Optimal play

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Optimal play on the whole board

For the purpose of the following definitions, by a position, we mean an arrangement of pieces on the Hex board, together with the information whose turn it is.

Given any position on the Hex board, either Red or Blue has a winning strategy. A position in which a player P has a winning strategy is called a winning position for P; otherwise it is called a losing position. A move is called a winning move if it results in a winning position for the player who made the move; otherwise it is called a losing move. Note that this definition of winning move is theoretical: in practice, a player who makes a winning move may still lose the game if they later make a mistake; and a player who makes a losing move may still win the game if the opponent later makes a mistake.

A player is said to play optimally if the player makes a winning move whenever a winning move is available.

Optimal play in a region

When considering play on a portion of the board (rather than the whole board), the situation is more complicated. In this case, there isn't necessarily a notion of "winning"; rather, the outcome is measured by what Red and Blue want to achieve in that particular region. For example, Red may want to achieve a 2nd row ladder, but Blue may prefer it if Red only gets a 3rd row ladder. Or Red may want to achieve a 3rd row ladder escape, but Blue may prefer it if Red only gets a 2nd row ladder escape.

In this case, we say that a player plays optimally if they achieve the maximum of what they could have achieved. For example, if we say "under optimal play, Red will get a 3rd row ladder", it means that if Red plays optimally, Red will get at least a 3rd row ladder (but maybe something better), and if Blue plays optimally, Red will get at most a 3rd row ladder (but maybe something worse). If both players play optimally, Red will get exactly a 3rd row ladder.

Among all of the outcomes that a player can achieve in a region of the board, there isn't necessarily always a "best" one. What's best may also depend on what is going on in other regions of the board, which may be partly or fully under the opponent's control. If a player must choose between one of several potentially desirable outcomes for a region, it may be better to play elsewhere until it becomes clearer which of these outcomes is most beneficial for the player. In that case, there is no "optimal" play in the region; we say that the region is not settled.