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Equitable Distribution in a Three Players Problem

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Szopa M.

Studies in Logic, Grammar and Rhetoric

2014

vol. 37

issue 1

239-252

Jazz band is a 3 player superadditive game in characteristic function form. Three players have to divide the payoff they can get, while being in a grand coalition, provided their individual and duo coalitions payoffs are known. Assumptions of individual and collective rationality lead to the notion of the core of the game. We discuss offers that cannot readily be refused [OCRR] as the solutions of the game in case of an empty core, when duo coalitions are the best options but only for two out of three players. The experiment shows that even in case of an empty core the most probable results are three-way coalitions and the share of the weakest player usually exceeds his OCRR. The Shapley value is introduced and its fairness is discussed as it lies at the side of the core while, on the other hand, the nucleolus lies exactly at the center of the core. We conclude that, in spite of that, the Shapley value is the best candidate for a fair sharing solution of the jazz band game and other similar games as, opposite to the other values, it is dependent both on individual and duo coalitions payoffs.

Dlaczego w dylemat więźnia warto grać kwantowo?

100%

Szopa M.

Studia Ekonomiczne

2014

vol. 178

174-189

The Prisoner's Dilemma [PD] is the best known example of a two-person, simultaneous game, for which the Nash equilibrium is far from Pareto-optimal solutions. In this paper we define a quantum PD, for which player's strategies are defined as rotations of the SU(2) group, parameterized by three angles. Quantum strategies are correlated through the mechanism of quantum entanglement and the result of the game is obtained by the collapse of the wave function. Classic PD is a particular case of the quantum game for which the set of rotations is limited to one dimension. Each quantum strategy can be, by appropriate choice of counter-strategy, interpreted as a "cooperation" or "defection". Quantum PD has Nash equilibria that are more favorable than the classic PD and close to the Pareto optimal solutions. With proper selection of strategies, quantum PD can be reduced to the classic, zero-sum, "matching pennies" game. In this paper we show examples of economic phenomena (price collusion, the chess strategy) that mimics the Nash equilibria of quantum PD.

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1 Studia Ekonomiczne

1 Studies in Logic, Grammar and Rhetoric

2 2014

2 Szopa M.

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Equitable Distribution in a Three Players Problem

Dlaczego w dylemat więźnia warto grać kwantowo?