By Haurie A., Krawczyk J.
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Extra info for An Introduction to Dynamic Games
M ) ∈ 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; ·, . . , ·) : S1 × . . × Sm → IR, j ∈ M. 77) A prior probability distribution p(θ1 , . . θm ) on Θ is given as common knowledge. We assume that all the marginal probabilities are nonzero pj (θj ) > 0, ∀j ∈ M. When Player j observes that he is of type θj ∈ Θj he can construct his revised conditional probality distribution on the types θM −j of the other players through the Bayes formula p([θj , θM −j ]) p(θM −j |θj ) = .
Iii) Q D(Q) is bounded and strictly concave for all Q such that D(Q) > 0. 7). There exists an equilibrium. Let us consider the uniqueness issue in the duopoly case. Consider the pseudo gradient (there is no need of weighting (r1 , r2 ) since the constraints are not coupled) g(q1 , q2 ) = D(Q) + q1 D (Q) − C1 (q1 ) D(Q) + q2 D (Q) − C2 (q2 ) and the jacobian matrix G(q1 , q2 ) = 2D (Q) + q1 D (Q) − C1 (q1 ) D (Q) + q1 D (Q) D (Q) + q2 D (Q) 2D (Q) + q2 D (Q) − C1 (q2 ) The negative definiteness of the symmetric matrix 1 [G(q1 , q2 ) + G(q1 , q2 )T ] = 2 2D (Q) + q1 D (Q) − C1 (q1 ) D (Q) + 12 Q D (Q) 2D (Q) + q2 D (Q) − C1 (q2 ) D (Q) + 12 Q D (Q) .
6). In the above example we have seen that, by expanding the game via the adjunction of a first stage where “Nature” plays and gives information to the players, a new class of equilibria can be reached that dominate, in the outcome space, some of the original Nash-equilibria. If the random device gives an information which is common to all players, then it permits a mixing of the different pure strategy Nash equilibria and the outcome is in the convex hull of the Nash equilibrium outcomes. If the random device gives an information which may be different from one player to the other, then the correlated equilibrium can have an outcome which lies out of the convex hull of Nash equilibrium outcomes.