LectureFeb17_MATH4321_12S

LectureFeb17_MATH4321_12S - Part II Two-Person Zero-Sum...

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Part II. Two-Person Zero-Sum Games Example: Odd or Even Players I and II simultaneously call out one of the numbers one or two. Player I’s name is Odd; he wins if the sum of the numbers if odd. Player II’s name is Even; she wins if the sum of the numbers is even.
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1. The Strategic Form of a Game. The individual most closely associated with the creation of the theory of games is John von Neumann, one of the greatest mathematicians of last century. Although others preceded him in formulating a theory of games - notably Emile Borel - it was von Neumann who published in 1928 the paper that laid the foundation for the theory of two- person zero-sum games, i.e. games with only two players in which one player wins what the other player loses. In Part II, we restrict attention to such games. We will refer to the players as Player I and Player II.
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1.1 Strategic Form. The simplest mathematical description of a game is the strategic form. For a two-person zero-sum game, the payoff function of Player II is the negative of the payoff of Player I, so we may restrict attention to the single payoff function of Player I.
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Definition: The strategic form, or normal form, of a two- person zero-sum game is given by a triplet (X, Y,A), where (1) X is a nonempty set, the set of strategies of Player I (2) Y is a nonempty set, the set of strategies of Player II (3) A is a real-valued function defined on X × Y. (Thus, A (x, y) is a real number for every x X and every y Y.) The interpretation is as follows. Simultaneously , Player I chooses x X and Player II chooses y Y, each unaware of the choice of the other. Then their choices are made known and I wins the amount A (x, y) from II.
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If A is negative, I pays the absolute value of this amount to II. Thus, A (x, y) represents the winnings of I and the losses of II.
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1.2 Example: Paper-Scissors-Rock Players I and II simultaneously display one of the three objects: paper, scissors, or rock. If they both choose the same object to display, there is no payoff. If they choose different objects, then scissors win over paper (scissors cut paper), rock wins over scissors (rock breaks scissors), and paper wins over rock (paper covers rock). If the payoff upon winning or losing is one unit, then the matrix of the game is as follows.
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0 1 1 1 0 1 1 1 0 P S R P S R
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Matching Pennies Matching Pennies : : Two players simultaneously choose heads or tails.
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