First-player and second-player win

Diagram showing optimal strategy for tic-tac-toe. With perfect play, and from any initial move, both players can always force a draw.

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. 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.^[1] 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 (such as with all humans and all current chess engines). However, with perfect play, it remains unsolved as to whether the game is a first-player win (White), a second player win (Black), or a forced draw.^[2]^[3]^[4]

References[edit]

^ "Checkers Is Solved". Science. 317: 1518–1522. doi:10.1126/science.1144079. Retrieved 2008-11-24.
^ J.W.H.M. Uiterwijk, H.J. van den Herik. "The Advantage of the Initiative". (August 1999).
^ Shannon, C. (March 1950). "Programming a Computer for Playing Chess". Philosophical Magazine. 7. 41 (314). Archived from the original (pdf) on 2010-03-15. Retrieved 2008-06-27.
^ Victor Allis (1994). "PhD thesis: Searching for Solutions in Games and Artificial Intelligence" (pdf). Department of Computer Science. University of Limburg. Retrieved 2012-07-14.

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[1] "Checkers Is Solved". Science. 317: 1518–1522. doi:10.1126/science.1144079. Retrieved 2008-11-24.

[2] J.W.H.M. Uiterwijk, H.J. van den Herik. "The Advantage of the Initiative". (August 1999).

[3] Shannon, C. (March 1950). "Programming a Computer for Playing Chess". Philosophical Magazine. 7. 41 (314). Archived from the original (pdf) on 2010-03-15. Retrieved 2008-06-27.

[4] Victor Allis (1994). "PhD thesis: Searching for Solutions in Games and Artificial Intelligence" (pdf). Department of Computer Science. University of Limburg. Retrieved 2012-07-14.

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v t e Topics in game theory
Definitions	Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game
Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium
Strategies	Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy
Classes of games	Symmetric game Perfect information Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global game Strictly determined game Potential game
Games	Chess Infinite chess Checkers Tic-tac-toe Prisoner's dilemma Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock–paper–scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Prisoners and hats puzzle Trust game Princess and Monster game Rendezvous problem
Theorems	Arrow's impossibility theorem Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Purification theorem Revelation principle Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith Antoine Augustin Cournot John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition First-move advantage in chess Game mechanics Glossary of game theory List of game theorists List of games in game theory No-win situation Solving chess Topological game Tragedy of the commons Tyranny of small decisions

First-player and second-player win

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