By Haurie A., Krawczyk J.
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Extra info for An Introduction to Dynamic Games
BIMATRIX GAMES 41 game (A, B). t. 13) v 1 1m v 2 1n 0 1 1 IR. 19) (ii) (x, y) is an equilibrium for the bimatrix game. Proof: From the constraints it follows that xT Ay ≤ v 1 and xT By ≤ v 2 for any feasible (x, y, v 1 , v 2 ). Hence the maximum of the program is at most 0. Assume that (x, y) is an equilibrium for the bimatrix game. e. 1) a value 0. 19). 19). We know that an equilibrium exists for a bimatrix game (Nash theorem). 19) with optimal value 0. Hence the optimal program (x∗ , y∗ , v∗1 , v∗2 ) must also give a value 0 to the objective function and thus be such that xT∗ Ay∗ + xT∗ By∗ = v∗1 + v∗2 .
E. t. u∗ ∈ Γ(u∗ , r). Hence a coupled equilibrium exists. 8. One is required to show that the point to set mapping is upper semicontinuous. 2 This existence theorem is very close, in spirit, to the theorem of Nash. e. it does not provide a computational method. However, the definition of a normalised equilibrium introduced by Rosen establishes a link between mathematical programming and concave games with coupled constraints. 4. 39) can be defined by a set of inequalities hk (u) ≥ 0, k = 1, . .
M are given weights. The precise role of this weighting scheme will be explained later. For the moment we could take as well rj ≡ 1. Notice that, even if u and v are in U, the combined vectors (u1 , . . , vj , . . , um ) are element of a larger set in U1 × . . × Um . This function is continuous in u and concave in v for every fixed u. We call it a reaction function since the vector v can be interpreted as composed of the reactions of the different players to the given vector u. This function is helpful as shown in the following result.
An Introduction to Dynamic Games by Haurie A., Krawczyk J.