Domination
In game theory, a move dominates another move if it is at least as good. In Hex, we say that a cell X dominates another cell Y in a given position (and from a particular player's point of view) if playing at X is at least as good as playing at Y for that player. If there is a set of cells in which one cell dominates all of the others, the player can eliminate the dominated cells from consideration, because moving in the dominating cell will be at least as good. This can often simplify the analysis of Hex positions.
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Capture-domination
In general, it can be difficult to determine whether one move dominates another. But there are many situations where the concept of capturing can be used to reason about domination.
If moving at X would capture Y, then X always dominates Y. In this case, we say that X capture-dominates Y. It is often possible to figure this out locally, i.e., by looking at a few nearby cells, rather than having to consider the whole board.
Examples of dominated patterns
In all of the following examples, we assume that Red is the player to move. In each case, the move marked "*" capture-dominates the moves marked "+", because moving at "*" would capture the cells marked "+". The cells that are left white can be any color.
Mutually dominating moves
It is possible for two or more cells to dominate each other. In this case, they are all equally good moves. However, the player can still eliminate all but one of these moves from consideration.
For example, in the following position, each of the cells marked "*" dominates the other two. So any of these moves are equally good for Red (but of course there may be other moves on the board that are even better). When Red considers possible moves, she can concentrate on whichever of the three marked cells is the most convenient, and ignore the two others.
Usage examples
Example 1
Consider the following position, with Red to move. What is the best way for Red to connect her two groups?
The answer is c. A move at f would also connect Red's groups, but c dominates f. In fact, Red c captures b, d, f, and g, and therefore dominates all of these moves.
To see why c is strictly better than f, note that if Red connects using f, then Blue still has a forcing move at a; but if Red connects using c, Blue has no such forcing move.
Example 2
Consider the following position, where Red has captured cells a and b. If Blue actually plays in one of the cells a or b, what should Red do? Red could certainly play in the other of the two cells, enforcing the capture. However, it is better for Red to play at c, capturing all 5 cells. This response capture-dominates the solid connection at a or b (although it may in some cases be inferior with regard to the secondary objective of minimizing the number of moves needed to win).
Star decomposition domination
Sometimes it is possible to conclude that X dominates Y, even if moving at X would not exactly capture Y. One can reason as follows: suppose we can show that having two pieces at X and Y would be no better than just having a piece at X. Then X dominates Y, because moving at X is no worse than moving at X and Y, which in turn is at least as good as moving at Y alone.
As an example of this, consider the following position, with Blue to move. Which of the marked moves dominates the others?
The answer is c. For example, to see that c dominates b, we reason as follows. Suppose there is a blue piece at c.
Within the shaded region, the only thing that can affect the game's outcome is whether Red's piece connects to the bottom edge or not. Red can achieve a connection in one move, and Blue can prevent a connection in one move. Now suppose that Blue also had a piece at b.
In the shaded region, the situation is exactly the same as before: Red can connect in one move, and Blue can disconnect Red in one move. Therefore, the additional piece at b is not helpful to Blue, provided Blue has a piece at c. It follows that c dominates b. A similar argument also shows that c dominates a, d, and e (but not O).
This example is a special case of star decomposition domination. See Henderson and Hayward, "Captured-reversible moves and star decomposition domination in Hex".
Path-domination
Another way to find dominated moves is by path-domination. We say that a cell X path-dominates a cell Y, from a player's point of view, if every minimal winning path for that player that passes though Y also passes through X. By a "minimal winning path", we mean a set of pieces, added to the given position, that yields a chain between the player's edges, and such that no proper subset of the pieces has this property. If X path-dominates Y, then X dominates Y.
To prove it, assume the player in question is Red. Consider all ways of completely filling all cells of the board, starting from the given position. For each such completely-filled board, there are four possiblities: (1) X and Y are both red; (2) X is red and Y is blue; (3) X is blue and Y is red; (4) X and Y are both blue. They key observation is that if (3) is winning for Red, then so is (2). Consider a position where (3) is winning for Red. Change the red piece at Y to a blue piece. If the resulting position is winning, then so is (2), because the extra red piece at X can only help Red. If the resulting position is losing, then Y must have been on some minimal winning red path. But we assumed that every such path contains X, a contradiction.
Now given any winning strategy for Red that starts by putting a red piece at Y, we can obtain another winning strategy by simply exchanging the roles of X and Y. I.e., whenever the original strategy said to move at X, move at Y instead and vice versa. Therefore, moving at X is at least as good as moving at Y for Red, i.e., X dominates Y.
Examples of path-domination
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Example: the 10-cell acute corner
Consider the following cells in the acute corner:
Assume that the 10 indicated corner cells are still empty, and Red wants to make a move in the corner. Also assume that it is not the first move of the game (so Red does not need to worry about her move being swapped). Where should Red consider playing? It turns out that there are only 3 moves that dominate all other possibilities. Red should move at e, h, or j.
Caveat: Please note that this does not mean that Red should never make a move other than e, h, or j in the acute corner. It only means that if the corner is empty, Red's first move in the corner should be e, h, or j. In the presence of other (red or blue) stones, Red might of course need to move elsewhere in the corner. By symmetry, Blue's first move in the empty corner should be at a, e, or h.
Red's move at e
A red stone at e captures a, b, c, d, f, and g.
Therefore, e capture-dominates a, b, c, d, f, and g.
Red's move at h
Red's move at h dominates i. To see why, we note that the following two corner positions are equivalent:
is equivalent toClearly the right position is at least as good for Red as the left one, since Red has an additional stone. To see why the left position is at least as good as the right one, we consider two cases. Case 1: Red is the next player to make a move in the region. In this case, Red will play at e, capturing the entire region. Since the whole region (including i) is red-captured, the red stone at i no longer matters. Case 2: Blue is the next player to make a move in the region. If Blue moves anywhere other than e, Red responds at e, capturing the entire region (including i) and killing Blue's stone. If Blue moves at e, then the two positions are:
andIn this case, the only thing left to determine the outcome of the region {b,c,d,e,f,g,h,i} is whether Red connects h to the red edge or Blue connects e to the blue edge. In both the left and the right region, whichever player moves next achieves this outcome, so the two regions are equivalent.
Since having h is as good for Red as having both h and i, the move at h dominates i.
Red's move at j
The remaining possibility for Red in the empty 10-cell corner is to move at j.
e, h, and j are not dominated
We have seen above that all red moves other than e, h, and j in the corner are dominated. Could it be that one of these three remaining moves is also dominated? This is not the case. To see that e, h, and j are not dominated, it suffices to give examples of positions in which e (respectively h and j) is the only winning move.
Here are such positions. In each position, it is Red's turn, and the only winning move is indicated by the letter e, h, or j:
See also
External links
Henderson and Hayward, "Captured-reversible moves and star decomposition domination in Hex".
Ryan Hayward's publication page contains research articles on dead cells.
