A solved game is a game whose outcome (win, lose, or draw) can be correctly predicted from any position, given that both players play perfectly. Games which have not been solved are said to be “unsolved”. Games for which only some positions have been solved are said to be “partially solved”. This article focuses on two-player games that have been solved.
A two-player game can be “solved” on several levels:
- Prove 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 astrategy-stealing argument) that need not actually determine any moves of the perfect play.
- Provide 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. That is, produce at least one complete ideal game (all moves start to end) with proof that each move is optimal for the player making it. It does not necessarily mean a computer program using the solution will play optimally against an imperfect opponent. For example, the checkers program Chinook will never turn a drawn position into a losing position (since the weak solution of checkers proves that it is a draw), but it might possibly turn a winning position into a drawn position because Chinook does not expect the opponent to play a move that will not win but could possibly lose, and so it does not analyze such moves completely.
- Provide an algorithm that can produce perfect play (moves) from any position, even if mistakes have already been made on one or both sides.
Despite the name, many game theorists believe that “ultra-weak” are the deepest, most interesting and valuable proofs. “Ultra-weak” proofs require a scholar to reason about the abstract properties of the game, and show how these properties lead to certain outcomes if perfect play is realized.
By contrast, “strong” proofs often proceed by brute force — using a computer to exhaustively search a game tree to figure out what would happen if perfect play were realized. The resulting proof gives an optimal strategy for every possible position on the board. However, these proofs aren’t as helpful in understanding deeper reasons why some games are solvable as a draw, and other, seemingly very similar games are solvable as a win.
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.
As an example of a strong solution, 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 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 its 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.
In non-perfect information games, one also has the notion of essentially weakly solved. A game is said to be essentially weakly solved if a human lifetime of play is not sufficient to establish with statistical significance that the strategy is not an exact solution. As an example, the poker variation heads-up limit Texas hold ‘em have been essentially weakly solved by the poker bot Cepheus.
In game theory, perfect play is the behavior or strategy of a player that 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 backward 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.
- Awari (a game of the Mancala family)
- The variant of Oware 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.
- See “Draughts, English”
- 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 4, 1995). Weakly solved for all boardsizes where width+height is at most 15 (Feb 18, 2006).
- Draughts, English (Checkers)
- This 8×8 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 5×1020. The number of calculations involved was 1014, which were done over a period of 18 years. The process involved from 200 desktop computers at its peak down to around 50.