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 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.[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]

## See also

* Solved game
* Strategy-stealing argument
* Zugzwang
* Determinacy
* Combinatorial game theory
* First-move advantage in chess

## References

1. ↑ Schaeffer, J.; Burch, N.; Bjornsson, Y.; Kishimoto, A.; Muller, M.; Lake, R.; Lu, P.; Sutphen, S. (2007). "Checkers Is Solved". Science 317 (5844): 1518–1522. doi:10.1126/science.1144079. PMID 17641166. Bibcode: 2007Sci...317.1518S. http://www.sciencemag.org/cgi/content/abstract/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). http://archive.computerhistory.org/projects/chess/related_materials/text/2-0 and 2-1.Programming_a_computer_for_playing_chess.shannon/2-0 and 2-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf. Retrieved 2008-06-27.
4. ↑ Victor Allis (1994). "PhD thesis: Searching for Solutions in Games and Artificial Intelligence". Department of Computer Science. University of Limburg. http://www.dphu.org/uploads/attachements/books/books_3721_0.pdf.

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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|

* Go
* 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

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