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A safe position is a concept from game theory. If a player finds themselves in a safe position, they can always win unless they make a mistake, regardless of the actions of their opponents. For a simple illustration of safe positions, see Wythoff's game.

Any fair turn-based game has safe positions. For example in chess, king and queen against king will always be a safe position (unless it first arises in an unusual position on the board). I believe that any such game places one of the players in a safe position from the start; chess does, I believe, have a perfect strategy, although we currently have nothing like the computational power required to find it.

An illustration of this, first suggested by Gritchka, is to think of the different moves of the game as a kind of tree with different outcomes at the end of it's branches. For example, the tree of the (2,1) from the point of view of player 1 is:

P1:             (2,1)
               /  |  \
              /   |   \
             /    |    \
P2:       (1,1) (1,0) (0,1)
           /|\    |     |
          / | \   L     L
         /  |  \
        /   |   \
P1: (1,0) (0,0) (0,1)
      |     |     |
      L     W     L

On this diagram, nodes that only link down to others of one type adopt this type themselves: that is, positions which can only lead to a win for one side are considered a win in themselves; the rest of the game need not be played at all because no strategy can result in anything other than a win. Gritchka suggests thinking of it like painting: we paint upwards from the bottom, and we only paint nodes that have all the nodes below the same colour. Otherwise the outcome depends on strategy.

All decent games have very few nodes that can be painted beyond the bottom few rungs, because that would take the skill out of the game. The definition of a "safe position" in this system is one that links to at least one win regardless of the moves of the opponent, and thus winning is entirely dependent on you playing the right strategy.

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