 22
0 + 22 = 22
As long as the Forest squares remain connected to the corresponding Forest crowns, the total value is always 22. Therefore, the value of a Forest crown is fixed at 22 points.
This is the key principle of the model.
The value of a crown can be determined from the total number of squares of its type in the game, regardless of how those squares are eventually divided between the two players.
This is also why we can assign the crown its value from the moment it is drafted. We do not need to wait until the crown is actually connected to its squares, provided that the relevant connections can eventually be made.
Using this principle, the crown values are:
M* = 6
M** = 12
M*** = 18
S* = 10
S** = 20
G* = 14
G** = 28
L* = 18
F* = 22
W* = 26
These values can then be added directly to the normalized values of the two squares.
Section 6: A simplified version of the model
For practical calculations, I make one further simplification. I treat Wheat as 0 instead of -1.
The reason is that the -1 contribution is relatively small compared with the strategic importance of wheat squares in maintaining all connections, and ignoring it makes mental calculations easier.
The simplified square values therefore become:
W = 0
F = 0
L = 0
G = 0
S = 0
M = +4
In this simplified model, an ordinary domino without crowns has exactly zero value unless it contains Mines. All value comes from crowns and mine squares.
This makes the model much easier to use during an actual game.
Section 7: Mines and domino values summary
Every Mines square is worth +4 after normalization, while Mines crowns are worth 6 points each. So the crowned mine tiles worth:
M* = 6 + 4 = 10
M** = 12 + 4 = 16
M*** = 18 + 4 = 22
And:
W*M = 26 + 4 = 30
To give the full list in descending order:
W*M = 30
G** = 28
W* = 26
F* = M*** = 22
S** = 20
L* = 18
M** = 16
G* = 14
M* = S* = 10
no crown = 0
Section 8: The value of selection order
The next problem is the value of the drafting position. In the simplest version of the model, an ordinary domino without crowns has exactly zero normalized value.
So, in the simplest model, all the value comes from crowns and Mines squares.
However, the drafting position itself should also have a value that we cannot disregard.
Based on both empirical observation and a small simulation, the difference between adjacent drafting positions is approximately 6 to 7 points, leaning towards 7. I use 7 points by default.
For example, suppose an 18-point crown domino (L*) is in position 4. An ordinary domino in position 1 is three positions earlier, so I would assign it roughly: 3 x 7 = 21 additional points of positional value.
This makes the position 1 ordinary domino effectively stronger than the 18-point crown domino in position 4. A 22-point Forest crown in position 4, however, would already be stronger than that.
I also wrote a small program to estimate the average value difference between the four drafting positions when the available dominoes are randomly generated.
The result was approximately 6.5 to 6.8 points per position, which is close to the 7-point approximation.
There is one interesting detail in the result. The difference between positions 2 and 3 was about 0.2 points larger than the differences between positions 1 and 2 or positions 3 and 4.
This suggests that the distribution of crowned dominoes is such that, on average, around two of the four available dominoes are particularly valuable. As a result, the first two pick positions might have a slightly greater advantage than the simple 7-point-per-position model would suggest.
The calculation is not yet fully recursive, though. The 6.5 to 6.8 value is based on the raw value differences between dominoes and does not yet incorporate the positional value of the future positions into the value of the current dominoes.
A more complete model would need to account for that as well.
Section 9: Dynamic values during the game
In the last few sections I will describe some advanced concepts that I don't strictly use in real game scenarios. I do try to intuitively take them into account when the choice is close between two picks.
A more advanced calculation can be made if we dynamically adjust crown/square values during the game.
Whenever one square becomes disconnected from the crowns of its type, the value of every crown of that type decreases by 1.
Likewise, whenever one crown becomes disconnected, the value of every square of that type decreases by 1.
This follows directly from the crown valuation principle.
If one Forest square can no longer contribute to any connected Forest crown, then the total value represented by placed and future Forest crowns decreases by 1.
Conversely, if one Forest crown becomes disconnected, one fewer crown can benefit from every Forest square, so the value of each Forest square decreases by 1.
This provides a way of updating the values as the board develops. It has to be emphasized that the decrease in value affects both players in the same way regardless of who is responsible of the disconnection. Of course in nominal points only the player who made the disconnection will be penalized.
Another way to think about this is to treat taking crowned dominoes as taking on quests. The quest is to connect every square of that type to the crowns we take. If we take 0 crowns of a terrain type throughout the game then it does not matter whether we can connect the squares of that terrain.
Section 10: Placement and connection penalties
The next advanced adjustment is to account for whether a domino can actually connect where it needs to. This can be modeled using penalty values.
A square that can no longer connect to some crowns of its type should receive a penalty based on how many of those crowns it can no longer reach.
If a particular square cannot connect for either player, the combined penalty is 6 points (for F,L,G,S).
If a square cannot connect for one player, but can connect for the other the penalty value is anywhere between 0 and 6 depending on the crowns distribution. It's hard to account for crowns still in the deck in this case.
Section 11: Possibilities for improvements
As mentioned selection order value could be further fine-tuned with a randomized simulation with selection orders taken into account recursively.
There are still many cases that the model does not handle.
For example, suppose both players can place one remaining Forest crown, and there are two Forest crowns still available in the selection pool. Both players can therefore use one crown, but the player who takes the first one cannot take the second for its full value. For this the value of the first crown should already be decreased, but it's hard to determine by how much.
The value of a domino is therefore not always independent of the other available dominoes.
At that point, the model would need to account for the remaining crown counts, future drafting positions, possible blocking, connectivity, and the interaction between all of these factors.
Placement strategies have not been discussed in this post which is almost a completely separate aspect of the game.
I have been experimenting with a systematic way of assigning values to dominoes.
The main difficulty is that the value of a domino is not simply the number of points it immediately gives when placed. Several things interact.
- You may still acquire additional crowns of the terrain type later, increasing the value of the domino placed now.
- A crown you acquire now may become more valuable as more of its terrain type become connected to it.
- You can lose out on points when squares of a terrain type are left disconnected from the corresponding crowns.
- A placement that closes a connection (e.g. no further wheat-forest dominoes can be placed fully connected) can create future losses.
I will use the following notation throughout:
W = Wheat
F = Forest
L = Lake
G = Grass
S = Swamp
M = Mines
* = crown
I will call each half of a domino a square.
For example:
WM*** = Wheat/Mines with three crowns
FF = double Forest
Section 1: Scoring in the last round
The last round is the easiest place to start because, under certain conditions, the result can be calculated exactly.
Even here, however, there can be complicated interactions.
For example, there may be two forest crowns, but only one can still be placed in a high-scoring position. A crown may only be placeable in the desired location if the player avoids taking another domino that would block that location. There can be many similar dependencies.
What I want to analyze first is a more exact situation.
Both players can place every relevant domino in the desired location, and there are no conditional dependencies between the values of the individual dominoes.
In such a position, at the beginning of the round, both players can determine exactly how many points each domino will ultimately be worth to them, regardless of the combination in which they receive them.
Section 2: The value of a domino
If a domino can be placed by both players and there are no complicated dependencies, we can define its value as:
Value of the domino = points gained when placed by Player 1 + points gained when placed by Player 2
We can see this as follows: if we pick a domino, we score points from it, while the other player loses out on the points they would have scored from it. For the point difference these two effects should be considered with the same weight.
The order of the dominoes does not matter in the last round. Since there is no future round, the only thing that matters is who gets which domino. We can therefore compare the total value of the available dominoes directly.
Consider an FF domino in the last round. Assume that both players have all their Forest crowns connected, both players can place the domino, and all Forest crowns have already been taken.
Depending on the Forest crown distribution between the two players, the points gained by the players are:
Forest crown distribution (Player 1 / Player 2)
0/6: Player 1 = 0, Player 2 = 12, Total = 12
1/5: Player 1 = 2, Player 2 = 10, Total = 12
2/4: Player 1 = 4, Player 2 = 8, Total = 12
3/3: Player 1 = 6, Player 2 = 6, Total = 12
4/2: Player 1 = 8, Player 2 = 4, Total = 12
5/1: Player 1 = 10, Player 2 = 2, Total = 12
6/0: Player 1 = 12, Player 2 = 0, Total = 12
The key observation is that the domino has a fixed total value of 12, regardless of the crown distribution. If we take a domino for 0 points in order to prevent the opponent from getting 12, that is equivalent, in terms of total domino value, to taking a domino worth 12 points for ourselves which is worth 0 points for the opponent.
The scoring outcome in terms of total points will be different, but the point delta between the players is identical. The domino is worth 12 points between the two players regardless of whether those points come from scoring ourselves or denying points to the opponent.
Section 3: The same principle in the early/mid-game
The same principle also applies earlier in the game. The crowns do not have to be placed yet for us to assign a value to a domino.
Suppose we take an FF domino in the first round. At that moment, neither player may receive any immediate points from it. However, if we assume that whenever a player later takes a Forest crown, that crown can eventually be connected to this Forest square, then the domino already has its eventual value at the moment it is drafted.
In other words, if we know that the Forest squares and Forest crowns will remain connected, an FF domino can already be treated as a 12-point domino when it is taken, even though the points will only be realized later.
Section 4: Normalizing the square values
The value of a domino consists of the value of its two squares plus the value of any crowns present on them.
At this point, we have not yet discussed the value of different types of crowns. First, we can determine the base value of each square.
A Forest, Lake, Grass, or Swamp square has a value of 6 because of the 6 respective crowns present in the deck.
A Wheat square has a value of 5.
A Mines square has a value of 10.
We can normalize these values by subtracting 6 from every square:
W = -1
F = 0
L = 0
G = 0
S = 0
M = +4
This means that an ordinary domino without crowns can be evaluated by adding the normalized values of its two squares. For example:
FF = 0
WF = -1
The purpose of this normalization is simply to make the calculations easier. Most of the dominoes can be treated as zero-value dominoes. The actual scoring values have not changed.
The remaining part of the domino value comes from crowns.
Section 5: How much is a crown worth?
We can use similar reasoning to determine the value of crowns. Assume that, for a given type, both players have a single connected group of that type. In particular, no square of that type is disconnected from the corresponding crowns of its player. Then the value of one crown of that type is equal to the total number of squares of that type belonging to both players.
The important point is that this total does not depend on how those squares are distributed between the two players.
For example, there are 22 Forest squares in the game. If Player 1 eventually has 18 Forest squares and Player 2 has 4, then one Forest crown is worth 18 points to Player 1 and 4 points to Player 2, assuming both players managed to connect all of their Forest squares.
18 + 4 = 22
The distribution could instead be 11/11, 5/17, or 0/22.
11 + 11 = 22
5 + 17 = 22
0 + 22 = 22
As long as the Forest squares remain connected to the corresponding Forest crowns, the total value is always 22. Therefore, the value of a Forest crown is fixed at 22 points.
This is the key principle of the model.
The value of a crown can be determined from the total number of squares of its type in the game, regardless of how those squares are eventually divided between the two players.
This is also why we can assign the crown its value from the moment it is drafted. We do not need to wait until the crown is actually connected to its squares, provided that the relevant connections can eventually be made.
Using this principle, the crown values are:
M* = 6
M** = 12
M*** = 18
S* = 10
S** = 20
G* = 14
G** = 28
L* = 18
F* = 22
W* = 26
These values can then be added directly to the normalized values of the two squares.
Section 6: A simplified version of the model
For practical calculations, I make one further simplification. I treat Wheat as 0 instead of -1.
The reason is that the -1 contribution is relatively small compared with the strategic importance of wheat squares in maintaining all connections, and ignoring it makes mental calculations easier.
The simplified square values therefore become:
W = 0
F = 0
L = 0
G = 0
S = 0
M = +4
In this simplified model, an ordinary domino without crowns has exactly zero value unless it contains Mines. All value comes from crowns and mine squares.
This makes the model much easier to use during an actual game.
Section 7: Mines and domino values summary
Every Mines square is worth +4 after normalization, while Mines crowns are worth 6 points each. So the crowned mine tiles worth:
M* = 6 + 4 = 10
M** = 12 + 4 = 16
M*** = 18 + 4 = 22
And:
W*M = 26 + 4 = 30
To give the full list in descending order:
W*M = 30
G** = 28
W* = 26
F* = M*** = 22
S** = 20
L* = 18
M** = 16
G* = 14
M* = S* = 10
no crown = 0
Section 8: The value of selection order
The next problem is the value of the drafting position. In the simplest version of the model, an ordinary domino without crowns has exactly zero normalized value.
So, in the simplest model, all the value comes from crowns and Mines squares.
However, the drafting position itself should also have a value that we cannot disregard.
Based on both empirical observation and a small simulation, the difference between adjacent drafting positions is approximately 6 to 7 points, leaning towards 7. I use 7 points by default.
For example, suppose an 18-point crown domino (L*) is in position 4. An ordinary domino in position 1 is three positions earlier, so I would assign it roughly: 3 x 7 = 21 additional points of positional value.
This makes the position 1 ordinary domino effectively stronger than the 18-point crown domino in position 4. A 22-point Forest crown in position 4, however, would already be stronger than that.
I also wrote a small program to estimate the average value difference between the four drafting positions when the available dominoes are randomly generated.
The result was approximately 6.5 to 6.8 points per position, which is close to the 7-point approximation.
There is one interesting detail in the result. The difference between positions 2 and 3 was about 0.2 points larger than the differences between positions 1 and 2 or positions 3 and 4.
This suggests that the distribution of crowned dominoes is such that, on average, around two of the four available dominoes are particularly valuable. As a result, the first two pick positions might have a slightly greater advantage than the simple 7-point-per-position model would suggest.
The calculation is not yet fully recursive, though. The 6.5 to 6.8 value is based on the raw value differences between dominoes and does not yet incorporate the positional value of the future positions into the value of the current dominoes.
A more complete model would need to account for that as well.
Section 9: Dynamic values during the game
In the last few sections I will describe some advanced concepts that I don't strictly use in real game scenarios. I do try to intuitively take them into account when the choice is close between two picks.
A more advanced calculation can be made if we dynamically adjust crown/square values during the game.
Whenever one square becomes disconnected from the crowns of its type, the value of every crown of that type decreases by 1.
Likewise, whenever one crown becomes disconnected, the value of every square of that type decreases by 1.
This follows directly from the crown valuation principle.
If one Forest square can no longer contribute to any connected Forest crown, then the total value represented by placed and future Forest crowns decreases by 1.
Conversely, if one Forest crown becomes disconnected, one fewer crown can benefit from every Forest square, so the value of each Forest square decreases by 1.
This provides a way of updating the values as the board develops. It has to be emphasized that the decrease in value affects both players in the same way regardless of who is responsible of the disconnection. Of course in nominal points only the player who made the disconnection will be penalized.
Another way to think about this is to treat taking crowned dominoes as taking on quests. The quest is to connect every square of that type to the crowns we take. If we take 0 crowns of a terrain type throughout the game then it does not matter whether we can connect the squares of that terrain.
Section 10: Placement and connection penalties
The next advanced adjustment is to account for whether a domino can actually connect where it needs to. This can be modeled using penalty values.
A square that can no longer connect to some crowns of its type should receive a penalty based on how many of those crowns it can no longer reach.
If a particular square cannot connect for either player, the combined penalty is 6 points (for F,L,G,S).
If a square cannot connect for one player, but can connect for the other the penalty value is anywhere between 0 and 6 depending on the crowns distribution. It's hard to account for crowns still in the deck in this case.
Section 11: Possibilities for improvements
As mentioned selection order value could be further fine-tuned with a randomized simulation with selection orders taken into account recursively.
There are still many cases that the model does not handle.
For example, suppose both players can place one remaining Forest crown, and there are two Forest crowns still available in the selection pool. Both players can therefore use one crown, but the player who takes the first one cannot take the second for its full value. For this the value of the first crown should already be decreased, but it's hard to determine by how much.
The value of a domino is therefore not always independent of the other available dominoes.
At that point, the model would need to account for the remaining crown counts, future drafting positions, possible blocking, connectivity, and the interaction between all of these factors.
Placement strategies have not been discussed in this post which is almost a completely separate aspect of the game.