By Y. Narahari, Dinesh Garg, Ramasuri Narayanam, Hastagiri Prakash
With the arrival of the net and different smooth info and verbal exchange applied sciences, an impressive chance has spread out for introducing new, cutting edge versions of trade, markets, and enterprise. growing those suggestions demands major interdisciplinary interplay between researchers in computing device technological know-how, communique networks, operations examine, economics, arithmetic, sociology, and administration technology. within the rising period of recent difficulties and demanding situations, one specific software that has came upon frequent purposes is mechanism design.
The concentration of this booklet is to discover online game theoretic modeling and mechanism layout for challenge fixing in net and community economics. It offers a valid origin of appropriate strategies and concept, to aid follow mechanism layout to challenge fixing in a rigorous means.
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Extra resources for Game Theoretic Problems in Network Economics and Mechanism Design Solutions
Sample text
There are two companies 1 and 2 that wish to maximize their profits by choosing their prices p1 and p2 . The utilities of the two companies are: u1 (p1 , p2 ) = (p1 − c) x1 (p1 , p2 ) u2 (p1 , p2 ) = (p2 − c) x2 (p1 , p2 ). Note that u1 (c, c) = 0 and u2 (c, c) = 0. Also, it can be easily noted that u1 (c, c) ≥ u1 (p1 , c) ∀p1 ∈ S1 u2 (c, c) ≥ u2 (c, p2 ) ∀p2 ∈ S2 . Therefore the strategy profile (c, c) is a pure strategy Nash equilibrium. The implication of this result is that in the equilibrium, the companies set their prices equal to the marginal cost.
The books [6, 1] may be consulted for a proof of this theorem. Nash’s theorem, however, does not guarantee the existence of a Nash equilibrium for an infinite game. 8), which is an infinite game, does not have a mixed strategy Nash equilibrium. 4 Computation of Nash Equilibria The problem of computing Nash equilibria is one of the fundamental computational problems in game theory. In fact, this problem has been listed as one of the important current challenges in the area of algorithms and complexity.
Such a mixed strategy is said to be a degenerate mixed strategy and is denoted by e(si ) or simply by si . If Si = {si1 , si2 , . . , sim }, then clearly, the set of all mixed strategies of player i is the set of all probability distributions on the set Si . In other words, it is the simplex: Δ (Si ) = (σi1 , . . , σim ) ∈ Rm : σi j ≥ 0 for j = 1, . . , m and m ∑ σi j = 1 . j=1 The above simplex is called the mixed extension of Si . Using the mixed extensions of strategy sets, we would like to define a mixed extension of the pure strategy game Γ = N, (Si ), (ui ) .
