First-player and second-player win

In game theory, a two-player deterministic perfect information turn-based game is a first-player-win if with perfect play the first player to move can always force a win. Similarly, a game is second-player-win if with perfect play the second player to move can always force a win. With perfect play, if neither side can force a win, the game is a draw.

Some games with relatively small game trees have been proven to be first or second player wins. For example, the game of nim with the classic 3–4–5 starting position is a first-player-win game. However, Nim with the 1-3-5-7 starting position is a second-player-win game. The classic game of Connect Four has been mathematically proven to be first-player-win.

The first-player in checkers, can only guarantee themselves a draw under perfect play. Another example of a draw game is tic-tac-toe.

It remains a matter of conjecture as to whether other games such as chess are first-player-wins; see the article first-move advantage in chess for more on this.

See also

• Strategy-stealing argument
• Forced draw
• Zugzwang
• Determinacy
• Combinatorial game theory
• First-mover advantage