Lecture_March7_MATH4321_12S

Lecture_March7_MATH4321_12S - Overview: The analysis of...

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Overview: The analysis of two-person games is necessarily more complex for general-sum games than for zero-sum games. When the sum of the payoffs is no longer zero (or constant), maximizing one’s own payoff is no longer equivalent to minimizing the opponent’s payoff. The minimax theorem does not apply to bimatrix games!
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One can no longer expect to play “optimally” by simply looking at one’s own payoff matrix and guarding against the worst case. Clearly, one must take into account the opponent’s payoff matrix and the reasonable strategy options of the opponent. In doing so, we must remember that the opponent is doing the same. The general-sum case requires other more subtle concepts of solution.
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The theory is generally divided into two branches, the noncooperative theory and the cooperative theory. In the noncooperative theory, either the players are unable to communicate before decisions are made, or if such communication is allowed, the players are forbidden or are otherwise unable to make a binding agreement on a joint choice of strategy. The main noncooperative solution concept is the strategic equilibrium (SE).
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In the cooperative theory, it is assumed that the players are allowed to communicate before the decisions are made. They may make threats and counter-threats, proposals and counter- proposals, and hopefully come to some compromise. They may jointly agree to use certain strategies, and it is assumed that such an agreement can be made binding.
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The cooperative theory itself breaks down into two branches, depending on whether or not the players have comparable units of utility and are allowed to make monetary side payments in units of utility as an incentive to induce certain strategy choices. The
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Lecture_March7_MATH4321_12S - Overview: The analysis of...

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