Mišin, Mia (2014) Pregovaranje. Diploma thesis, Faculty of Science > Department of Mathematics.

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In this these we are studying bargaining and cooperative games. In order to understand that, we should be familiar with the basic terms of the utility theory of von Neumann and Morgenstern, based on the axioms on the page 8. They are weak condition, non-triviality, reduction, substitution, monotony and continuity. Proofs of those two results are not given in this paper, since they were given in class. The main part in this study is given by Nash: The Barganing Problem, Econometrica 18 (1950), 155-162 and Two-Person Cooperative Games, Econometrica 21 (1953), br. 1 , 128-140. Nash’s axioms provides a unique solution to the negotiation problem as an argument of the maximum product of the utility function, if the axioms of invariance, symmetry, independence and Pareto axiom are satisfied. Each of these axioms is processed and reasoned, and the example is given for every axiom that shows that all axioms are necessary and that none can be mitigated. Nash’s solution is illustrated and explained also graphically so it is more convenient to understand it. The main example in the paper is ”Prisoner’s Dilemma”. The example shows the paradox of playing solely for the purpose of achieving the maximum utility for yourself. The same can be seen in the example of two traders who both want to maximize their profits, but also throw another trader out of business; or the example of two great powers that have the power to decide the quantity of the armaments, which is shown through the history with the example of the USA and the USSR. The arms race ended when the USSR was left without funds, because of a lack of cooperation. This gave me the motivation to get to generalizations of the Nash theory. ” Prisoner’s Dilemma ” served as assistance in obtaining basic ideas which were later generalized. It is given the dismissal of ” Prisoner’s Dilemma” and it was precisely explained how the players should play in order to reach the highest total maximum utility. Also, in this example it is found that we can observe the sum of the utility function, which is often more simple than the product, and get a solution identical to Nash’s. This was the motivation to show the same result in the general case, with the premise that the function is obtained by summing the utility function as strictly monotone. The final chapter discusses the Kalai-Smorodinsky bargaining solution to the problem. The theorem which gives the necessary and su fficient conditions for the existence of Kalai-Smorodinsky solution is proved. Those conditiones are the axioms of scale invariance, symmetry, homogeneous ideal independence of irrelevant alternatives and non-triviality. In addition, an example that provides a comparison of Kalai-Smorodinsky and Nash bargaining solutions was processed and it shows that there are other ways of solving the problem of bargaining, such as calculating Walras equilibrium.

Item Type: Thesis (Diploma thesis)
Supervisor: Čaklović, Lavoslav
Date: 2014
Number of Pages: 44
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 11 Jun 2015 10:08
Last Modified: 11 Jun 2015 10:08

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