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By backward reasoning, one can recursively evaluate a non-final position as identical to the position that is one move away and best valued for the player whose move it is. Based on the rules of a game, every possible final position can be evaluated (as a win, loss or draw). Perfect play for a game is known when the game is solved. 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. An ultra-weak solution (e.g., Chomp or Hex on a sufficiently large board) generally does not affect playability.

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).

Whether a game is solved is not necessarily the same as whether it remains interesting for humans to play. Games like nim also admit a rigorous analysis using combinatorial game theory. Many algorithms rely on a huge pre-generated database and are effectively nothing more.Īs a simple example of a strong solution, the game of tic-tac-toe is easily solvable as a draw for both players with perfect play (a result manually determinable). 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. 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, these proofs are not 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. The resulting proof gives an optimal strategy for every possible position on the board. īy 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. '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. Strong solution Provide an algorithm that can produce perfect moves from any position, even if mistakes have already been made on one or both sides.ĭespite their name, many game theorists believe that 'ultra-weak' proofs are the deepest, most interesting and valuable. Weak solution 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. This can be a non-constructive proof (possibly involving a strategy-stealing argument) that need not actually determine any moves of the perfect play. Solving such a game may use combinatorial game theory and/or computer assistance.Ī two-player game can be solved on several levels: Ultra-weak solution Prove whether the first player will win, lose or draw from the initial position, given perfect play on both sides.

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This concept is usually applied to abstract strategy games, and especially to games with full information and no element of chance

Game whose outcome can be correctly predictedĪ solved game is a game whose outcome (win, lose or draw) can be correctly predicted from any position, assuming that both players play perfectly.