First-player and second-player win: Difference between revisions

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[[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.]]

{{Refimprove|date=March 2009}}

In [[combinatorial game theory]], a two-player deterministic [[perfect information]] [[Sequential game|turn-based game]] is a '''first-player-win''' if with [[Solved game#Perfect play|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 (tie)|draw]].

Some games with relatively small [[game tree]]s have been proven to be first or second -player wins. For example, the game of [[Nimnim]] 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.

In [[game theory]], a two-player deterministic perfect-information [[turn-based game]] is '''first-player-win''' if a [[optimal play|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.

When winning is not possible with perfect play by both opposing sides, the game is a '''draw'''.

With perfect play, [[English draughts|checkers]] has been determined to be a draw; neither player can force a win.<ref>{{cite journal|title=Checkers Is Solved |doi=10.1126/science.1144079 |pmid=17641166 |volume=317 |issue=5844 |journal=Science |pages=1518–1522|year=2007 |last1=Schaeffer |first1=J. |last2=Burch |first2=N. |last3=Bjornsson |first3=Y. |last4=Kishimoto |first4=A. |last5=Muller |first5=M. |last6=Lake |first6=R. |last7=Lu |first7=P. |last8=Sutphen |first8=S. |bibcode=2007Sci...317.1518S |s2cid=10274228 |doi-access=free }}</ref> 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.

Some games with relatively small [[game tree]]s 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.

Significant theory has been completed in the effort to [[Solving chess|solve chess]]. It has been speculated that there may be [[First-move advantage in chess|first-move advantage]] which can be detected when the game is played imperfectly (such as with all humans and all current [[chess engine]]s). 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.<ref>J.W.H.M. Uiterwijk, H.J. van den Herik. [https://web.archive.org/web/20180109064212/https://pdfs.semanticscholar.org/55dd/2fee1f0981fbfabd4b158a6584eefaacbcea.pdf "The Advantage of the Initiative]". (August 1999).</ref><ref>{{cite journal

The first player in [[English draughts|Checkers]], can only guarantee themselves a draw under perfect play.<ref>{{cite web|url=http://www.sciencemag.org/cgi/content/abstract/1144079 |title=Checkers Is Solved - Schaeffer et al. 317 (5844): 1518 - Science |publisher=Sciencemag.org |date= |accessdate=2008-11-24}}</ref>

|author-link=Claude Shannon

Another example of a draw game is [[tic-tac-toe]].

|last=Shannon

|first=C.

|series=7

|volume=41

|number=314

|date=March 1950

|title=Programming a Computer for Playing Chess

|url=http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf

|journal=[[Philosophical Magazine]]

|access-date=2008-06-27

|archive-url=https://web.archive.org/web/20100706211229/http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf

|archive-date=2010-07-06

|url-status=dead

}}</ref><ref>{{cite web

|author=Victor Allis

|url=http://www.dphu.org/uploads/attachements/books/books_3721_0.pdf

|title=PhD thesis: Searching for Solutions in Games and Artificial Intelligence

|work=Department of Computer Science

|publisher=[[University of Limburg]]

|year=1994

|access-date=2012-07-14

|author-link=Victor Allis

|archive-date=2020-11-22

|archive-url=https://web.archive.org/web/20201122211341/https://www.dphu.org/uploads/attachements/books/books_3721_0.pdf

|url-status=dead

}}</ref>

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==

*[[Solved game]]

*[[Strategy-stealing argument]]

*[[Forced draw]]

*[[Zugzwang]]

*[[Determinacy]]

*[[Combinatorial game theory]]

*[[First-movermove advantage in chess]]

==References==

{{Reflist}}

[[Category:GameNon-cooperative theorygames]]

[[Category:Mathematical games]]

{{Game theory}}

{{Mathapplied-stub}}