Exploring Games with Pre-determined Optimal Strategies

When discussing game theory and player strategy, a fundamental question arises: which games have a pre-determined optimal strategy? In this article, we will delve into the world of solved games, where the best possible moves can be calculated and applied before any actual gameplay begins.

Introduction to Solved Games

A solved game is one in which the optimal strategy can be determined before any moves are made, ensuring a perfect outcome for one of the players. This concept is fascinating for both mathematicians and gamers alike. Historically, several classic games have been solved, revealing the optimal strategy for both players.

Classic Examples: Tic-Tac-Toe, Connect Four, and Checkers

One of the most well-known examples of a solved game is Tic-Tac-Toe. In its simplest form, when both players play optimally, the game will always result in a draw. This is because the first player (cross) can always force a draw if the second player (circle) does not make any critical mistakes. The strategies here are well-documented and can be applied by the players before the game begins.

Connect Four is another solved game, similar to Tic-Tac-Toe in that it guarantees a draw when both players use their optimal strategies. The game's rules are such that it ensures that any moves made by one player can be perfectly countered by the other player. Thus, the first player can only force a win if the second player makes a mistake, otherwise, the game is a guaranteed draw.

Another popular solved game is Checkers, now known as International Checkers. In 2007, a team of scientists at the University of Alberta managed to solve the game, proving that there is an optimal strategy that can be applied before the game starts. This means that with the right sequence of moves, a player can theoretically always win or at least force a draw from the opening position if both players follow the optimal strategy.

Complex Games and Decision Trees

While the above games are well-defined and have clear, pre-determined optimal strategies, not all games can be solved in this way. Chess and Go, for instance, are complex games with extensive and incomparable decision trees, making it nearly impossible to determine a pre-determined optimal strategy.

In Chess, the vast number of possible moves and permutations means that even the most advanced algorithms struggle to calculate every move to its end. The complexity of the game means that players must continually adapt and respond to their opponent's moves. Thus, while chess geniuses can develop grand strategies, no single predetermined strategy can guarantee victory.

Go is even more complex, with millions of potential positions and moves. This game's depth and breadth have stumped even modern AI, often leading to unexpected and unpredictable outcomes. While some patterns and strategies can be learned and applied, the game fundamentally lacks a pre-determined optimal strategy that can be used from the opening move.

Conclusion

Solved games are a unique subset of games where the optimal strategy can be determined and applied before any moves are made. These include classic games like Tic-Tac-Toe, Connect Four, and Checkers. However, as we've seen, the complexity of games like Chess and Go presents insurmountable challenges for finding such strategies. The field of game theory continues to be a rich area for exploration and learning, providing insights into both strategy and probability.