By Tamer Basar, Geert Jan Olsder
Meant for postgraduates and researchers in economics, information, maths and engineering, this publication offers an in depth and up to date therapy of static and dynamic non-cooperative online game concept. It emphasizes the interaction among dynamic info styles and the structural houses of a number of sorts of equilibria. positive aspects of this moment variation contain new theoretical advancements, a number of illustrative examples and workouts, and an intensive record of references.
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Additional info for Dynamic Noncooperative Game Theory
Further introducing the notation y = Y/Vl (y) and recalling the definition of Y from (l Sa), we observe that the optimization problem faced by PI in determining his mixed security strategy is minimize v 1 (y) over R m subject to A'y::; In Y'lm = y ~ 0, [Vl(y)r 1 y = YV1(Y)' This is further equivalent to the maximization problem max y'/ m (25a) subject to (25b) (25c) which is a standard LP problem. The solution of this LP problem will give the mixed security strategy of PI, normalized with the average saddle-point value of the game, Vm(A).
PROPOSITION I A single-act zero-sum two-person finite game in extensiveform admits a (pure-strategy) saddle-point solution if, and only if, each matrix game corresponding to the information sets of the second-acting player has a saddle point in pure strategies. 0 PROPOSITION 2 Every single-act zero-sum two-person finite game in extensive form, in which the information sets of the second-acting player are singletons, t admits a pure-strategy saddle-point solution. 0 If the matrix game corresponding to an information set does not admit a saddle-point solution in pure strategies, then it is clear that the strategy spaces of the players have to be enlarged, in a way similar to the introduction of mixed strategies in section 2 within the context of matrix games.
Then, (i) every mixed saddle-point strategy pair for matrix game A also constitutes a mixed saddle-point solution for the matrix game B, and vice versa. (ii) Vm(A) = Vm(B) + c. Proof Let (y*,z*) be a saddle-point solution for A, thus satisfying inequalities (16). If A is replaced by B + clm/~ in (16), then it is easy to see that (y*, z*) also constitutes a saddle-point solution for B, since y'/m/~z = 1 for every y E Y, Z E Z. Since the reverse argument also applies, this completes the proof of (i).
Dynamic Noncooperative Game Theory by Tamer Basar, Geert Jan Olsder