Download PDF by Silviu Guiasu: Coalition and Connection in Games. Problems of Modern Game

By Silviu Guiasu

ISBN-10: 0080230334

ISBN-13: 9780080230337

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Additional info for Coalition and Connection in Games. Problems of Modern Game Theory Using Methods Belonging to Systems Theory and Information Theory

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1 it also follows that, for any two disjoint subsets of players of the triad, C and D (Cf\D = 0), we have v(CUD)^v(C)+v(D), where C(JD denotes the coalition formed by the players belonging to C with the players belonging to D. If isolation is taken as a principle of conduct in a triad, each player will act on his own against the others. Another principle of conduct is that of coalition formation. The latter principle proves the more advantageous. 1, if, for instance, players 1 and 2 form a coalition, then K{1,2}) = K0}U{2})» K{l})+K{2», which shows that in such a coalition the two players can expect a higher average payoff than if each played against the others.

As U1+U2 < K{1»2}) = 1, there obviously is an infinity of real numbers w*, w^ such that W1+W2 < W1+W2 ^ 1. Hence, in any case, there is an imputation (w*, z/*, ul) dominating the imputation (wi, w2, W3). Consider now an essential triad in reduced form. We have only three possibilities to analyse in terms of the domination relation between imputations, corresponding to the three coalitions proper that can form. e. a utility equal to + 1 , while player 3 obtains — 1. The triad being completely symmetrical, so that neither of players 1 and 2 has a more advantageous position, we can with good reason expect that they will equally share the payoff.

Thus, for instance, u 2,3, H = u \x2> y& z n ) = * ~ 2 , because, as results from the way in which they were defined, the three strategies x29 ya, and znlead to a definite variant of the game, viz. 100. But the utility of this variant to player 2 is - 2 . As for the three players' random strategies, they are introduced without difficulties exactly as in a two-person game. e. 2 /=1 4 j=l 16 *=1 where ξί is the probability that player 1 will choose the pure strategy xt\ r\} is the probability that player 2 will choose the pure strategy yy9 and ζ]ζ is the probability that player 3 will choose the pure strategy zk.

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Coalition and Connection in Games. Problems of Modern Game Theory Using Methods Belonging to Systems Theory and Information Theory by Silviu Guiasu


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