Solved game

A solved game is a game whose outcome can be correctly predicted from any position when each side plays optimally. Games which haven't been solved are said to be "unsolved". This article focuses on two-player games that have been solved.

A two-player game can be "solved" on several levels:

Ultra-weak

In the weakest sense, solving a game means proving whether the first player will win, lose, or draw from the initial position, given perfect play on both sides. This can be a non-constructive proof (possibly involving a strategy stealing argument) that may not actually help determine this perfect play.

Weak

More typically, solving a game means providing an algorithm that secures a win for one player, or a draw for either, against any possible moves by the opponent, from the beginning of the game.

Strong

The strongest sense of solution requires an algorithm which can produce perfect play from any position, i.e. even if mistakes have already been made on one or both sides.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree. However, since for many non-trivial games such an algorithm would require an infeasible amount of time to generate a move in a given position, a game is not considered to be solved weakly or strongly unless the algorithm can be run by existing hardware in a reasonable time. Many algorithms rely on a huge pre-generated database, and are effectively nothing more than that.

As an example, the game of tic-tac-toe is solvable as a draw for both players with perfect play (a result even manually determinable by schoolchildren). Games like nim also admit of a rigorous analysis using combinatorial game theory.

Whether a game is solved is not necessarily the same as whether it remains interesting for humans to play. Even a strongly solved game can still be interesting if the solution is too complex to be memorized; conversely, a weakly solved game may lose its attraction if the winning strategy is simple enough to remember (e.g. Maharajah and the Sepoys). An ultra-weak solution (e.g. Chomp or Hex on a sufficiently large board) generally does not affect playability.

Perfect play

In game theory, perfect play is the behavior or strategy of a player which leads to the best possible outcome for that player regardless of the response by the opponent. Based on the rules of a game, every possible final position can be evaluated (as a win, loss or draw). By backwards reasoning, one can recursively evaluate a non-final position as identical to that of the position that is one move away and best valued for the player whose move it is. Thus a transition between positions can never result in a better evaluation for the moving player and a perfect move in a position would be a transition between positions that are equally evaluated. As an example, a perfect player in a drawn position would always get a draw or win, never a loss. If there are multiple options with the same outcome, perfect play is sometimes considered the fastest method leading to a good result, or the slowest method leading to a bad result.

Perfect play can be generalized to non-perfect information games, as the strategy that would guarantee the highest minimal expected outcome regardless of the strategy of the opponent. As an example, the perfect strategy for Rock, Paper, Scissors would be to randomly choose each of the options with equal (1/3) probability. The disadvantage in this example is that this strategy will never exploit non-optimal strategies of the opponent, so the expected outcome of this strategy versus any strategy will always be equal to the minimal expected outcome.

Although the optimal strategy of a game may not (yet) be known, a game-playing computer might still benefit from solutions of the game from certain endgame positions (in the form of endgame tablebases), which will allow it to play perfectly after some point in the game. Computer chess programs are well-known for doing this.

Solved games

Awari (a game of the Mancala family)

The variant allowing game ending "grand slams" was strongly solved by Henri Bal and John Romein at the Vrije Universiteit in Amsterdam, Netherlands (2002). Either player can force the game into a draw.

Chopsticks

The second player can always force a win.

Connect Four

Solved first by James D. Allen (Oct 1, 1988), and independently by Victor Allis (Oct 16, 1988). First player can force a win. Strongly solved by John Tromp's 8-ply database (Feb 22, 1995). Weakly solved for all boardsizes where width+height is at most 15 (Feb 18, 2006).

Draughts, English (i.e. checkers)

This 8x8 variant of draughts was weakly solved on April 29, 2007 by the team of Jonathan Schaeffer, known for Chinook, the "World Man-Machine Checkers Champion". From the standard starting position, both players can guarantee a draw with perfect play. Checkers is the largest game that has been solved to date, with a search space of 5x10²⁰. The number of calculations involved was 10¹⁴, and those were done over a period of 18 years. The process involved from 200 desktop computers at its peak down to around 50.

Fanorona

Weakly solved by Maarten Schadd. The game is a draw.

Free Gomoku

Solved by Victor Allis (1993). First player can force a win without opening rules.

Ghost

Solved by Alan Frank using the Official Scrabble Dictionary in 1987.

Hex

• A strategy-stealing argument (as used by John Nash) will show that all square board sizes cannot be lost by the first player. Combined with a proof of the impossibility of a draw this shows that the game is ultra-weak solved as a first player win.
• Strongly solved by several computers for board sizes up to 6×6.
• Jing Yang has demonstrated a winning strategy (weak solution) for board sizes 7×7, 8×8 and 9×9.
• A winning strategy for Hex with swapping is known for the 7×7 board.
• Strongly solving hex on an N×N board is unlikely as the problem has been shown to be PSPACE-complete.
• If Hex is played on an N × N+1 board then the player who has the shorter distance to connect can always win by a simple pairing strategy, even with the disadvantage of playing second.

Kalah

Most variants solved by Geoffrey Irving, Jeroen Donkers and Jos Uiterwijk (2000) except Kalah (6/6). Strong first-player advantage was proven in most cases.

L Game

Easily solvable. Either player can force the game into a draw.

Maharajah and the Sepoys

This asymmetrical game is a win for the sepoys player with correct play.

Nim

Completely solved for all configurations.

Nine Men's Morris

Solved by Ralph Gasser (1993). Either player can force the game into a draw

Ohvalhu

Weakly solved by humans, but proven by computers. (Dakon is, however, not identical to Ohvalhu, the game which actually had been observed by de Voogt)

Pentominoes

Weakly solved by H. K. Orman. It is a win for the first player.

Quarto

Solved by Luc Goossens (1998). Two perfect players will always draw.

Qubic

Weakly solved by Oren Patashnik (1980) and Victor Allis. The first player wins.

Free Renju

(Without opening rules) claimed to be solved by János Wagner and István Virág (2001). A first-player win.

Sim

Weakly solved: win for the second player.

Teeko

Solved by Guy Steele (1998). Depending on the variant either a first-player win or a draw.

Three Men's Morris

Trivially solvable. Either player can force the game into a draw.

Tic-tac-toe

Trivially solvable. Either player can force the game into a draw.

Tigers and Goats

Weakly solved by Yew Jin Lim (2007). The game is a draw.

Partially solved games

Chess

Solved by retrograde computer analysis for all three- to six-piece, and some seven-piece endgames, counting the two kings as pieces. It is solved for all 3–3 and 4–2 endgames with and without pawns, where 5-1 endgames are assumed to be won with some trivial exceptions (see endgame tablebase for an explanation). The full game has 32 pieces. Chess on a 3x3 board is strongly solved by Kirill Kryukov (2004). The question of whether or not chess can be perfectly solved in the future is controversial.

International Draughts

All positions with two through seven pieces were solved. Positions with 4x4 and 5x3 pieces where each side had one king or less. Positions with five men versus four men, five men versus three men and one king, and four men and one king versus four men. Solved in 2007 by Ed Gilbert of the United States, computer analysis show that highly likely it always end in a draw if both players play perfectly.

Go

Board sizes up to 4×4 are strongly solved. The 5×5 board is weakly solved for all opening moves. Humans usually play on a 19×19 board which is over 145 orders of magnitude more complex than 7x7. For all board sizes without Komidashi, the game is either a win or a draw for the first player, by the strategy stealing argument.

Othello

Strongly solved on a 4×4 and 6×6 board as a second player win in July 1993 by Joel Feinstein. Far from solved on 8×8 board (the standard one), yet computer analysis shows a very likely draw: there are thousands of likely draw lines although not a single line has been fully proven. No strongly supposed estimates other than increased chances for the starting player (black) on 10×10 and greater boards exist.

m,n,k-game

It is trivial to show that the second player can never win; see strategy-stealing argument. Almost all cases have been solved weakly for k ≤ 4. Some results are known for k = 5. The games are drawn for k ≥ 8.

See also

• Game complexity
• God's algorithm
• Zermelo's theorem (game theory)

Further reading

• Allis, Beating the World Champion? The state-of-the-art in computer game playing. in New Approaches to Board Games Research.

External links

• Computational Complexity of Games and Puzzles by David Eppstein.
• GamesCrafters - on solving two-person games with perfect information and no chance