Difference between revisions of "Inferiority"

Latest revision as of 02:05, 26 April 2024

A move is inferior if there is another move that is at least as good.

Technically, every losing move is inferior (unless it is the only available move), and every winning move is also inferior, unless it is the only winning move. However, it is often possible to figure out inferiority locally, i.e., by looking at a few nearby cells, rather than having to consider the whole board. In particular, it is often possible to figure out whether a move is inferior without actually knowing whether it is winning or losing.

The concept of inferiority is closely related to the concept of domination: By definition, a move is inferior if and only if it is dominated by some other move. However, it is sometimes possible to know that a move is inferior without knowing which specific other move dominates it. This knowledge is still useful, because inferior moves should never be played and can be eliminated from consideration (unless there are no other options).

Inferiority by domination

If a move X dominates a move Y, then Y is inferior. See the article on domination for examples, and for many different methods of proving domination.

Inferiority by strong reversibility

We say that a Red move at X is strongly reversed by a Blue move at Y if (X,Y) = (Red,Blue) is strategically equivalent to (X,Y) = (Empty,Blue). In other words, after Blue's response, Red's stone could be removed without changing the value of the position.

Theorem. If a move X is strongly reversed by some move Y, then X is inferior.

Proof. Consider the position (X,Y) = (Empty,Empty), and suppose X is a winning move for Red. We must show that there exists another winning move elsewhere. By assumption, (Red,Empty) is a second-player win for Red, so (Red,Blue) is a first-player win for Red. By the assumption of strong reversibility, (Empty,Blue) is also a first-player win for Red, so Red has some winning move. If that winning move is somewhere other than X, then by monotonicity, the same move is also winning for the position (Empty,Empty), proving the claim. If that winning move is X, then (Red,Blue) is a second-player win for Red. By strategic equivalence, (Empty,Blue) is also a second-player win for Red. Therefore (Blue,Blue) is a first-player win for Red, so Red has a winning move somewhere. By monotonicity, the same move is also winning from (Empty,Empty), proving the theorem. □

Example 1.

Consider the position

We claim that both cells marked "X" are inferior for Red. Indeed, after Blue Y, both X's are captured by Red: Red can get at least one of them, killing the other. Therefore, any red stone on either X is made irrelevant after Blue's Y, so that Y strongly reverses X.

It follows that X is an inferior move for Red, but the proof does not tell us what other move is better. It doesn't necessarily have to be Y.

Example 2.

Consider the position

We claim that X is an inferior move for Red because X is strongly reversed by Y. To show this, we must show that the following two positions are strategically equivalent:

In both positions, Blue is connected to the edge by template IV-2b, and Red has an escape fork for 2nd row ladders into the corner. With more work, one can show that the two positions are in fact equivalent, and therefore X is inferior for Red. Note that the proof does not indicate a specific move that is better.

For a concrete example, consider this position, with Red to move:

X is losing, but there are two winning moves: c3 and f2.

The term "strongly reversed", with the meaning given here, appears in the source code of MoHex. Example 1 is one of MoHex's strong reversibility patterns.