Tonight's math lecture was quite interesting, as John from UW Madison came down and spoke about game theory. He said that every game can be split up until two piles: Games of Chance and Games of No-Chance, meaning there could be a winning strategy involved to win. He later corrected himself that they are also Games of Skill, Physics, and etc. Games of Chance would be games like Yatzee and Poker, and Games of No-Chance would be Checkers, Chess, Go, Othello, Tic-Tac-Toe, Battleship, and Stratego.
He further expanded upon the No-Chance category into games where all the information is known to both players, and games where not all the information is known to both players. Obvious games where some information is hidden would be Battleship and Stratego. All the information is known in games like Checkers and Chess because you can see all the pieces on the board.
He further expanded the known category into games where players have the same moves, and where players have different moves. Players that have different moves would be Chess and Checkers again because one player can only move their pieces. For games like Tic-Tac-Toe and Othello, you have the same moves because you lay pieces.Games of this nature are known as non-partisan games.
Further explaining the structures of games, you can't split every game into these finite categories. There are other categories that games can fall under as well, like 2 player v. multiplayer, winning conditions, and finite moves v. infinite moves. He said that he would be explaining winning strategies for 2 player non-partisan games with finite number of moves where the last person to move wins. This is called combinatorial game theory.
He went through 3 main games: 2-player Nim, Hex, and Chocolate Bars. He explained that there isn't a definite winning strategy for each player, but there are ad hoc strategies that do work. For 2-player Nim the second player will always win if you match the first player's moves. For Hex, if you're the first player, assuming there is a winning strategy for the second player, you can use that strategy, and if you come across your original move, you have the advantage. For Chocolate Bars, depending on the amount of total pieces the bar can be broken into, if the total pieces are odd than the first player will always win regardless of strategy, moves, or anything. If the total pieces are even, the same is true for the second player.
Pretty interesting stuff. ^_^ Unfortunately, the guy ran out of time when he was about 70% through his presentation, which skipped over further strategies for NIM and discussion of nimbers in other factions of mathematics. His short presentation also skipped over discussion on Dots, which is a game i love to play, and kick ass at. But, he had to skip over it... sigh.
I think I should look into that "surreal numbers" book he reference at the end of his lecture, as well as "the theory of dots" ^_^