Games Theory III

"But the answers provided by the theory of games are sometimes very puzzling and ambiguous. In many situations, no single course of action dominates all the others; instead, a whole set of possible solutions are all equally consistent with the postulates of rationality." (Herbert A Simon et al, "Decision Making and Problem Solving", Interfaces Vol. 17 (5), 1987)

"Allowing more than two players into the game and/or postulating payoff structures in which one player's gain does not necessarily equal the other player's loss brings us much closer to the type of games played in real life. Unfortunately, it's generally the case that the closer you get to the messiness of the real world, the farther you move from the stylized and structured world of mathematics. Game theory is no exception." (John L Casti, "Five Golden Rules", 1995)

"The Minimax Theorem applies to games in which there are just two players and for which the total payoff to both parties is zero, regardless of what actions the players choose. The advantage of these two properties is that with two players whose interests are directly opposed we have a game of pure competition, which allows us to define a clear-cut mathematical notion of rational behavior that leads to a single, unambiguous rule as to how each player should behave." (John L Casti, "Five Golden Rules", 1995)

"What's important about a saddle point is that it represents a decision by the two players that neither can improve upon by unilaterally departing from it. In short, either player can announce such a choice in advance to the other player and suffer no penalty by doing so. Consequently, the best choice for each player is at the saddle point, which is called a 'solution' to the game in pure strategies. This is because regardless of the number of times the game is played, the optimal choice for each player is to always take his or her saddle-point decision. […] the saddle point is at the same time the highest point on the payoff surface in one direction and the lowest in the other direction. Put in algebraic terms using the payoff matrix, the saddle point is where the largest of the row minima coincides with the smallest of the column maxima." (John L Casti, "Five Golden Rules", 1995)

"Chess, as a game of zero sum and total information is, theoretically, a game that can be solved. The problem is the immensity of the search tree: the total number of positions surpasses the number of atoms in our galaxy. When there are few pieces on the board, the search space is greatly reduced, and the problem becomes trivial for computers’ calculation capacity." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"Game theory proposes a method called minimization-maximization (minimax) that determines the best possibility that is available to a player by following a decision tree that minimizes the opponent’s gain and maximizes the player’s own. This important algorithm is the basis for generating algorithms for chess programs." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"The two essential components in decision making in chess are recognizing patterns stored in long-term memory (which requires an exhaustive knowledge database) and searching for a solution within the problem space. The first component uses perception and long-term memory, and the second leans mainly on the calculation of variations, which in turn has its foundations in logical reasoning." (Diego Rasskin-Gutman, "Chess Metaphors: Artificial Intelligence and the Human Mind", 2009)

"We can find the minimax strategy by exploiting the game’s symmetry. Roughly speaking, the minimax strategy must have the same kind of symmetry." (Ian Stewart, "Symmetry: A Very Short Introduction", 2013)