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By Haurie A., Krawczyk J.

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7. 5 x2 = 0, y = 1. 5 A general definition of Bayesian equilibria We can generalize the analysis performed on the previous example and introduce the following definitions. Let M be a set of m players. Each player j ∈ M may be of different possible types. Let Θj be the finite set of types for Player j. Whatever his type, Payer j has the same set of pure strategies Sj . If θ = (θ1 , . . θm ) ∈ Θ = Θ1 × . . × Θm is a type specification for every player, then the normal form of the game is specified by the payoff functions uj (θ; ·, .

R2 ) by Player 1 when he implements a mixed strategy, knowing that he is of type θi . Call y (resp. 1 − y) the probability of choosing c1 (resp. c2 ) by Player 2 when he implements a mixed strategy. We can define the optimal response of Player 1 to the mixed strategy (y, 1 − y) of Player 2 by solving6 1 1 max aθi1 y + aθi2 (1 − y) i=1,2 if the type is θ1 , 2 2 max aθi1 y + aθi2 (1 − y) i=1,2 if the type is θ2 . We can define the optimal response of Player 2 to the pair of mixed strategy (xi , 1− xi ), i = 1, 2 of Player 1 by solving 1 1 2 2 max p1 (x1 bθ1j + (1 − x1 )bθ2j ) + p2 (x2 bθ1j + (1 − x2 )bθ2j ).

More precisely it indicates the sensitivity of the optimum solution to marginal changes in this right-handside. The multiplier permits also a price decentralization in the sense that, through an ad-hoc pricing mechanism the optimizing agent is induced to satisfy the constraints. In a normalized equilibrium, the shadow cost interpretation is not so apparent; however, the price decomposition principle is still valid. Once the common multiplier has been defined, with the associated weighting rj > 0, j = 1, .

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An Introduction to Dynamic Games by Haurie A., Krawczyk J.


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