First-player and second-player win

In game theory, a two-player turn-based game is first-player-win if a perfect player who plays first can always force a win. Similarly, a game is second-player-win if a perfect player who plays second can always force a win.

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.

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. The first player in Checkers, however, can only guarantee themselves a draw under perfect play.

See also

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