So that Player I wins the same amount on the average whether II calls ‘one’ or ‘two’. Let p denote the proportion of times that Player I calls ‘one’. Player I fix it so that he wins a positive amount no matter what II calls? Strategy, I is assured of at least breaking even on the average no matter what II does. That is, if I mixes his choices in the given way, the game is even every time II callsĬ on the average every time II calls ‘two’. If II call ‘two’, I wins 3 dollars 3/5ths of the time and loses 4 dollars 2/5ths of the time Time on the average, he wins −2(3/5) + 3(2/5) = 0 (he breaks even in the long run).Ģ. If II calls ‘one’, I loses 2 dollars 3/5ths of the time and wins 3 dollars 2/5ths of the Of the time and ‘two’ 2/5ths of the time at random. Let us analyze this game from Player I’s point of view. It turns out that one of the players has a distinct advantage in this game. Strategic form we must specify X, Y and A. The winner by the loser is always the sum of the numbers in dollars. Player II’s name is Even she wins if the sum of the numbers is even. Player I’s name is Odd he wins if the sum of the numbers if odd. Players I and II simultaneously call out one of the As an example, take the game calledġ.2 Example: Odd or Even. To illustrate the notions involved in games, let us consider the simplest non-trivialĬase when both X and Y consist of two elements. Other types of games may be modeled and described in strategic form. Later, when we study the extensive form of a game, we will see that many Is rather small, so that it is possible to study all strategies and find an optimal strategyįor each player. On the other hand, the number of games of tic-tac-toe Possible strategies since there are too many in fact, there are more strategies than thereĪre atoms in the known universe. Naturally, in the game of chess it is physically impossible to describe all The set of all such strategies for Player I isĭenoted by X. In a match in 1997, represents one strategy. The program Deep Blue, that beat then world chess champion Gary Kasparov Instructing a machine to play chess well have been written. One strategy, good or bad, for the game of chess. It is rather time-consuming to write down even
ONTO VS ONE TO ONE MATRIX HOW TO
A strategy for a game of chess,įor example, is a complete description of how to play the game, of what move to make inĮvery possible situation that could occur. Sufficiently broadminded about the definition of a strategy. finite combinatorial games and games such as tic-tac-toe and chess. This is a very simple definition of a game yet it is broad enough to encompass the Thus, A(x, y) represents the winnings of I and the losses of II. If A is negative, I pays the absolute value Depending on the monetary unit involved,Ī(x, y) will be cents, dollars, pesos, beads, etc. Known and I wins the amount A(x, y) from II. II chooses y ∈ Y, each unaware of the choice of the other. Simultaneously, Player I chooses x ∈ X and Player (3) A is a real-valued function defined on X × Y. (2) Y is a nonempty set, the set of strategies of Player II (1) X is a nonempty set, the set of strategies of Player I The strategic form, or normal form, of a two-person zero-sum game is given To the single payoff function of Player I, which we call here L.ĭefinition 1. For a two-person zero-sum game, the payoffįunction of Player II is the negative of the payoff of Player I, so we may restrict attention The simplest mathematical description of a game is the strategic form, mentioned in the introduction. Refer to the players as Player I and Player II.ġ.1 Strategic Form. In Part II, we restrict attention to such games. games with only two players in which one player The theory of von Neumann and Morgenstern is most complete for the class of gamesĬalled two-person zero-sum games, i.e. Mathematical Association of America, 1993. The expository book, Game Theory and Strategy by Philip D. Text book, Game Theory by Guillermo Owen, 2nd edition, Academic Press, 1982, and Other more current books on the theory of games may be found in the Written in collaboration with Oskar Morgenstern entitled Theory of Games and Economicīehavior, 1944. Von Neumann’s work culmin ated in a fundamental book on game theory Who published in 1928 the paper that laid the foundation for the theory of two-person Preceded him in formulating a theory of games - notably Émile Borel - it was von Neumann John von Neumann, one of the greatest mathematicians of this century. The individual most closely associated with the creation of the theory of games is 1.3 Pure Strategies and Mixed Strategies.Ĥ.3 Invariance Under Change of Location and Scale.Ĥ.4 Reduction to a Linear Programming Problem.Ĥ.5 Description of the Pivot Method for Solving Games.ĥ.4 The Representation of a Strategic Form Game in Extensive Form.ĥ.5 Reduction of a Game in Extensive Form to Strategic Form.Ħ.1 Matrix Games with Games as Components.Ħ.3 Recursive Games.