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

Sample text

Let B = (Blu . . e. Sik > 0, £ X 0* = 1, /= l*= and l 0 = (0 U , . . e. i = l y = l Let also η = (ηι, . , rjs) and £ = (£i, . . , £,) be two random strategies associated with the sets F and, respectively, Z corresponding to player 2 and, respectively, player 3. By the definition of the characteristic function, we have r K{2, 3}) = max min £ 0 [ r «feW/*+min Σ i4 Ä 1J £ /Σ = ι jΣ = i /c» t r s -i t min Σ Σ Σ Uvk£iVjZk+min Σ Σ Σ I / = 1y= 1 Ä = 1 ί ί= 1;=Η=1 r 5 t r s t s* max min Σ II s t £ (w,% +ιφΛ) ξ,-θ,* /= i y= i *= i ξ Σι 7·=ι Σ *=Σι s* max ί™π L ι /* 0 s £ Σ ξ / = iy = u *■ max min Σ = K{2})+K{3}).

E. xh yp zk) we have the equality u\xh yj, Zk) +u2(xh yh zk) -f w3(*/, yj9 zk) = 0, or, equivalently, u}jk +ufjk +u}jk = 0. The von Neumann-Morgenstern theory assumes in general that the utilities u\jk can be distributed among the players. The central question is, What combinations of players (coalitions) will form, and what changes of utilities (or payoffs) will be made to form these coalitions? The possibilities of forming coalitions are peculiar to triads and, in general, to games with more than two players.

Let us now define the fourth triad that we are going to use. 4. Triad R2 (Rapoport, 1970) This is a non-zero-sum, even a non-constant-sum, triad. Only two possible actions (or moves) are available to each player: 1 represents the action, "player co-operates", and 0 represent "player does not co-operate". We also assume, as in the case of triad JRI, that each player makes one move. In the extensive form the triad looks like that in Fig. 2. The graph shows that if all three players co-operate, the utility to each is equal to 1.

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