Solved game

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A solved game is a game whose results (win, lose or series) can be predicted correctly from any position, assuming that both players play perfectly.

Video Solved game

Overview

The game of two players can be "completed" on several levels:

Ultra-weak

Prove whether the first player will win, lose or draw from the starting position, given the perfect game on both sides. This can be non-constructive evidence (perhaps involving strategy-stealing arguments) that need not really determine every perfect game movement.

Weak

Provide an algorithm that secures victory for one player, or a draw for one, against any movement the opponent may have made, from the start of the game. That is, it produces at least one complete ideal game (all movements begin to end) with evidence that every movement is optimal for the player who made it. That does not mean the computer program that uses the solution will play optimally against an imperfect opponent. For example, Chinook's pengetian program will never change the position drawn into a losing position (because the weaker solution of the checkers proves that it is a series), but may change the position of victory to a withdrawn position because Chinook does not expect the opponent to play a movement that will not win but may lose, and therefore do not analyze such movements completely.

Strong

Give an algorithm that can produce perfect movement from any position, even if an error has been made on one or both sides.

Regardless of their name, many game theorists believe that the "ultra-weak" evidence is the deepest, most interesting and valuable one. Evidence of "ultra-weakness" requires a scholar to think about the abstract nature of the game, and show how this property leads to certain results if the perfect game materializes.

Conversely, the "strong" evidence is often continued with violence - using computers to search the game tree in depth to find out what would happen if a perfect game materialized. The resulting proof provides an optimal strategy for every possible position on the board. However, these proofs do not help in understanding the deeper reasons why some games can be solved as a series, and others, games that seem very similar can be solved as a victory.

Given the rules of a two-person game with a limited number of positions, one can always build a minimax algorithm that will traverse the overall game tree. However, since for many non-trivial games, such an algorithm will require an insufficient amount of time to produce motion in a particular position, the game is not considered to be solved weakly or strongly unless the algorithms can be executed by existing hardware within a reasonable time. Many algorithms rely on very large databases that were created before, and are effectively nothing more.

As an example of a powerful solution, the tic-tac-toe game can be solved as a draw for both players with perfect game (results that can even be determined manually by schoolchildren). Games like nim also recognize rigorous analysis using combinatorial game theory.

Whether a game is solved is not always the same as whether it remains attractive for humans to play. Even highly resolved games can still be interesting if the solution is too complicated to memorize; on the contrary, weakly resolved games can lose their appeal if winning strategies are simple enough to remember (eg Maharajah and Sepoys). The ultra-weak solutions (eg Chomp or Hex on a fairly large board) generally do not affect playback.

Furthermore, even if the game is not solved, it is possible that the algorithm yields a good approximate solution: for example, an article in Science of January 2015 claims that their heads to the limit of Texas holding 'em poker bot Cepheus ensures that the age of human play is not enough to establish with statistical significance that its strategy is not the right solution.

Maps Solved game

Perfect play

In game theory, perfect game is a player's behavior or strategy that leads to the best possible outcome for the player regardless of the opponent's response. Perfect play for a game is known when the game is over. Under the rules of the game, any possible end position may be evaluated (as a win, lose or series). With backward reasoning, one can recursively evaluate non-final positions as identical to positions that are one step away and most valuable to a moving player. So the transition between positions will never produce a better evaluation for the moving player, and the perfect step in position will be a transition between positions that are equally evaluated. For example, the perfect player in a drawn position will always get a draw or win, never lose. If there are multiple choices with similar results, a perfect game is sometimes regarded as the fastest method that leads to good results, or the slowest method that leads to poor results.

Perfect play can be generalized for incomplete information games, because a strategy that will guarantee the expected results is at least the highest regardless of the opponent's strategy. For example, the perfect strategy for rock-paper-scissors is to choose each option with the same probability (1/3) randomly. The disadvantage in this example is that this strategy will never exploit opponent strategies that are not optimal, so the expected outcomes of this strategy versus any strategy will always be the same as the expected minimum.

Source of the article : Wikipedia