First-player and second-player win
Image:Tictactoe-X.svg|thumb|right|Diagram showing optimal strategy for tic-tac-toe. With perfect play, and from any initial move, both players can always force a draw.
In combinatorial [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. The classic game of Connect Four has been mathematically proven to be first-player win.
With perfect play, checkers has been determined to be a draw; neither player can force a win. Another example of a game which leads to a draw with perfect play is tic-tac-toe, and this includes play from any opening move.
Significant theory has been completed in the effort to solve chess. It has been speculated that there may be first-move advantage which can be detected when the game is played imperfectly. However, with perfect play, it remains unsolved as to whether the game is a first-player win, a second-player win, or a forced draw.